Cardinal Sequences and Combinatorial Principles Lajos Soukupreal-d.mtak.hu/402/4/Doktori_mű.pdf ·...

Cardinal Sequences and Combinatorial Principles

Lajos Soukup

D. Sc. Thesis

Submitted to the Hungarian Academy of Sciences

Budapest, 2010

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Contents

Chapter 1. Cardinal sequences 11.1. Introduction and Summary 11.2. A tall space with small bottom 81.3. Cardinal sequences of length < ω2 under GCH 131.4. Cardinal sequences of length ≥ ω2 under GCH 221.5. Universal spaces 271.6. A lifting theorem 301.7. Wide scattered spaces and morasses 341.8. Spaces constructed from strongly unbounded function 441.9. Regular and zero-dimensional spaces 551.10. Initially ω1-compact spaces 581.11. First countable, initially ω1-compact spaces 70

Chapter 2. Combinatorial principles 832.1. Combinatorial principles from adding Cohen reals 832.2. LCS∗ spaces in Cohen real extension 962.3. Weak Freeze-Nation property of posets 1012.4. Weak Freeze-Nation property of Boolean algebras 1102.5. Weak Freeze-Nation property of P (ω) 1152.6. Stick and clubs 120

Publications of the author 127

Other publications 129

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CHAPTER 1

Cardinal sequences

1.1. Introduction and Summary

Scattered spaces. Definitions and basic facts. A topological space is called scattered ifits every non-empty subspace has an isolated point. Since

• the union of two scattered spaces is scattered, and• the increasing union of scattered, open subspaces is scattered,

using Zorn lemma one can prove that every topological space X has a unique partition into twosubspaces,

X = P ∪ S, (1.1)such that S is an open, scattered subspace, and P is perfect, i.e., closed and dense-in-itself. (Ofcourse, P or S can be empty.)

This observation explains why it is natural to investigate the structures of scattered spaces.In the first chapter of the dissertation we study the structure of scattered spaces. To do so, weassign invariants to the scattered spaces, and we investigate the possible values of these invariants.

The Cantor-Bendixson Hierarchy. All topological spaces will be assumed to be infiniteHausdorff in this thesis.

Since a space is scattered iff it is right-separated, we have |X| ≤ w(X) for scattered spaces.Given a topological space X, for each ordinal number α, the α-th derived set of X, X(α), is

defined as follows: X(0) = X, X(α+1) is the derived set of X(α), i.e. the collection of all limitpoints of X(α), and if α is limit then X(α) = ∩ν<αX(ν). Since X(α) ⊇ X(β) for α < β we havea minimal ordinal α such that X(α) = X(α+1). This ordinal α, denoted by ht(X), is called theCantor-Bendixson height, or just the height of X. Clearly the subspace X(α) does not have anyisolated points, it is dense-in-itself. The derived sets are all closed, so X(α) is perfect. Moreover,Y = X \X(α) is scattered and so it has cardinality ≤ w(X). (So P = X(α) and S = X \X(α)

in 1.1) This observation yields the Cantor-Bendixson theorem: every space of countable weightcan be represented as the union of two disjoint subspaces, of which one is perfect and the otheris countable. G. Cantor and I. Bendixson proved this fact independently in 1883 for subsets ofthe real line.

Historically, the investigation of scattered spaces was started by Cantor. He proved, in [32],that if the partial sums of a trigonometric series

a0/2 +∞∑n=1

(an cosnx+ bn sinnx)

converge to zero except possibly on a set of points of finite scattered height, then all coefficientsof the series must be zero.

Cardinal sequences of scattered spaces. We will assign invariants to scattered spaces.Denote by I(Y ) the isolated points of a topological space Y . For each ordinal α define the

α-th Cantor-Bendixson level of a topological space X, Iα(X), as follows:

Iα(X) = I(X \ ∪Iβ(X) : β < α

).

Clearly Iα(X) = X(α) \X(α+1), and so if X is scattered, then

ht(X) = minα : Iα(X) = ∅. (1.2)

Write I<λ(X) =⋃α<λ Iα(X).

1

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2 1. CARDINAL SEQUENCES

Observe that X is scattered iff X = I<ht(X)(X) iff X(ht(X)) = ∅.Given a scattered space X, define the width of X, wd(X), as follows:

wd(X) = sup|Iα(X)| : α < ht(X). (1.3)

The cardinal sequence of a scattered space X, CS(X), is the sequence of the cardinalities ofits Candor-Bendixson levels, i.e.

CS(X) =⟨|Iα(X)| : α < ht(X)

⟩. (1.4)

We shall use the notation 〈κ〉α to denote the constant κ-valued sequence of length α. Asusual, the concatenation of a sequence f of length α and of a sequence g of length β is denotedby f _g. So the domain of h = f _g is α + β, h(ξ) = f(ξ) for ξ < α, and h(α + η) = g(η) forη < β.

By definition, if CS(X) = 〈κ〉α then X has height α and width κ.

Cardinal sequences of Boolean algebras. A Boolean algebra B is superatomic iff everyhomomorphic image of B is atomic. We abbreviate "superatomic Boolean algebra" to "sBA".

The essential questions concerning sBA-s were asked by Tarski and Mostowski, [87], in 1939.The main categories of results are: cardinal invariants, isomorphism types and automorphismgroups. The major cardinal invariants of sBA-s were also introduced in [87] in the following way.

Definition 1.1.1. Let A be a Boolean algebra.(a) Let At(A) be the atoms of A.(b) If J is an ideal in A, then let J ∗ be the ideal generated by

J ∪ x ∈ A : x/J ∈ At(A/J ).

(c) Let J0(A) = 0A, Jα+1(A) = Jα(A)∗, if α is limit then Jα(A) =⋃α′<α Jα′(A).

(d) The height of A, ht(A), is the least δ with Jδ(A) = Jδ+1(A).(e) wdα(A) = |At(A/Jα(A)|,(f) The cardinal sequence of A, CS(A), is defined as follows:

CS(A) = 〈wdα(A) : α < ht(A)〉 . (1.5)

By [68, Proposition 17.8], A is superatomic iff there is an ordinal α such that A = Jα(A).A Boolean space is a compact, 0-dimensional, Hausdorff space. The Stone duality establishes

a 1–1 correspondence between Boolean spaces and Boolean algebras.Under Stone duality, homomorphic images of a Boolean algebra A correspond to closed sub-

spaces of its Stone space S(A), and atoms of A correspond to isolated points of S(A). Moreover,if B is a superatomic Boolean algebra, then the dual space of B(α) is (S(B))(α) (see [68, Con-struction 17.7]). So we have the following fact.

Fact* 1.1.2.A Boolean algebra B is superatomic iff its dual space S(B) is scattered. Moreover,ht(B) = ht(S(B)), and CS(B) = CS(S(B)).

Historically, the cardinal sequences of Boolean algebras were defined earlier: this notion wasintroduced by Day, [33], in 1967. The notion of cardinal sequences of topological spaces wasintroduced by LaGrange.

The beginning. By a classical result of S. Mazurkiewicz and J. Sierpiński, (see [86] or [68,Theorem 17.11]) a countable, superatomic Boolean algebra B is determined completely by itscardinal sequence!

Theorem* 1.1.3. If B is a countable, superatomic Boolean algebra, then CS(B) = 〈β〉ω _〈n〉,where β + 1 = ht(B), and n = |wdβ(X)| is a natural number. Moreover, in this case B ishomeomorphic to Boolean algebra of the clopen subsets of the compact ordinal space ωβ · n + 1,i.e. S(B) = ωβ · n+ 1.

What about uncountable sBA-s? An sBA is called thin-tall iff it has width ω and height atleast ω1. The question of whether thin-tall sBA can exist was asked by Telgarsky in 1968. Thisquestion was the actual starting point of the modern investigation of sBA-s.

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1.1. INTRODUCTION AND SUMMARY 3

Cardinal sequences of LCS∗ spaces. As we remarked, the study of cardinal sequences wasactually originated in the theory of superatomic Boolean algebras. Since superatomic Booleanalgebras correspond to compact scattered spaces via Stone duality, the efforts were concentratedon the study of cardinal sequences of compact scattered spaces.

Reduction. We say that a space X is an LCS∗-space iff it is locally compact, scattered, butnot compact. 1

A locally compact scattered space X is non-compact iff either ht(X) is a limit ordinal, orht(X) = δ + 1 and the top level of X, Iδ(X), is infinite.

The height of a compact scattered space X is a successor ordinal, ht(X) = δ + 1, and Iδ(X)is finite. So Y = X \Iδ(X) is an LCS∗ space, moreover CS(X) = CS(Y )_〈n〉, where n = |Iδ(X)|.

On the other hand, if Y is an LCS∗ space, and X is the disjoint union of n copies of the onepoint compactification αY of Y , then X is compact and CS(X) = CS(Y )_〈n〉. So we have

CS(X) : X is a compact space = CS(Y )_〈n〉 : Y is an LCS∗ space, n ∈ ω. (1.6)

Hence, instead of compact, scattered spaces we can study the cardinal sequences of LCS∗ spaces.So Telgarsky’s question can be reformulated as follows:

Telgarsky’s Question: Is there an LCS∗ space with cardinal sequence 〈ω〉ω1?

His question was answered by Rajagopalan.

Theorem* 1.1.4 (Rajagopalan, 1976, [91]).There is an LCS∗ space X with CS(X) = 〈ω〉ω1.

There are strong limitations on cardinal sequences of LCS∗ spaces.

Fact* 1.1.5.The cardinality of a scattered T3, in particular of an LCS∗ space X is at most 2 | I(X)|,hence ht(X) < (2| I(X)|)+. Hence if s = 〈κα : α < δ〉 is the cardinal sequence of a scattered regularspace, then for each α < β < δ we have κβ ≤ 2κα and |δ \ α| ≤ 2s(α).

The next definition simplifies the formulation of certain results.

Definition 1.1.6.We let C(α) denote the class of all cardinal sequences of length α of LCS∗-spaces. We also put, for any fixed infinite cardinal λ,

Cλ(α) = s ∈ C(α) : s(0) = λ ∧ ∀β < α [s(β) ≥ λ]. (1.7)

In 1978, Juhász and Weiss improved the above mentioned result of Rajagopalan.

Theorem* 1.1.7 (Juhász, Weiss, [59]). 〈ω〉α ∈ C(α) for each α < ω2.

By Fact 1.1.5, if CH holds then 〈ω〉ω2/∈ C(ω2). Juhász and Weiss, [59], asked if it is consistent

that 〈ω〉α ∈ C(α). Just proved that the failure of CH is not enough to get this consistency.

Theorem* 1.1.8 (Just, [64]). In the Cohen model there is no LCS∗-space X with CS(X) =〈ω〉ω2

.

In 1985, Baumgartner and Shelah gave an affirmative answer to the question of Juhász andWeiss.

Theorem* 1.1.9 (Baumgartner, Shelah, [25]). It is consistent that there is an LCS∗-spaceX with CS(X) = 〈ω〉ω2

.

Their proof was based on the construction of a ∆-function.

Definition 1.1.10 ([25]). Let f :[ω2

]2 → [ω2

]≤ω be a function with fα, β ⊂ α ∩ β forα, β ∈

[ω2

]2. (1) We say that two finite subsets x and y of ω2 are good for f provided that forα ∈ x ∩ y, β ∈ x \ y and γ ∈ y \ x we always have(a) α < β, γ =⇒ α ∈ fβ, γ,(b) α < β =⇒ fα, γ ⊂ fβ, γ,(c) β < γ =⇒ fα, β ⊂ fα, γ.

1In the literature an LCS space is defined as a locally compact, scattered space.

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(2) The function f is a ∆-function if every uncountable family of finite subsets of ω2 contains twoelements x and y which are are good for f .(3) The function f is a strong ∆-function if every uncountable family A of finite subsets of ω2

contains an uncountable subfamily B such that any two sets x and y from B are good for f .

To get Theorem 1.1.9 actually the following statements were proved:

• It is consistent that there is a ∆-function.• If there is a ∆-function, the in some c.c.c generic extension of the ground model there

is an LCS∗-space X with CS(X) = 〈ω〉ω2.

It is worth to mention that there is a strong ∆-function in L.

Theorem* 1.1.11 (Velickovic). If ω1 holds then there is a strong ∆-function.

Two types of questions were considered in connection with cardinal sequences:

(1) Given a sequence s of cardinals of length α decide whether s ∈ C(α) in a certain/in somemodel of ZFC.

(2) Characterize C(α) or Cλ(α) in certain models of ZFC.

There are many results concerning Type 1 problems in the literature. In Section 1.2 we alsoconsider such a problem. By Fact 1.1.5, ht(X) < (2|I(X)|)+ for any LCS∗ space X. Especially, ifI0(X) is countable, then ht(X) < (2ω)+. It is not hard to prove (see e.g. Theorem 1.2.3 below)that the estimate above is sharp for LCS∗ spaces with countably many isolated points:

• for each α < (2ω)+ there is an LCS∗ space X with ht(X) = α and | I0(X)| = ω.

Much less is known about LCS∗ spaces with ω1 isolated points, for example it is a long standingopen problem whether there is, in ZFC, an LCS∗ space of height ω2 and width ω1. In fact, as wasnoticed by Juhász in the mid eighties, even the much simpler question if there is a ZFC exampleof an LCS∗ space of height ω2 with only ω1 isolated points, turned out to be surprisingly difficult.In section 1.2 we prove the main result of [9], and in Theorem 1.2.1 we give an affirmative answerto the above question of Juhász: There is, in ZFC, an LCS∗ space of height ω2 and having ω1

isolated points. Although it is a ZFC result, but the main ingredient of the proof, Theorem 1.2.2,is a construction under CH!

To prove this result we had to develop a new amalgamation method of construction of LCS∗spaces.

In Sections 1.3, 1.4, 1.5 and 1.7 we consider “Type 2” problems, i.e. we try to give fullcharacterization of the elements of certain classes Cλ(α).

For countable sequences LaGrange [79], for sequences of length ω1 Juhász and Weiss, [61],give full characterization:

Theorem* 1.1.12 ([61]). 〈κξ : ξ < ω1〉 ∈ C(ω1) iff κη ≤ κωξ holds whenever ξ < η < ω1.

It follows that cardinal arithmetic alone decides whether a sequence of cardinals of length ω1

belongs to C(ω1) or not. The situation changes dramatically for longer sequences, in fact alreadyfor sequences of length ω1 + 1. For example, the question if 〈ω〉ω1

_〈ω2〉1 ∈ C(ω1 + 1) is notdecided by the following cardinal arithmetic: 2ω = ω2 and 2κ = κ+ for all κ > ω (see Just[64]and Roitman[93]).

However, as we showed in [14], the elements of C(α) can be characterized for all α < ω2 if weassume GCH. Section 1.3 contains this result. In order to characterize those sequences of length< ω2 which are cardinal sequences of LCS spaces, it suffices to characterize the classes Cλ(α) forany ordinal α < ω2 and any infinite cardinal λ. In fact, this follows from the general reductionTheorem 1.3.2 that is valid in ZFC.

The promised GCH characterization of the classes Cλ(α) is given in Theorem 1.3.4. Themain ingredient of the proof is Theorem 1.3.5 which is generalization of the main construction ofTheorem 1.2.2.

In Section 1.4 we consider the classes Cλ(α) for α ≥ ω2 under GCH. In 1.4.1 we define afamily Dλ(α) of sequences of cardinals. Using elementary topological considerations, this familyis a natural “upper bound” of Cλ(α), i.e. GCH implies that Cλ(α) ⊆ Dλ(α).

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We conjecture that GCH implies that Cλ(α) = Dλ(α), but in [15] we could prove just a weakerstatement. Namely, as we prove it in Section 1.4, for each regular cardinal λ it is consistent thatGCH holds and Cλ(α) = Dλ(α), see Theorem 1.4.3.

Using this result, in Theorem 1.4.4 we characterize those sequences of regular cardinals whichcan be obtained as a cardinal sequence of an LCS∗ space in some model of ZFC + GCH.

To prove our theorems we should introduce the notion of universal spaces in 1.4.2 which isthe main tool of some proofs in the forthcoming sections.

In Sections 1.5 and 1.7 we investigate the class Cω(ω2).Baumgartner and Shelah proves that it is consistent that 〈ω〉ω2

∈ Cω(ω2). Building on theirmethod, Bagaria, [21], proved that Cω(ω2) ⊇ s ∈ ω2ω, ω1 : s(0) = ω is also consistent.However, he used MAℵ2 in his argument, and MAℵ2 implies 2ω0 ≥ ω3, and if 2ω0 = ωα, then thenatural “upper bound” of Cω(ω2) is a much larger family of sequences:

Cω(ω2) ⊆ s ∈ ω2ων : ν ≤ α : s(0) = ω. (1.8)

These results naturally raised the questions whether we may have equality in (1.8). In Theorem1.5.10 we answer in the positive: it is consistent that 2ω = ω2 and

Cω(ω2) = s ∈ ω2ω, ω1, ω2 : s(0) = ω.

In Section 1.7 we improve the above mention result: we prove Theorem 1.6.1 which claimsthat it is consistent that 2ω is as large as you wish and every sequence s = 〈sα : α < ω2〉 of infinitecardinals with sα ≤ 2ω is the cardinal sequence of some LCS space.

For a long time ω2 was a mystique barrier in both height and width. In that section we canconstruct wider spaces. However we should pay a price: by forcing we should introduce a morasstype structure which helps us to obtain a very strong ∆-like functions in a generic extensionwhich makes possible to force our desired space.

In Section 1.6 we prove certain stepping up theorems.

In a classical theorem, Roitman proved that it is consistent that 〈ω〉ω1

_〈ω2〉 ∈ Cω(ω1 + 1).Combining Koszmider’s strongly functions and Martinez’s orbit methods in Section 1.8 we improveRoitman’s result. We show e.g. that for each β < ω3 with cf(β) = ω2 it is consistent that 2ω1 isas large as you wish and 〈ω1〉β _〈2ω1〉 ∈ Cω1(β + 1).

Since the classes of the regular, of the zero-dimensional, and of the locally compact scatteredspaces are different, it was a natural question what is the relationship between their cardinalsequences. Actually, Juhász raised the question whether there is a regular space with cardinalsequence 〈ω〉2ω in ZFC. In [25, Theorem 10.1] the authors answered his question positively. foreach α < (2ω)+ there is a regular (even zero-dimensional) scattered space with cardinal sequence〈ω〉α.

In Section 1.9 we succeeded in giving a complete characterization of the cardinal sequencesof both T3 and zero-dimensional T2 scattered spaces. Although the classes of the regular and ofthe zero-dimensional scattered spaces are different, it will turn out that they yield the same classof cardinal sequences.

E. van Douwen and, independently, A. Dow [34] have observed that under CH an initiallyω1-compact T3 space of countable tightness is compact. (A space X is initially κ-compact ifany open cover of X of size ≤ κ has a finite subcover, or equivalently any subset of X of size≤ κ has a complete accumulation point). Naturally, the question arose whether CH is neededhere, i.e. whether the same is provable just in ZFC. The question became even more intriguingwhen in [23] , D. Fremlin and P. Nyikos proved the same result from PFA. Quite recently, A.V. Arhangel’skiıhas devoted the paper [20] to this problem, in which he has raised many relatedproblems as well.

In [90] M. Rabus has answered the question of van Douwen and Dow in the negative. He con-structed by forcing a Boolean algebra B such that the Stone space St(B) includes a counterexam-ple X of size ω2 to the van Douwen–Dow question, in fact St(B) is the one point compactificationof X, hence X is also locally compact.

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In Sections 1.10, we give an alternative forcing construction of counterexamples to the vanDouwen–Dow question, We directly force a topology τ on ω2 such that in the generic extensionV P out space X = 〈ω2, τ〉 is normal, locally compact, 0-dimensional, Frechet-Urysohn, initiallyω1-compact and non-compact space X of size ω2 having the following property: for every open(or closed) set A in X we have |A| ≤ ω1 or |X \A| ≤ ω1.

In [20, problem 12] Arhangel’skii asks if it is provable in ZFC that a normal, first countableinitially ω1-compact space is necessarily compact. We could not completely answer this question,but we show that the Frechet-Urysohn property (which is sort of half-way between countabletightness and first countability) in not enough to get compactness.

To achieve that we constructed a further generic extension of the model V P in which Xbecomes Frechet-Urysohn but its other properties are preserved, for example, X remains initiallyω1-compact and normal.

Arhangel’skii raised the question, [20, problem 3], whether CH can be weakened to 2ω <2ω1 in the theorem of van Douwen and Dow? We shall answer this question in the negative:theorem 1.10.34 implies that the existence of a counterexample to the van Douwen–Dow questionis consistent with practically any cardinal arithmetic that violates CH.

Improving the results of section 1.10, in Section 1.11 we answer a question of Arhangel’skii,[20,problem 12]: we force a first countable, normal, locally compact, initially ω1-compact and non-compact space X = 〈ω2 × 2ω, τ〉.

Actually, Alan Dow conjectured that applying the method of [74] (that "turns" a compactspace into a first countable one) to the space of Rabus in [90] yields an ω1-compact but non-compact first countable space. How one can carry out such a construction was outlined by thesecond author in the preprint [71]. However, [71] only sketches some arguments as the languageadopted there, which follows that of [90], does not seem to allow direct combinatorial controlover the space which is forced. This explains why the second author hesitated to publish [71].

One missing element of [71] was a language similar to that of [11] which allows working withthe points of the forced space in a direct combinatorial way. In section 1.11 we combine theapproach of [11] with the ideas of [71] to obtain directly an ω1-compact but non-compact firstcountable space. Consequently, our proofs follow much more closely the arguments of [11] thanthose of [90] or their analogues in [71].

As before, we again use a ∆-function to make our forcing CCC but we need both CH and a∆-function with some extra properties to obtain first countability.

It is immediate from the countable compactness of X that its one-point compactification X∗is not first countable. In fact, one can show that the character of the point at infinity ∗ in X∗ isω2. As X is initially ω1-compact, this means that every (transfinite) sequence converging from Xto ∗ must be of type cofinal with ω2. Since X is first countable, this trivially implies that there isno non-trivial converging sequence of type ω1 in X∗. In other words: the convergence spectrumof the compactum X∗ omits ω1. As far as we know, this is the first and only (consistent) exampleof this sort.

Combinatorial principles. In the second chapter of the dissertation we consider combina-torial principles.

The last 40 years have seen a furious activity in proving results that are independent ofthe usual axioms of set theory, that is ZFC. As the methods of these independence proofs (e.g.forcing or the fine structure theory of the constructible universe) are often rather sophisticated,while the results themselves are usually of interest to “ordinary” mathematicians (e.g. topologistsor analysts), it has been natural to try to isolate a relatively small number of principles, i.e.independent statements that a) are simple to formulate and b) are useful in the sense that theyhave many interesting consequences. Most of these statements, we think by necessity, are ofcombinatorial nature, hence they have been called combinatorial principles. The best knownexample is the Martin’s Axiom: to prove the consistency of this axiom you need to know theiterated forcing, but to apply this axiom in algebra, topology or combinatorics it is enough toknow elementary set-theory.

The above formulated criteria a) and b) as to what constitutes a combinatorial principle areoften contrary to each other: for more usefulness one often has to sacrifice some simplicity. It is

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not clear whether an ideal balance exists between them. It is up to the reader to judge if we havecome close to this balance.

As we mentioned, Just proved that in the Cohen model neither 〈ω〉ω2nor 〈ω〉ω1

_〈ω2〉 is acardinal sequence of an LCS∗ space. The two proofs are based on similar arguments. Usingthe same ideas we could prove other statements in the Cohen model, so it was a natural ideato investigate if we can combine these arguments into a “combinatorial principle”. That was thestarting point of the investigation of combinatorial principles in the Cohen model. As we will see,we could find principles which are (a) true in the Cohen model, and (b) which implies the abovementioned results of Just.

In section 2.1 we present several new combinatorial principles that are all statements aboutP(ω), the power set of the natural numbers. In fact, they all concern matrices of the form〈A(α, n) : 〈α, n〉 ∈ κ× ω〉, where A(α, n) ⊂ ω for each 〈α, n〉 ∈ κ × ω, and, in the interestingcases, κ is a regular cardinal with c = 2ω ≥ κ > ω1.

We show that these statements are valid in the generic extensions obtained by adding anynumber of Cohen reals to any ground model V , assuming that the parameter κ is a regular andω-inaccessible cardinal in V ( i.e. λ < κ implies λω < κ).

Then we present a large number of consequences of these principles, some of them com-binatorial but most of them topological, mainly concerning separable and/or countably tighttopological spaces. (This, of course, is not surprising because these are objects whose structuredepends basically on P(ω).)

Recently we could find an application of our principles in the theory of Banach spaces, [1].Using a much finer analysis of the Cohen model, in Section 2.2 we recall the main result of

[10]: in the Cohen model an LCS∗ may have at most ω1 many countable Cantor-Bendixson level.We conjecture that in ZFC an LCS∗ may have at most ω2 many countable Cantor-Bendicty

level. In his celebrated theorem, Shelah proves that max pcfℵn : n < ω < ℵω4 . One ofthe central question of set theory is to improve this result. If this conjecture is true, thenmax pcfℵn : n < ω < ℵω3 .

I jointed to the investigation of the weak Freeze-Nation (wFN) property independently fromthe research of the combinatorial principles. However, at some point we realized that the wFNproperty of the poset 〈P,⊂〉 itself has numerous interesting combinatorial and topological conse-quence, so it can be considered as a combinatorial principle. This observation explains why wediscuss the wFN property of certain posets in 3 sections.

Definition 1.1.13.A poset P = 〈P,≤〉 has the weak Freese-Nation (wFN) property iff there isa function f : P →

[P]ω such that

• if p, q ∈ P and p ≤ q then there is r ∈ f(p) ∩ f(q) with p ≤ r ≤ q.Let Q ⊆ P . Write Q ≤σ P if for each p ∈ P the set q ∈ Q : q ≤ p has a countable cofinalsubset, and the set q ∈ Q : q ≥ p has a countable coinitial subset.

These notions were introduced by Heindorf and Shapiro [51].By [46], the poset P has the wFN property iff the family Q ∈

[P]ω1 : Q ≤σ P contains a

club subfamily. In [7], we prove that a weaker assumption on the class Q ∈[P]ω1 : Q ≤σ P is

enough to derive the wFN property of P . This result is included in section 2.3.We introduce a very weak box principle ∗∗∗µ , and prove Theorem 2.3.14: assume that λ is a

cardinal with(i) cf([µ]ω,⊆) = µ if ω1 < µ < λ and cf(µ) ≥ ω1,(ii) ∗∗∗µ holds for each µ < λ with cofinality ω.

Then for each poset P of cardinality ≤ λ the following are equivalent:(w1) P has the wFN property,(w2) for each large enough regular cardinal χ, if M ≺ H(χ), P , ω1 ∈ M , and M is the union of

an ω1 chain of countable elementary submodels of H(χ) then P ∩M ≤σ P .We proved that the assumption of ∗∗∗ can not be omitted: if GCH holds and the Chang

conjecture (ℵω+1,ℵω) (ℵ1,ℵ0). holds then the poset ([ℵω]ℵ0 ,⊆) does not have the wFNproperty.

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Let us remark that if 2ω = ω1 then (w2) holds for ([ℵω]ℵ0 ,⊆). So GCH is not enough toprove the equivalence of (w1) and (w2)

In section 2.4 we solve some questions concerning wFN properties of Boolean algebras. Whilein section 2.3 we proved that 〈P(ω),⊂〉 has the wFN property if we add arbitrary number of Cohenreals to a “nice‘” model of ZFC, (e.g. to, L), in Section 2.5 we proved that L can not be replacedby an arbitrary model of ZFC which satisfies GCH.

In Section 2.5 we show that the assumption 〈P(ω),⊂〉 has the wFN property can be consideredas a useful combinatorial principle. Indeed, we proved, that if 〈P(ω),⊂〉 has the wFN property,then(a) there is no ℵ2-Luzin gap, and ω2 is not embeddable into 〈P(ω),⊆∗〉;(b) 〈ω〉ω1

_〈ω2〉 /∈ C(ω1 + 1) and 〈ω〉ω2/∈ C(ω2) ;

(c) non(M) = ω1 and cov(M) > ω1;(d) a = ω1,(e) any ω1-fold cover of the real real by closed sets can be partitioned into ω1 disjoint subcover.The last application is from [2].

So far we investigated principles which hold in the Cohen model. However, there are otherprinciples which can not hold in the Cohen model, e.g. ♣ or •| .

Our starting point in the last section was to find a generic extension which allows to blow upthe continuum without collapsing cardinals in such a way that •| holds in the generic extension.

To handle this problem we developed a new type of product of posets.As it turned out, this new product is suitable to obtain models in which seemingly inconsistent

statements hold. The basic idea is to add Cohen reals in such a way that in the generic extensionthere is no generic filter for the poset Fn(ω1, 2;ω).

We formulate just our most interesting result. As usual, MA(countable) denotes the state-ment that Martin’s Axiom holds for countable posets. So we proved that it is consistent that youcan obtain a model without collapsing cardinals in such a way that in the generic extension 2ω isas large as you wish, and both MA(countable) and •| hold.

1.2. A tall space with small bottom

(This section is based on [9])

By theorems 1.1.7, 1.1.8 and 1.1.9,• for each α < ω2 there is an LCS∗ space X with CS(X) = 〈ω〉α,• ZFC+ ¬CH does not decide if there is an LCS∗ space X with CS(X) = 〈ω〉ω2

.

The estimate ht(X) < (2|I(X)|)+ is sharp for LCS∗ spaces with countably many isolated pointsby theorem 1.2.3 below:

• for each α < (2ω)+ there is an LCS∗ space X with ht(X) = α and | I0(X)| = ω.Much less is known about LCS∗ spaces with ω1 isolated points, for example it is a long standingopen problem whether there is, in ZFC, an LCS∗ space of height ω2 and width ω1. In fact, as wasnoticed by Juhász in the mid eighties, even the much simpler question if there is a ZFC exampleof an LCS∗ space of height ω2 with only ω1 isolated points, turned out to be surprisingly difficult.On the other hand, Martínez in [84, theorem 1] proved that

• it is consistent that for each α < ω3 there is a LCS∗ space X with CS(X) = 〈ω1〉α.As the main result of [9], we gave an affirmative answer to the above question of Juhász:

Theorem 1.2.1.There is, in ZFC, an LCS∗ space of height ω2 and having ω1 isolated points.

The main ingredient of the proof of Theorem 1.2.1 is the following result.

Theorem 1.2.2. If κ<κ = κ then there is an LCS∗ space X of height κ+ with | I0(X)| = κ.

In particular, if 2ω = ω1 then the above result yields an LCS∗ space X with ht(X) = ω2 and| I0(X)| = ω1. That such a space also exist under ¬CH, hence in ZFC, follows from the followingresult.

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1.2. A TALL SPACE WITH SMALL BOTTOM 9

Theorem 1.2.3. For each α < (2ω)+ there is a locally compact, scattered space Xα with |Xα| =|α|+ ω, ht(Xα) = α and | I0(Xα)| = ω.

Proof. We do induction on α. If α = β + 1 then we let Xα be the 1-point compactification ofthe disjoint topological sum of countably many copies of Xβ .

If α is limit then we first fix an almost disjoint family Aβ : β < α ⊂[ω]ω, which is

possible by |α| ≤ 2ω. Applying the inductive hypothesis, for each β < α we can also fix a locallycompact scattered space Xβ of cardinality ω + |β| and height β such that I0(Xβ) = Aβ andXβ ∩ Xγ = Aβ ∩ Aγ for β, γ ∈

[α]2. Now amalgamate the spaces Xβ as follows: consider

the topological space X = 〈∪β<αXβ , τ〉 where τ is the topology generated by ∪β<ατXβ . SinceAβ ∩ Aγ is a finite and open subspace of both Xβ and Xγ it follows that each Xβ is an opensubspace of X. Consequently, X is LCS∗ with countably many isolated points, and ht(X) =supβ<α htXβ = α.

Proof of Theorem 1.2.1 using Theorem 1.2.2 . If 2ω = ω1, then theorem 1.2.2 gives sucha space.

If 2ω > ω1 then (2ω)+ ≥ ω3 and so according to theorem 1.2.3 for each α < ω3 there is locallycompact, scattered space of height α and with countably many isolated points.

The rest of this section is devoted to the proof of Theorem 1.2.2.

Definition 1.2.4.Given a family of sets A we define the topological space X(A) = 〈A, τA〉 asfollows: τA is the coarsest topology in which the sets UA(A) = A ∩ P(A) are clopen for eachA ∈ A, in other words: UA(A),A \UA(A) : A ∈ A is a subbase for τA.

We shall write U(A) instead of UA(A) if A is clear from the context.

Clearly X(A) is a 0-dimensional T2-space. A family A is called well-founded iff 〈A,⊂〉 iswell-founded. In this case we can define the rank-function rk : A −→ On as usual:

rk(A) = suprk(B) + 1 : B ∈ A ∧ B ( A,and write Rα(A) = A ∈ A : rk(A) = α.

The family A is said to be ∩-closed iff A ∩B ∈ A ∪ ∅ whenever A,B ∈ A.It is easy to see that if A is ∩-closed, then a neighbourhood base in X(A) of A ∈ A is formed

by the sets

W(A;B1, . . . , Bn) = U(A) \n⋃1

U(Bi),

where n ∈ ω and Bi ( A for i = 1, . . . , n. ( For n = 0 we have W(A) = U(A).)The following simple result enables us to obtain LCS∗ spaces from certain families of sets.

Let us point out, however, that not every LCS∗ space is obtainable in this manner, but we donot dwell upon this because we will not need it.

Lemma 1.2.5 ([9]).Assume that A is both ∩-closed and well-founded. Then X(A) is an LCS∗space.

To simplify notation, if X(A) is scattered then we write Iα(A) = Iα(X(A)).Clearly each minimal element of A is isolated in X(A); more generally we have α ≤ rk(A) if

A ∈ Iα(A), as is shown by an easy induction on rk(A).

Example 1.2.6.Assume that 〈T,≺〉 is a well-ordering, tp 〈T,≺〉 = α, and let A be the familyof all initial segments of 〈T,≺〉, i. e. A = T ∪ Tx : x ∈ T, where Tx = t ∈ T : t ≺ x.Then A is well-founded, ∩-closed and it is easy to see that X(A) ∼= α+ 1, i.e. the space X(A) ishomeomorphic to the space of ordinals up to and including α.

Example 1.2.6 above shows that, in general, Rα(A) and Iα(A) may differ even for α = 0.Indeed, if x is the successor of y in 〈T,≺〉 then Tx is isolated in X(A) because Tx = W (Tx;Ty) =UA(Tx) \UA(Ty) is open, but rk(Tx) = tp(Tx) > 0. However, for a wide class of families, the twokinds of levels do agree. Let us call a well-founded family A rk-good iff the following condition issatisfied:

∀A ∈ A ∀α < rk(A) |A′ ∈ A : A′ ⊂ A ∧ rk(A′) = α| ≥ ω.

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Then we have the following result.

Lemma 1.2.7 ([9]). If A is a well-founded, ∩-closed and rk-good family, then Iα(A) = Rα(A)for each α.

Example 1.2.8. For a fixed cardinal κ and any ordinal γ < κ+ we define the family Eγ ⊂ P(κγ)as follows:

Eγ =[κ1+α · ξ, κ1+α · (ξ + 1)

): α ≤ γ, κ1+α · (ξ + 1) ≤ κ1+γ

.

Of course, throughout this definition exponentiation means ordinal exponentiation.Eγ is clearly well-founded, ∩-closed, moreover rk

([κ1+α · ξ, κ1+α · (ξ + 1)

))= α, hence Eγ

is also rk-good. Consequently X(Eγ) is an LCS space of height γ + 1 in which the αth level is[κ1+α · ξ, κ1+α · (ξ + 1)

): κ1+α · (ξ + 1) ≤ κ1+γ

, i. e. all levels except the top one are of size

κ.

To get an LCS∗ space of height κ+ with “few” isolated points, our plan is to amalgamatethe spaces X(Eγ) : γ < κ+ into one LCS∗ space X in such a way that | I0(X)| ≤ κ<κ. Thefollowing definition describes a situation in which such an amalgamation can be done.

Definition 1.2.9.A system of families Ai : i ∈ I is called coherent iff A ∩ B ∈ Ai ∪ ∅whenever i, j ∈

[I]2, A ∈ Ai and B ∈ Aj .

To simplify notation, we introduce the following convention. Whenever the system of familiesAi : i ∈ I is given, we will write Ui(A) for UAi(A), and τi for τAi . If the family A is definedthen we will write U(A) for UA(A), and τ for τA.

Lemma 1.2.10 ([9]. ).Assume that Ai : i ∈ I is a coherent system of well-founded, ∩-closedfamilies and A = ∪Ai : i ∈ I. Then for each i ∈ I and A ∈ Ai we have U(A) = Ui(A), A isalso well-founded and ∩-closed, moreover τi | U(A) = τ | U(A). Consequently each X(Ai) is anopen subspace of X(A) and thus X(Ai) : i ∈ I forms an open cover of X(A).

Given a system of families Ai : i ∈ I we would like to construct a coherent system offamilies Ai : i ∈ I such that Ai and Ai are isomorphic for all i ∈ I. A sufficient condition forwhen this can be done will be given in lemma 1.2.12 below.

First, however, we need a definition. While reading it, one should remember that an ordinalis identified with the family of its proper initial segments.

Definition 1.2.11.Given a limit ordinal ρ and a family A with ρ ⊂ A ⊂ P(ρ), let us definethe family A as follows. Consider first the function kA on ρ determined by the formula kA(η) =UA(η + 1) for η ∈ ρ and put

A = k′′AA : A ∈ A.Since ρ ⊂ A, for each η ∈ ρ we clearly have ∪UA(η) = η and so kA(η) = UA(η + 1) 6=

UA(ξ + 1) = kA(ξ) whenever η, ξ ∈[ρ]2. Consequently, kA is a bijection that yields an

isomorphism between A and A (and so the spaces X(A) and X(A) are homeomorphic).If the system of families Ai : i ∈ I is given, then we write ki for kAi for each i ∈ I.If A ⊂ P(ρ) and ξ ≤ ρ then we let

A | ξ = A ∩ ξ : A ∈ A.

For A0 6= A1 ⊂ P(ρ) we let

∆(A0,A1) = minδ : A0 | δ 6= A1 | δ.

Clearly we always have ∆(A0,A1) ≤ ρ. If, in addition, ρ + 1 ⊂ A0 ∩ A1, moreover both A0

and A1 are ∩-closed then we also have

∆(A0,A1) = minδ : U0(δ) 6= U1(δ),

because then Ai | δ = Ui(δ) whenever i ∈ 2 and δ ≤ ρ.

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1.2. A TALL SPACE WITH SMALL BOTTOM 11

Lemma 1.2.12 ([9]).Assume that κ is a cardinal, Ai : i ∈ I ⊂ PP(κ) are ∩-closed families,κ

.+ 1 ⊂ Ai for each i ∈ I, and ∆(Ai,Aj) is a successor ordinal whenever i, j ∈

[I]2. Then

the system Ai : i ∈ I is coherent.

More is needed still if we want the "amalgamated" family to provide us a space with a smallbottom, i.e. having not too many isolated points. This will be made clear by the following lemma.

Lemma 1.2.13. Let κ be a cardinal and Ai : i ∈ I ⊂ PP(κ) be a system of families such that

(i) κ.+ 1 ⊂ Ai and Ai is well-founded and ∩-closed for each i ∈ I,

(ii) ∆(Ai,Aj) is a successor ordinal for each i, j ∈[I]2.

Then(a) the system Ai : i ∈ I is coherent and thus A =

⋃Ai : i ∈ I is well-founded, ∩-closed and

X(A) is covered by its open subspaces X(Ai) : i ∈ I.If, in addition, we also have

(iii) I0(Ai) ⊂[κ]<κ for each i ∈ I,

and(iv) |Ui(ξ)| < κ for each i ∈ I and ξ ∈ κ,then

(b) I0(A) ⊂[[

[κ]<κ]<κ]<κ

.

Proof of lemma 1.2.13. The system Ai : i ∈ I is coherent by lemma 1.2.12, thus (a) holdsby lemma 1.2.10.

Consequently we have

I0(A) =⋃I0(Ai) : i ∈ I.

Now if A ∈ I0(Ai) and η ∈ κ then |A| < κ by (iii) and Ui(η) ∈[[κ]<κ]<κ by (iv), hence

k′′i A = Ui(η + 1) : η ∈ A ∈[[[

κ]<κ]<κ]<κ

.

This, by I0(Ai) = k′′i A : A ∈ I0(Ai), proves (b).

Definition 1.2.14. If ρ is an ordinal and A ⊂ P(ρ) let us put

A∗ = A ∩ ξ : A ∈ A ∧ ξ ≤ ρ = A ∪ A ∩ ξ : A ∈ A ∧ ξ < ρ.Definition 1.2.15.A family A is called chain-closed if for each non-empty B ⊂ A if B is orderedby ⊂ (i.e. if B is a chain) then ∪B ∈ A.Lemma 1.2.16 ([9]). If ρ is an ordinal and A0,A1 ⊂ P(ρ) are chain-closed, ∩-closed and well-founded families such that A0

∗ 6= A1∗ then ∆(A0

∗,A1∗) is a successor ordinal.

The last result shows us that the operation * is useful because its application yields us familiesthat satisfy condition (ii) of lemma 2.10. On the other hand, the following result tells us thatthe LCS spaces associated with certain families modified by * do not differ significantly from thespaces given by the original families, moreover they also satisfy condition (iii) of lemma 2.10.

Lemma 1.2.17. Let κ be a cardinal and A ⊂[κ]κ be well-founded and ∩-closed. Then so is A∗,

moreover(a) X(A) is a closed subspace of X(A∗),(b) I0(A) ⊆ Iκ(A∗).(c) ht(A∗) ≥ κ

.+ ht(A),

(d) I0(A∗) ⊂[κ]<κ.

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Proof of lemma 1.2.17. We shall write U(A) for UA(A), and U∗(A) for UA∗(A).First observe that because

U∗(A) ∩ A =

U(A) if A ∈ A,∅ if A ∈ A∗ \ A,

X(A) is a closed subspace of X(A∗), hence (a) holds.Now let A ∈ I0(A). Then there are B1, . . . , Bn ∈ U(A) \ A such that

A = W(A;B1, . . . , Bn) = U(A) \n⋃i=1

U(Bi).

Since here Bi ( A and |A| = κ, we can fix η ∈ A such that (A ∩ η) 6⊂ Bi for every i = 1, . . . , n.Now consider the basic neighbourhood

Z = W∗(A;A ∩ η,B1, . . . , Bn) = U∗(A) \U∗(A ∩ η) \n⋃i=1

U∗(Bi)

of A in X(A∗). We claim that Z = A ∩ ξ : η < ξ ≤ κ. The inclusion ⊃ is clear from thechoice of η. On the other hand, if C ∩ ξ ∈ Z with C ∈ A and ξ ≤ κ, then C ∩ ξ ⊂ A henceC ∩ ξ = A ∩ C ∩ ξ, so as A is ∩-closed we can assume that C ⊆ A. If we had C 6= A thenA = W(A;B1, . . . , Bn) would imply C ⊂ Bi for some i, hence C∩ξ ∈ U∗(Bi) and so C∩ξ /∈ Z,a contradiction, thus we must have C = A. Moreover, since U∗(A ∩ η) ⊃ A ∩ ν : ν ≤ η, wemust also have ξ > η.

By example 1.2.6 we have X(Z) ∼= κ + 1. Moreover, the topologies τZ and τA∗ | Z coincidebecause the above argument also shows that for each C ∈ A and ζ ≤ κ we have

U∗(C ∩ ζ) ∩ Z = U∗(A ∩ C ∩ ζ) ∩ Z =

UZ(A ∩ ζ) if A ⊂ C and ζ > η;∅ otherwise.

Hence X(Z) ∼= κ.+ 1 is a clopen subspace of X(A∗) and so A = Iκ(Z) = Iκ(A∗)∩Z, what

proves (b).(c) follows immediately from (a) and (b).Finally, I0(A∗) ⊂ I<κ(A∗) ⊂ (A∗ \ A) ⊂

[κ]<κ, as follows immediately from (b), proving

(d).

Before we could apply the amalgamation result to the families Eγ , however, we need somefurther preparation that will be useful in ensuring the fulfillment of condition (iv) in 1.2.13.

Definition 1.2.18.A family A is called tree-like iff A ∩ A′ 6= ∅ implies that A ⊂ A′ or A′ ⊂ A,whenever A,A′ ∈ A.

It is easy to see that the families Eγ given in example 1.2.8 are both tree-like and chain-closed.Also, tree-like families are clearly ∩-closed.Lemma 1.2.19 ([9]). If δ is an ordinal and A ⊂ P(δ) is tree-like, well-founded and chain-closedthen so is A | ξ for each ξ ≤ δ.

Definition 1.2.20.Given a family A ⊂ P(δ) and α, β ∈ δ let us put

SA(α, β) = ∪A ∈ A : α ∈ A and β /∈ A.Lemma 1.2.21.Assume that δ is an ordinal and A ⊂ P(δ) is a tree-like, well-founded andchain-closed family with δ ∈ A. Then

A \ ∅ = δ ∪ SA(α, β) : α, β ∈ δ \ ∅.Consequently, |A| ≤ |δ|2.

Proof of lemma 1.2.21. Given α, β ∈ δ, the family S = A ∈ A : α ∈ A, β /∈ A is orderedby ⊂ because A is tree-like. Thus either S = ∅ and so SA(α, β) = ∪S = ∅, or if S 6= ∅ thenSA(α, β) = ∪S ∈ A, for A is chain-closed.

Assume now that A ∈ A \ ∅, δ and let D = D ∈ A : A ( D. Clearly δ ∈ D. SinceA is tree-like, D is ordered by ⊂ , so it has a ⊂-least element, say D, because 〈A,⊂〉 is also

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1.3. CARDINAL SEQUENCES OF LENGTH < ω2 UNDER GCH 13

well-founded. Pick β ∈ D \A and let α ∈ A. We claim that A = SA(α, β). Clearly A ⊂ SA(α, β)because α ∈ A and β /∈ A. On the other hand, if A′ ∈ A, α ∈ A′ and β /∈ A′ then either A′ ⊂ Aor A ⊂ A′ because A is tree-like. But β /∈ A′ implies that A′ /∈ D, i.e. A ( A′ can not hold.Thus A′ ⊂ A and so SA(α, β) = A is proved.

Now we are ready to collect the fruits of all the preparatory work.

Proof of theorem 1.2.2. For each γ < κ+ consider the well-founded, ∩-closed, rk-good familyEγ constructed in example 1.2.8:

Eγ =[κ1+α · ξ, κ1+α · (ξ + 1)

): α ≤ γ, κ1+α · ξ < κγ

.

Fix a bijection fγ : κγ −→ κ, and let Fγ = fγ ′′E : E ∈ Eγ, i.e. Fγ is simply an isomorphiccopy of Eγ on the underlying set κ. As Eγ is also chain-closed and tree-like, hence so is Fγ .

We shall now show that the ∗-modified families Fγ∗ : γ < κ+ satisfy conditions (i)-(iv) oflemma 1.2.13. Since κ ∈ Fγ it follows that κ

.+ 1 ⊂ Fγ∗ and so (i) is true. For γ, δ ∈

[κ+]2, the

height of X(Eγ) is γ + 1 and the height of X(Eδ) is δ + 1, hence Eγ and Eδ are not isomorphic.Thus Fγ 6= Fδ and so Fγ∗ 6= Fδ∗ as well because Fγ = Fγ∗ ∩

[κ]κ and Fδ = Fδ∗ ∩

[κ]κ. Hence

∆(Fγ∗,Fδ∗) is a successor ordinal by lemma 1.2.16, i.e. (ii) is satisfied.(iii) holds by 1.2.17.(d.)To show (iv), let us fix ξ < κ. Then UFγ∗(ξ) = Fγ∗ | ξ = ∪Fγ | ζ : ζ ≤ ξ where |Fγ | ζ| ≤

|ζ|2 for all ζ ≤ ξ by lemmas 1.2.19 and 1.2.21, consequently |Fγ∗ | ξ| ≤ |ξ|3 < κ.Thus we may apply lemma 1.2.13 to the family F = ∪Fγ∗ : γ < κ+ and conclude that

the space X = X(F) is LCS, moreover | I0(X)| ≤((κ<κ)<κ

)<κ = κ. Since for every γ ∈ κ+

the space X(Fγ∗) is an open subspace of X, we have ht(X) ≥ ht(X(Fγ∗)) > γ, consequentlyht(X) ≥ κ+.

1.3. Cardinal sequences of length < ω2 under GCH


The cardinal sequences of LCS∗ spaces with height ω1 has the following characterization in ZFC:

Theorem 1.3.1 (Juhasz, Weiss, [61]). 〈κξ : ξ < ω1〉 ∈ C(ω1) iff κη ≤ κωξ holds wheneverξ < η < ω1.

It follows that cardinal arithmetic alone decides whether a sequence of cardinals of length ω1

belongs to C(ω1) or not. The situation changes dramatically for longer sequences, in fact alreadyfor sequences of length ω1 + 1. For example, the question if 〈ω〉ω1

_ 〈ω2〉1 ∈ C(ω1 + 1) is notdecided by the following cardinal arithmetic: 2ω = ω2 and 2κ = κ+ for all κ > ω (see Just[64]and Roitman[93]).

However, as we showed in [14], the elements of C(α) can be characterized for all α < ω2 ifwe assume GCH. This section contains that result.

In order to characterize those sequences of length < ω2 which are cardinal sequences of LCS∗spaces, it suffices to characterize the classes Cλ(α) for any ordinal α < ω2 and any infinite cardinalλ. In fact, this follows from the following general reduction theorem that is valid in ZFC.

Theorem 1.3.2 ([14]). For any ordinal α and any sequence f of cardinals of length α the followingare equivalent:(1) f ∈ C(α)(2) for some natural number n there is a decreasing sequence λ0 > λ1 > · · · > λn−1 of infinite

cardinals and there are ordinals α0, . . . αn−1 such that α = α0 + · · · + αn−1 and f = f0_

f1_ · · · _ fn−1 with fi ∈ Cλi(αi) for each i < n.

We prove Theorem 1.3.2 at the end of this section.From here on let us assume GCH. Our aim is to characterize the classes Cλ(α) with α < ω2.

(For an ordinal α and a set B, as usual, we let αB denote the set of all sequences of length α

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taking values in B.) Now, for any s ∈ αλ, λ+ we write

Aλ(s) = β ∈ α : s(β) = λ = s−1λ.Definition 1.3.3. If α is any ordinal, a subset L ⊂ α is called κ-closed in α, where κ is an infinitecardinal, iff sup 〈αi : i < κ〉 ∈ L ∪ α for each increasing sequence 〈αi : i < κ〉 ∈ κL. The set Lis < λ-closed in α provided it is κ-closed in α for each cardinal κ < λ. We say that L is successorclosed in α if β + 1 ∈ L ∪ α for all β ∈ L.

We are now ready to present the promised GCH characterization of the classes Cλ(α) andconsequently, in view of 1.3.2, the characterization of C(α) for all α < ω2.

Theorem 1.3.4.Assume GCH and fix α < ω2.(i) Cω(α) = s ∈ αω, ω1 : s(0) = ω.(ii) If λ > cf(λ) = ω,

Cλ(α) = s ∈ αλ, λ+ : s(0) = λ and Aλ(s) is ω1-closed in α.(iii) If cf(λ) = ω1,

Cλ(α) = s ∈ αλ, λ+ : s(0) = λ andAλ(s) is both ω-closed and successor-closed in α.

(iv) If cf(λ) > ω1,Cλ(α) = 〈λ〉α.

The main ingredient of the proof is the following statement:

Theorem 1.3.5. Let λ be a cardinal with µ = cf(λ) > ω and satisfying λ = λ<µ. Then for anycardinal κ with λ < κ ≤ λµ and for every ordinal α < µ+ with cf(α) = µ we have 〈λ〉α _ 〈κ〉µ+ ∈C(µ+).

To prove theorems above we need some preparation.If X is a scattered space and x ∈ X then we write ht(x,X) = α iff x ∈ Iα(X). Trivially, then

ht(X) = minβ : ∀x ∈ X[ht(x,X) < β].It is obvious that if Y ⊂ X then ht(x,X) ≥ ht(x, Y ) whenever x ∈ Y , and if Y is also open in Xthen actually ht(x,X) = ht(x, Y ). On the other hand, for the points of X outside of Y one canget the following upper bound.

Fact 1.3.6. If Y is an open subspace of the scattered space X then for every point x ∈ X \ Y wehave ht(x,X) ≤ ht(Y ) + ht(x,X \ Y ). Consequently, ht(X) ≤ ht(Y ) + ht(X \ Y ).

Indeed, this can be proved by a straight-forward transfinite induction on ht(x,X), usingY ⊂ I<ht(Y )(X).

It is well-known that any ordinal, as an ordered topological space, is locally compact andscattered. It is easy to see that if α < β are ordinals then ht(α, β) = γ iff α can be written inthe form ωγ · (2δ + 1), or equivalently, γ is minimal such that α can be written as α = ε + ωγ .Note that in the notation ht(α, β) the ordinals play a double role: α is considered as a "point" inthe set β. Using the above characterization of the Cantor-Bendixson levels of ordinal spaces, itis easy to show that for any infinite cardinal λ and for any ordinal α < λ+ we have 〈λ〉α ∈ C(α).

This section is based on [14] which is is a natural sequel to [61], so now we recall a fewgeneral statements concerning cardinal sequences from [61] that will be needed later.

Fact* 1.3.7 ([61, lemma 1]). If s ∈ C(β) then |β| ≤ 2s(0) and s(α) ≤ 2s(0) for each α < β.

Fact* 1.3.8 ([61, lemma 2]). If s ∈ C(β) and α+ 1 < β then s(α+ 1) ≤ s(α)ω.

Fact* 1.3.9 ([61, lemma 3]). If s ∈ C(β), δ < β is a limit ordinal and C is any cofinal subset ofδ, then

s(δ) ≤∏s(α) : α ∈ C .

We shall also need the following general construction from [61] that is used to obtain anLCS∗ space by gluing together certain others.

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Lemma 1.3.10 ([61, lemma 7]). Let X be an LCS∗ space with a closed discrete subset S suchthat for each s ∈ S there is given a sequence 〈Us,n : n ∈ ω〉 of pairwise disjoint compact opensubsets of X \S, converging to the point s. Also, for each s ∈ S let Ys be a separable LCS∗ spacesuch that the collection of spaces X ∪ Ys : s ∈ S is disjoint. Then there is an LCS∗ space Zwith the following three properties (i) - (iii):

(i) Z = (X \ S) ∪⋃Ys : s ∈ S with X \ S as an open subspace and each Ys as a closed

subspace. Moreover, Ys : s ∈ S forms a discrete collection in Z.(ii) ht(x, Z) = ht(x,X) for x ∈ X \ S.(iii) ht(y, Z) = δs + ht(y, Ys) for y ∈ Ys, where δs is the least ordinal δ such that the set

n < ω : Us,n ∩ Iδ(X) 6= ∅ is finite. (Clearly, δs ≤ ht(s,X).)

Definition 1.3.11. For any family of sets A we define the topological space X(A) = 〈A, τA〉 asfollows: τA is the coarsest topology on A such that the sets UA(A) = A ∩ P(A) are clopen foreach A ∈ A. In other words: UA(A),A \UA(A) : A ∈ A is a subbase for τA.

Clearly X(A) is a 0-dimensional T2-space.A family A is called well-founded iff the partial order 〈A,⊂〉 is well-founded. A is said to be

∩-closed iff A ∩B ∈ A ∪ ∅ whenever A,B ∈ A.It is easy to see that if A is ∩-closed, then a neighbourhood base of A ∈ A in the space X(A)

is formed by the clopen sets

WA(A;B1, . . . , Bn) = UA(A) \n⋃1

UA(Bi),

where n ∈ ω and Bi ( A for i = 1, . . . , n. ( For n = 0 we have WA(A) = UA(A).) We shall writeU(A) instead of UA(A) if A is clear from the context, and similarly for the W’s.

The following statement, that was proved in [9, Lemma 2.2], shows the relevance of the aboveconcepts to the subject matter of this section.

Fact 1.3.12.Assume that A is both ∩-closed and well-founded. Then X(A) is an LCS∗ space.

To simplify notation, if X(A) is scattered then we write Iα(A) instead of Iα(X(A)), andI<α(A) instead of ∪Iζ(A) : ζ < α. In the same spirit, for A ∈ A we sometimes write ht(A,A)instead of ht(A,X(A)).

We shall say that A is an ordinal family if all members of A are sets of ordinal numbers,moreover A is both ∩-closed and well-founded. (As usual, we shall denote by On the class ofall ordinals.) The following definition makes sense for any ordinal family A and will play animportant role in our construction.

If A is an ordinal family and ξ is any ordinal then we let

A ξ = A ∩ ξ : A ∈ Aand

A∗ =⋃A ξ : ξ ∈ On.

So A∗ is simply the family consisting of all initial segments of all members of A. Clearly,A∗ = A ∪ A ∩ ξ : A ∈ A ∧ ξ ∈ A.

It is easy to see that if A is an ordinal family then so is A∗, hence both X(A) and X(A∗)are LCS∗ spaces. A key ingredient in our construction is, just like in [9], the clarification of therelationship between these two spaces. The following technical lemma will play a significant rolein this. As indicated above, we shall write U(A) instead of UA(A), U∗(A) instead of UA∗(A),and similarly for the W’s.

Lemma 1.3.13 ([14]). Let A be an ordinal family. Then for any A ∈ A we have

ht(U∗(A)) ≤ supht(U∗(A′)) : A′ ∈ U(A) \ A)+ ht(tpA+ 1)

What we shall really need in our construction is the following corollary of lemma 1.3.13.

Lemma 1.3.14 ([14]). Let A be an ordinal family such that, for a fixed indecomposable α ∈ On,we have |A| < cf(α) and ht(tpA) < α for all A ∈ A. Then ht(X(A∗)) < α.

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Now we shall prove a result showing that, for certain ordinal families H, the space X(H) is avery special subspace of X(H∗). This result naturally corresponds to [9, lemma 2.14]. If ρ is anordinal and L ⊂ On then we write

(L)ρ = K ⊂ L : tpK = ρ,

moreover(L)<ρ =

⋃(L)α : α < ρ.

Lemma 1.3.15 ([14]). Let ρ be an indecomposable ordinal such that ht(ρ) = α is also indecom-posable, cf(α) = cf(ρ), moreover ht(ξ) < α for all ξ < ρ. Let H ⊂ (ρ)ρ be an ordinal family suchthat

∣∣H ξ∣∣ < cf(α) for all ξ < ρ. Then X(H) forms a "tail" of X(H∗) in the following sense:(a) X(H) is a closed subspace of X(H∗),(b) Iβ(H) = Iα+β(H∗) for all β < ht(X(H)),(c) I<α(H∗) = H∗ \ H.

Definition 1.3.16.Two families of sets A0 and A1 are said to be coherent iff A0 ∩ A1 ∈ (A0 ∩A1)∪ ∅ whenever Ai ∈ Ai for i < 2. A system of families Ai : i ∈ I is coherent iff Ai and Ajare coherent for each pair i, j ∈

[I]2.

If Ai : i ∈ I is a coherent system of well-founded and ∩-closed families then we can“amalgamate” the spaces X(Ai) : i ∈ I as follows: According to [9, lemma 7], then A =∪Ai : i ∈ I is also well-founded and ∩-closed, hence X(A) is also an LCS∗ space; moreoverX(Ai) : i ∈ I forms an open cover of X(A).

Next we introduce a method that transforms an ordinal family A into another, isomorphicfamily A. We do this because, for certain systems of ordinal families Ai : i ∈ I, the new systemAi : i ∈ I turns out to be coherent. Since X(Ai) and X(Ai) are clearly homeomorphic, inthis way the spaces X(Ai) : i ∈ I may be amalgamated in two steps. The definition of theoperation A 7→ A given below is a slight generalization of the one given in definition 2.8 from [9].

The next definition is a generalization of Definition 1.2.11.

Definition 1.3.17. Let us be given a set of ordinals L ⊂ On of limit order type and a family Awith L ⊂ A ⊂ P(L). We first define the map kA on L by the formula kA(η) = A η + 1 forη ∈ L; then we define the map χA on A by putting χA(A) = k′′AA, i. e. χA(A) is the kA-imageof A ∈ A. Finally, we define the family A by

A = χA(A) : A ∈ A.Let us remark that L ∩ (η + 1) ∈ L ⊂ A for each η ∈ L, hence we have

kA(η) = A η + 1 = UA(L ∩ (η + 1)).

By the same token, for any η ∈ L we also have

max[∪ kA(η)] = max[L ∩ (η + 1)] = η,

consequently kA is a bijection between the sets L and χA(L). Therefore, χA is indeed an iso-morphism between the partial orders 〈A,⊂〉 and 〈A,⊂〉, and so the spaces X(A) and X(A) arehomeomorphic. Note, however, that the sets χA(A) from A are not sets of ordinals anymore.This is the price we have to pay for transforming the ordinal family A into the coherent familyA.

Next, if A0 6= A1 are two families of sets of ordinals then we let

∆(A0,A1) = minδ : A0 δ 6= A1 δ.

Since Ai∗ ξ = ∪Ai η : η ≤ ξ, we clearly have

∆(A0,A1) = ∆(A0∗,A1

∗),

provided that A0∗ 6= A1

∗ holds as well.The following lemma is a strengthening of [9, lemma 2.9]. It gives us a condition under which

the transforms A and B of two families A and B, respectively turn out to be coherent.

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Lemma 1.3.18 ([14]).Assume that L and M are two sets of ordinals of limit order type suchthat L ∩M is a proper initial segment of both L and M , moreover L ⊂ A ⊂ P(L) and M ⊂B ⊂ P(M) are ∩-closed families. If ∆(A,B) = δ + 1 is a successor ordinal then the families Aand B are coherent.

We can prove the result that was formulated at the beginning of this section.

Proof of 1.3.5. First we show that it suffices to prove the theorem in the case when α is alsoindecomposable. Indeed, let α < µ+ be arbitrary with cf(α) = µ. We may then write α = α′+α′′

where α′′ is indecomposable and, of course, cf(α) = cf(α′′) = µ. We may apply lemma 1.3.20 tothe sequences 〈λ〉α′ ∈ C(α′) and 〈λ〉α′′ _ 〈κ〉µ+ ∈ C(µ+) to conclude that

〈λ〉α_ 〈κ〉µ+ = 〈λ〉α′

_ 〈λ〉α′′_ 〈κ〉µ+ ∈ C(µ+).

So assume now that α is indecomposable and let ρ = ωα. Clearly then ρ is also indecompos-able, cf ρ = cf(α) = µ, moreover ht(ρ) = α and ht(ξ) < α for all ξ < ρ. Let 〈νζ : ζ < µ〉 be astrictly increasing sequence of limit ordinals cofinal in ρ. As ρ is indecomposable and cf(ρ) = µ,for every set a ∈ [µ]µ we have

∑νζ : ζ ∈ a = ρ.

Next we fix a disjoint family Kt : t ∈ <µλ of intervals of ordinals such that for any t, s ∈ <µλwe have(a) tp(Kt) = νdom t,(b) if s is a proper initial segment of t then supKs < minKt.We also choose a family of functions G ⊂ µλ with |G| = κ and for every g ∈ G put

Lg =⋃Kgξ : ξ ∈ µ.

Then we have(I) tpLg = ρ for each g ∈ G,(II) Lg ∩ Lh is a proper initial segment of both Lg and Lh whenever g, h ∈

[G]2.

In the proof of the main result of [9], namely Theorem 2.19, we constructed, for any fixedcardinal µ and for all ordinals γ < µ+, ordinal families Fγ such that the following five conditionswere satisfied:(i) µ ∈ Fγ ⊂

[µ]µ,

(ii) Fγ is well-founded and ∩-closed,(iii) htX(Fγ) = γ + 1,(iv) ∆(Fγ ,Fδ) = ∆(F∗γ ,F∗δ ) is a successor ordinal if γ 6= δ,(v)

∣∣Fγ ξ∣∣ ≤ |ξ|+ ω < µ for each ξ < µ.We shall also make use of these families Fγ : γ < µ+, more precisely some transformed

versions of them, in the present proof. To this end, we first fix a function Γ : G −→ µ+ such that|Γ−1γ| = κ for each γ ∈ µ+. This is possible because |G| = κ ≥ µ+.

Fix g ∈ G and for all F ⊂ µ put

ϕg(F ) = ∪Kgξ : ξ ∈ F;then we define

Hg = ϕg(F ) : F ∈ FΓ(g).Note that, by (i), for each g ∈ G we have Hg ⊂ (Lg)ρ.

It is obvious that the map ϕg induces an inclusion-preserving isomorphism between the fam-ilies FΓ(g) and Hg (i. e. between the partial orders

⟨FΓ(g),⊂

⟩and 〈Hg,⊂〉), consequently the

spaces X(FΓ(g)) and X(Hg) are homeomorphic. It is also easy to check that, for each g ∈ G, theordinal family Hg satisfies all the requirements of lemma 1.3.15, or actually of its more generalversion formulated in the remark made afterwards. In view of this and property (iii), we maysum up the relevant properties of the spaces X(Hg∗) as follows.

Fact 1.3.18.1.X(Hg) is a closed subspace of X(Hg∗) and we have(a) ht(X(Hg∗)) = α+ Γ(g) + 1,(b) Iα+β(Hg∗) = Iβ(Hg) for all β < ht(X(Hg)) = Γ(g) + 1,

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(c) I<α(Hg∗) = Hg∗ \ Hg.Our aim is to amalgamate the spaces X(Hg∗) : g ∈ G but to do that we shall have to

transform the families Hg∗ by means of the operation "hat" described in lemma 1.3.18.Since µ ∈ FΓ(g), we have Lg ⊂ Hg∗ ⊂ P(Lg) for each g ∈ G. Also, if g, h ∈

[G]2 then

Lg ∩ Lh is a proper initial segment of both Lg and Lh by (II). Consequently, by lemma 1.3.18,the system H∗g : g ∈ G will be proven to be coherent once the following claim is established.

Claim 1.3.18.1.∆(Hg∗,Hh∗) is a successor ordinal for each g, h ∈[G]2.

Proof of the claim. Let ξ < µ be minimal such that g ξ 6= h ξ, then we clearly have

η = min(Lg 4 Lh) = min(Kgξ ∪Khξ).

(As usual, we are using 4 to denote symmetric difference.) Now, if FΓ(g) ξ = FΓ(h) ξ (thishappens for instance if Γ(g) = Γ(h)) then Hg η = Hh η, and so we also have H∗g η = H∗h η.On the other hand, Kgξ ∩Khξ = ∅ implies that

(Lg ∩ Lh) ∪ η ∈ (H∗g η + 1)4 (H∗h η + 1),

hence ∆(Hg∗,Hh∗) = η + 1, and we are done.Thus we can assume that FΓ(g) ξ 6= FΓ(h) ξ. In this case, by property (iv), we know that

∆(FΓ(g),FΓ(h)) ≤ ξ is a successor ordinal, say δ + 1. But then there is a set A ∈ FΓ(g) δ =FΓ(h) δ such that

A ∪ δ ∈ (FΓ(g) δ + 1)4 (FΓ(h) δ + 1).

Write σ = minKgδ(= minKhδ) and put

D = ∪Kgζ : ζ ∈ A ∪ σ,

then clearlyD ∈ (Hg∗ σ + 1)4 (Hh∗ σ + 1).

On the other hand, FΓ(g) δ = FΓ(h) δ and g δ = h δ together imply Hg σ = Hh σ andso Hg∗ σ = Hh∗ σ. Thus we have ∆(Hg∗,Hh∗) = σ + 1, completing the proof.

Consequently, we may now apply lemma 1.3.18 to conclude that the family H∗g : g ∈ Gis coherent. Therefore, by [9, Lemma 2.7], the family H =

⋃H∗g : g ∈ G is well-founded and

∩-closed, and so the amalgamation X(H) is an LCS∗ space that is covered by its open subspacesX(H∗g) : g ∈ G. As a consequence of this we have

Iβ(H) =⋃Iβ(H∗g) : g ∈ G (‡)

for any ordinal β. Since X(H∗g) is homeomorphic to X(H∗g) we obviously have from fact 1.3.18.1that ht(X(H)) = µ+.

Our aim now is to determine the sizes of the levels Iβ(H) of the LCS∗ space X(H). Tosimplify notation, for each g ∈ G we shall denote the maps kH∗g and χH∗g , both defined in 1.3.17,by kg and χg, respectively.

Claim 1.3.18.2. | Iβ(H)| = κ whenever α ≤ β < µ+.

Proof. Let γ be the ordinal such that α + γ = β. Then for every g ∈ G with Γ(g) ≥ γ we canapply lemma 1.3.15 to the family Hg to conclude that Iβ(Hg∗) = Iγ(Hg) 6= ∅.

On the other hand, for any G ∈ Hg we have G ∈ (Lg)ρ and so G is cofinal in Lg, whileLg ∩Lh is bounded in both Lg and in Lh whenever g, h ∈

[G]2. So if G ∈ Hg and H ∈ Hh are

arbitrary thenχg(G) ∩ χh(H) ⊂ χg(Lg ∩ Lh) = k′′g (Lg ∩ Lh)

implies χg(G) 6= χh(H). In other words, the families

χg(G) : G ∈ Hg = I≥α(H∗g)

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are pairwise disjoint as g ranges over G. Since we have Iβ(H) ⊃ Iβ(H∗g) for g ∈ G by (‡), weconclude that

| Iβ(H)| ≥ |g ∈ G : Γ(g) ≥ γ| = κ.

But by |H| = κ we must have equality here.

Next we show, in a single step, that | Iβ(H)| ≤ λ for each β < α.

Claim 1.3.18.3. | I<α(H)| ≤ λ.

Proof of 1.3.18.3. To each S ∈ I<α(H) we may assign a quadruple F (S) as follows. First pickg ∈ G with S ∈ I<α(H∗g). Then we have S = χg(T ) for some T ∈ I<α(H∗g). By (1.3.15)(c) this setT must be bounded in Lg, so we can fix ξ < µ such that T ⊂

⋃Kgζ : ζ < ξ. Let us put then

F (S) =⟨ξ, g ξ, FΓ(g) ξ, T

⟩.

We shall now show that F is injective, i.e. S can be recovered from the assigned quadruple F (S).Indeed, both the sequence 〈Kgζ : ζ < ξ〉 and the ordinal η = minKgξ are obviously deter-

mined by the map g ξ. Next, the family H∗g η is determined by the sequence 〈Kgζ : ζ < ξ〉and the family FΓ(g) ξ because we clearly have

H∗g η =∪Kgζ : ζ ∈ A : A ∈ FΓ(g) ξ

∗.

It is easy to see that the family H∗g η determines the map kg η and consequently χg (H∗g η)as well. But S = χg(T ) where we have T ∈ H∗g η, and so we are done.

Therefore, to conclude, it suffices to prove that there are at most λ many quadruples of theform F (S). To see this, first note that we have µ many choices for ξ. Next, since λ<µ = λ wehave λ many choices for g ξ. By (v) we have

FΓ(g) ξ ∈[P(ξ)

]≤|ξ+ω|,

consequently there are at most 2|ξ+ω| ≤ λ|ξ+ω| = λ many choices for FΓ(g) ξ. Finally, it is easyto see that

|H∗g η| ≤ | ∪ Kgζ : ζ < ξ| · |FΓ(g) ξ| ≤ µ,hence, for fixed ξ, g ξ, and FΓ(g) ξ, there are at most µ many choices for T . All this togetherclearly gives us that

|F (S) : S ∈ I<α(H)| = | I<α(H)| ≤ λ.

We are now almost finished with the proof of theorem 1.3.5: the LCS∗ space X = X(H)satisfies | Iβ(X)| = κ for all α ≤ β < µ+ and | Iβ(X)| ≤ λ for all β < α. Thus if Y is the disjointtopological sum of λ many copies of X then Y is an LCS∗ space with

CS(Y ) = 〈λ〉α_ 〈κ〉µ+ .

Proof of Theorem 1.3.4. It follows immediately from 1.3.7 and GCH that

Cλ(α) ⊂ αλ, λ+.The first case, λ = ω, follows immediately from [61, Theorem 9] that actually implies

αω, ω1 ⊂ C(α)

in ZFC.Now consider the second case: λ > cf(λ) = ω. Let 〈λn : n < ω〉 be an increasing sequence

of cardinals cofinal in λ.Necessity: Assume s ∈ Cλ(α) and fix an LCS∗ space X with cardinal sequence s. Suppose

β < α, cf(β) = ω1, Aλ(s) ∩ β is cofinal in β. We have to show that β ∈ Aλ(s). Let βη : η <ω1 ⊂ Aλ(s) be an increasing sequence cofinal in β. For each x ∈ Iβ(X) let Ux be a compactopen neighbourhood of x such that

Ux \ x ⊂ ∪Iξ(X) : ξ < β.

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For each η < ω1 pick a point p(x, η) ∈ Ux∩Iβη (X). Then the sequence 〈p(x, η) : η < ω1〉 convergesto x. Now since | Iβη (X)| = s(βη) = λ for all η < ω1 we have that the set

S = ∪Iβη (X) : η < ω1

has size λ. Let S = ∪Sn : n < ω where |Sn| = λn for each n < ω . For each x ∈ Iβ(X) theremust be some n < ω such that

Sn ∩ p(x, η) : η < ω1is uncountable and so x ∈ Sn. However, according to the GCH, we have |Sn| ≤ λ+

n < λ for eachn < ω, consequently

s(β) = | Iβ(X)| ≤ | ∪ Sn : n ∈ ω| ≤ supn<ω

λ+n = λ,

i.e. β ∈ Aλ(s). This completes the necessary part of the second case.Sufficiency: We first handle some specific sequences s ∈ αλ, λ+. In particular, as was noted

in the introduction, the constant sequence 〈λ〉α is a member of Cλ(α).

Claim 1.3.18.4. If 0 < β, γ < ω2 and cf(β) < ω1 then

〈λ〉β_⟨λ+⟩γ∈ Cλ(β + γ).

Proof of the claim. First we do the case γ = 1, that is we construct an LCS∗ space Z withCS(Z) = 〈λ〉β _ 〈λ+〉1.

If cf(β) = ω let 〈βn : n < ω〉 be an increasing sequence converging to β. If β = ρ + 1 letβn = ρ for each n < ω. For each n < ω and each ordinal µ with λn ≤ µ < λn+1 let Zµ be a copyof the one point compactification of an LCS∗ space of height βn such that CS(Zµ) = 〈λ〉βn andZµ : µ < λ is disjoint. Let aη : η < λ+ be a collection of almost disjoint subsets of λ suchthat |aη ∩ (λn+1 \ λn)| = 1 for all η < λ+ and n < ω. (The existence of such an almost disjointfamily of size λ+ is well-known.) pη : η < λ+ be a set of new points and set

Z = pη : η < λ+ ∪⋃Zµ : µ < λ.

Let τ be the topology on Z generated by all sets which are open in any Zµ along with allsets of the form

pη ∪⋃Zµ : µ ∈ aη and µ > λm

for η < λ+ and m < ω. It is straightforward to show that 〈Z, τ〉 is an LCS∗ space of height β+ 1with CS(Z, τ) = 〈λ〉β _ 〈λ+〉1.

Now, if γ > 1, then we extend this space Z using lemma 1.3.10 with the choices S = pη : η <λ+, Upη,n = Zζ where ζ = aη∩[λn, λn+1), and each Ypη as an LCS∗ space of height γ satisfyingCS(Ypη ) = 〈ω〉γ , i. e. | Iξ(Ypη )| = ω for all ξ < γ. Note that for each η < λ+ we have δpη = β

here, consequently we thus obtain an LCS∗ space with cardinal sequence 〈λ〉β _ 〈λ+〉γ .

Now let s ∈ αλ, λ+ be an arbitrary sequence such that s(0) = λ and Aλ(s) is ω1-closed inα. Our aim is to construct an LCS∗ space X with s as its cardinal sequence.

Let C = α \Aλ(s) = γ < α : s(γ) = λ+. For each γ ∈ C let βγ = minβ ∈ C : [β, γ] ⊂ C.By the choice of s we have βγ > 0 and cf(βγ) < ω1.

By claim 1.3.18.4 above, for each γ ∈ C there is an LCS∗ space Xγ with CS(Xγ) = 〈λ〉βγ_

〈λ+〉(γ+1)

.−βγ

.The final X that we require is simply the disjoint topological sum of Xγ : γ ∈ C ∪ Y ,

where CS(Y ) = 〈λ〉α.Now consider the third case: cf(λ) = ω1.Necessity: Let s ∈ Cλ(α). By 1.3.8 and the GCH we see that if β + 1 < α and s(β) = λ then

s(β + 1) ≤ s(β)ℵ0 = λ as well, i. e. Aλ(s) is successor closed in α. Similarly, 1.3.9 and the GCHtogether guarantee that if β < α, cf(β) = ω, and Aλ(s) ∩ β is cofinal in β then s(β) ≤ λω = λ,that is Aλ(s) is indeed ω-closed in α. So we have completed the necessary part of this case.

The proof of sufficiency makes essential use of the following proposition that is an immediatecorollary to theorem 1.3.5 .

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Proposition 1.3.19. Let λ be a cardinal with cf λ = ω1 and satisfying λω = λ. Then for everyordinal γ < ω2 with cf γ = ω1 we have

〈λ〉γ_⟨λ+⟩ω2∈ Cλ(ω2).

Now let s ∈ αλ, λ+ be such that s(0) = λ and Aλ(s) is both ω-closed and successor closedin α. Let B = α \ Aλ(s) = β < α : s(β) = λ+. For β ∈ B let γβ = minγ ∈ B : [γ, β] ⊂ B.Since s(0) = λ and Aλ(s) is both ω-closed and successor closed, we have

cf(γβ) = ω1 = cf(λ)

for each β ∈ B.Thus, using proposition 1.3.19, we may fix for each β ∈ B an LCS∗ space Xβ with CS(Xβ) =

〈λ〉γβ_ 〈λ+〉νβ , where νβ is chosen so as to satisfy β + 1 = γβ + νβ . Now, let X be the disjoint

topological sum of the family of spaces

Y ∪ Xβ : β ∈ B ,

where Y is an LCS∗ space with CS(Y ) = 〈λ〉α. Since |B| ≤ ω1 ≤ λ, it is clear that CS(X) = s.Finally, consider the case when cf(λ) > ω1. For any s ∈ Cλ(α) we may then use facts 1.3.8

and 1.3.9 along with the GCH to inductively show that s(ξ) = λ for all ξ < α, hence s = 〈λ〉α.Since 〈λ〉α ∈ Cλ(α) is clear, we are done.

1.3.1. The reduction theorem.

Proof of Theorem 1.3.2. We first prove that the condition is necessary, so fix f ∈ C(α). Letus say that β < α is a drop point in f if for all γ < β we have f(γ) > f(β). Clearly f has onlyfinitely many, say n, drop points; let βi : i < n enumerate all of them in increasing order. (Inparticular, we have then β0 = 0.) For each i < n let us set λi = f(βi), then we clearly haveλ0 > λ1 > · · · > λn−1.

For each i < n−1 let αi = βi+1−βi be the unique ordinal such that βi+αi = βi+1, moreoverdefine the sequence fi on αi by the stipulation

fi(ξ) = f(βi + ξ)

for all ξ < αi. Similarly, let αn−1 = α− βn−1 be the unique ordinal such that βn−1 + αn−1 = α,and define fn−1 on αn−1 by the stipulation

fn−1(ξ) = f(βn−1 + ξ)

for all ξ < αn−1. Now, it is obvious that we have f = f0_ f1

_ · · · _ fn−1 where fi ∈ Cλi(αi)for each i < n.

We shall now prove that the condition is also sufficient. In fact, this will follow from the nextlemma.

Lemma 1.3.20. If f ∈ C(α), g ∈ C(β), and f(ν) ≥ g(0) for each ν < α then f _ g ∈ C(α+ β).

Indeed, given the sequences fi for i < n, this lemma enables us to inductively define for everyi = 0, . . . , n− 1 an LCS∗ spaces Zi with cardinal sequence f0

_ · · · _ fi because

fi(0) = λi < λi−1 = minfj(ν) : j < i, ν < αj.

In particular, then we have CS(Zn−1) = f0_ f1

_ · · · _ fn−1. 1.3.2

Proof of lemma 1.3.20. Let Y be an LCS∗ space with cardinal sequence g and satisfyingIβ(Y ) = ∅. Next fix LCS∗ spaces Xy for all y ∈ I0(Y ), each having the cardinal sequence fand satisfying Iα(Xy) = ∅. Assume also that the family Y

⋃Xy : y ∈ Y is disjoint.

We then define the space Z = 〈Z, τ〉 as follows. Let us first set

Z = Y ∪⋃Xy : y ∈ I0(Y ).

For any subset V ⊂ Y we let

Z(V ) = V ∪⋃Xy : y ∈ I0(Y ) ∩ V ,

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moreover we putT = W : W is compact open in some Xy.

Then the family

B = T ∪ Z(V ) \ ∪T ′ : V is compact open in Y and T ′ ∈[T]<ω

clearly covers Z and is closed under finite intersections, hence it forms a base for the topology τon Z that it generates .

Since T ⊂ τ and B ∩Xy is open in Xy for all B ∈ B, each Xy is an open subspace of Z. Wealso have

B ∩ Y : B ∈ B = V ⊂ Y : V is compact open in Y ,and the latter is a base of Y , hence Y is a closed subspace of Z. It easily follows then that anynon-empty subspace A ⊂ Z has an isolated point, hence Z is scattered. It is also easy to checkthat Z is Hausdorff because so are Y and all the Xy.

So to see that Z is LCS∗, it remains to check that it is locally compact. Now, let V becompact open in Y , we claim that then Z(V ) is compact in Z. Indeed, this can be proved by astraight-forward transfinite induction on

σ(V ) = maxht(z, Y ) : z ∈ V .

It clearly follows from this that all members of B are compact in Z, hence Z is locally compact.Note that for any isolated point y of Y we have that

Z(y) = y ∪Xy,

as a subspace of Z, is the one-point compactification of Xy. This clearly implies that ht(y, Z) = α.From this, with an easy transfinite induction, one can prove that for all points z ∈ Y we have

ht(z, Z) = α+ ht(z, Y ).

On the other hand, since each Xy is an open subspace of Z, it follows that for every pointx ∈ Xy we have

ht(x, Z) = ht(x,Xy) < α.

Consequently, for each ν < α we have

Iν(Z) =⋃Iν(Xy) : y ∈ I0(Y ).

This implies that ht(Z) = α+ β, and if ν < α then

| Iν(Z)| = | I0(Y )| · f(ν) = g(0) · f(ν) = f(ν).

Moreover, if η < β then | Iα+η(Z)| = | Iη(Y )| = g(η), consequently CS(Z) = f _ g. 1.3.20

1.4. Cardinal sequences of length ≥ ω2 under GCH


Having given a full characterization, under GCH, of the classes C(α) for all α < ω2, it is nownatural to raise the following question: Can this GCH characterization be extended to longersequences, i. e. to C(ω2) and beyond? This question, however, remains open. In fact, we do noteven know if GCH implies that the constant sequence 〈ω1〉ω2

belongs to C(ω2), although this isknown to be consistent with GCH.

A partial answer was given in [15].

Definition 1.4.1. For a cardinal λ and ordinal δ < λ++ we define Dλ(δ) as follows: if λ = ω,

Dω(δ) = f ∈ δω, ω1 : f(0) = ω,

and if λ is uncountable,

Dλ(δ) = s ∈ δλ, λ+ : s(0) = λ, s−1λ is < λ-closed and successor-closed in δ.

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1.4. CARDINAL SEQUENCES OF LENGTH ≥ ω2 UNDER GCH 23

In [14, Theorem 4.1] it was proved that if GCH holds then

Cω1(δ) ⊆ Dω1(δ),

and we have equality for δ < ω2. In Theorem 1.4.3 we show that it is consistent with GCH thatwe have equality for each δ < ω3.

To formulate our results we need to introduce some more notation.

Definition 1.4.2.An LCS∗ space X is called Cλ(α)-universal iff CS(X) ∈ Cλ(α) and for eachsequence s ∈ Cλ(α) there is an open subspace Y of X with CS(Y ) = s.

In this section we prove the following result:

Theorem 1.4.3. If κ is an uncountable regular cardinal with κ<κ = κ and 2κ = κ+ then for eachδ < κ++ there is a κ-complete κ+-c.c poset P of cardinality κ+ such that in V P

Cκ(δ) = Dκ(δ)

and there is a Cκ(δ)-universal LCS∗ space.

How do the universal spaces come into the picture? The first idea to prove the consistencyof Cλ(α) = Dλ(α) is to try to carry out an iterated forcing. For each f ∈ Dλ(α) we can try tofind a poset Pf such that

1Pf There is an LCS∗ space Xf with cardinal sequence f .

Since typically |Xf | = λ+, if we want to preserve the cardinals and CGH we should try to finda λ-complete, λ+-c.c. poset Pf of cardinality λ+. In this case forcing with Pf introduces λ+

new subsets of λ because Pf has cardinality λ+. However |Dλ(α)| = λ++! So the length of theiteration is at least λ++, hence in the final model the cardinal λ will have λ+ · λ++ = λ++ manynew subsets, i.e. 2λ > λ+.

A Cλ(δ)-universal space has cardinality λ+ so we may hope that there is a λ-complete, λ+-c.c. poset P of cardinality λ+ such that V P contains a Cλ(δ)-universal space. In this case(2λ)V

P ≤ ((|P |λ)λ)V = λ+. So in the generic extension we might have GCH.Theorem 1.4.3 yields the following characterization:

Theorem 1.4.4.Under GCH for any sequence f of regular cardinals of length α the followingstatements are equivalent:(A) f ∈ C(α) in some cardinal preserving and GCH-preserving generic-extension of the ground

model.(B) for some natural number n there are infinite regular cardinals λ0 > λ1 > · · · > λn−1 and

ordinals α0, . . . , αn−1 such that α = α0 + · · ·+αn−1 and f = f0_ f1

_· · · _fn−1 where eachfi ∈ Dλi(αi).

Proof. (A) clearly implies (B) by Theorem 1.3.2.Assume now that (B) holds. Without loss of generality, we may suppose that λn−1 = ω.

Since the notion of forcing defined in Theorem 1.4.3 preserves GCH, we can carry out a cardinal-preserving and GCH-preserving iterated forcing of length n− 1, 〈Pm : m < n− 1〉, such that form < n− 1

V Pm |= Cλm(αm) = Dλm(αm).Put k = n − 2, β = α0 + · · · + αk and g = f0

_f1_· · · _fk. Since fm ∈ Dλm(αm) ∩ V , in V Pk

we have fm ∈ Cλm(αm) for each m < n− 1. Hence in V Pk we have g ∈ C(β) by [14, Lemma 2.2].Also, by using [61, Theorem 9], we infer that fn−1 ∈ C(αn−1) in ZFC. Then as f = g_fn−1, inV Pk we have f ∈ C(α) again by [14, Lemma 2.2]. 1.4.4

For an ordinal δ < κ++ let Lδκ = α < δ : cf(α) ∈ κ, κ+.Definition 1.4.5.An LCS∗ space X is called Lδκ-good iff X has a partition X = Y ∪∗

⋃∗Yζ :ζ ∈ Lδκ such that(1) Y is an open subspace of X, CS(Y ) = 〈κ〉δ,(2) Y ∪ Yζ is an open subspace of X with CS(Y ∪ Yζ) = 〈κ〉ζ _〈κ+〉δ−ζ .

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Clearly Theorem 1.4.3 follows immediately from Theorem 1.4.6 and Proposition 1.4.7 below.

Theorem 1.4.6. If κ is an uncountable regular cardinal with κ<κ = κ then for each δ < κ++

there is a κ-complete κ+-c.c poset P of cardinality κ+ such that in V P there is an Lδκ-good space.

Proposition 1.4.7. Let κ be an uncountable regular cardinal, δ < κ++ and X be an Lδκ-goodspace. Then for each s ∈ Dκ(δ) there is an open subspace Z of X with CS(Z) = s. Especially,under GCH an Lδκ-good space is Cκ(δ)-universal.

Proof of Proposition 1.4.7 . Let J = s−1κ+ ∩ Lδκ. For each ζ ∈ J let

f(ζ) = min((δ + 1) \ (s−1κ+ ∪ ζ)).

LetZ = Y ∪

⋃I<f(ζ)(; )Y ∪ Yζ : ζ ∈ J.

Since Y ∪ Yζ is an open subspace of X it follows that I<f(ζ)(Y ∪ Yζ) is an open subspace of Z.Hence for every α < δ

Iα(Z) = Iα(Y ) ∪⋃Iα(I<f(ζ)(Y ∪ Yζ)) : ζ ∈ J

= Iα(Y ) ∪⋃Iα(Y ∪ Yζ) : ζ ∈ J, ζ ≤ α < f(ζ). (1.9)

Since [ζ, f(ζ)) ⊂ s−1κ+ for ζ ∈ J it follows that if s(α) = κ then Iα(Z) = Iα(Y ), and so

| Iα(Z)| = | Iα(Y )| = κ. (1.10)

If s(α) = κ+, let ζα = minζ ≤ α : [ζ, α] ⊂ s−1κ+. Then ζα ∈ J because s(0) = κ and s−1κis < κ-closed and successor-closed in δ. Thus ζα ≤ α < f(ζα) and so

| Iα(Z)| ≥ | Iα(Y ∪ Yζα)| = κ+. (1.11)

Since |Z| ≤ |X| = κ+ we have | Iα(Z)| = κ+. Thus CS(Z) = s. 1.4.7

The rest of this section is devoted to the proof of Theorem 1.4.6, so κ is an uncountableregular cardinal with κ<κ = κ, and δ < κ++ is an ordinal.

If α ≤ β are ordinals let[α, β) = γ : α ≤ γ < β. (1.12)

We say that I is an ordinal interval iff there are ordinals α and β with I = [α, β). Write I− = αand I+ = β.

If I = [α, β) is an ordinal interval let E(I) = εIν : ν < cf(β) be a cofinal closed subset of Ihaving order type cf β with α = εI0 and put

E(I) = [εIν , εIν+1) : ν < cf β (1.13)

provided β is a limit ordinal, and let E(I) = α, β′ and put

E(I) = [α, β′), β′ (1.14)

provided β = β′ + 1.Define In : n < ω as follows:

I0 = [0, δ) and In+1 =⋃E(I) : I ∈ In. (1.15)

Put I =⋃In : n < ω. Note that I is a cofinal tree of intervals in the sense defined in [84].

Then, for each α < δ we define

n(α) = minn : ∃I ∈ In with I− = α, (1.16)

and for each α < δ and n < ω we define

I(α, n) ∈ In such that α ∈ I(α, n). (1.17)

Proposition 1.4.8.Assume that ζ < δ is a limit ordinal. Then, there is a j(ζ) ∈ ω and aninterval J(ζ) ∈ Ij(ζ) such that ζ is a limit point of E(J(ζ)). Also, we have n(ζ)−1 ≤ j(ζ) ≤ n(ζ),and j(ζ) = n(ζ) if cf(ζ) = κ+.

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1.4. CARDINAL SEQUENCES OF LENGTH ≥ ω2 UNDER GCH 25

Proof. Clearly j(ζ) and J(ζ) are unique if defined.If there is an I ∈ In(ζ) with I+ = ζ then J(ζ) = I, and so j(ζ) = n(ζ). If there is no such I,

then ζ is a limit point of E(I(ζ,n(ζ)− 1)), so J(ζ) = I(ζ,n(ζ)− 1) and j(ζ) = n(ζ)− 1.Assume now that cf(ζ) = κ+. Then ζ ∈ E(I(ζ,n(ζ)− 1)), but |E(I(ζ,n(ζ)− 1))∩ζ| ≤ κ, so ζ

can not be a limit point of E(I(ζ,n(ζ)− 1)). Therefore, it has a predecessor ξ in E(I(ζ,n(ζ)− 1)),i.e [ξ, ζ) ∈ In(ζ), and so J(ζ) = [ξ, ζ) and j(ζ) = n(ζ).

Example 1.4.9. Put δ = ω2 · ω2 + 1. We defineE([0, δ)) = 0, ω2 · ω2,E([0, ω2 · ω2)) = ω2 · ξ : 0 ≤ ξ < ω2,E([ω2 · ξ, ω2 · (ξ + 1))) = ζ : ω2 · ξ ≤ ζ < ω2 · (ξ + 1),E(ζ) = ζ for each ζ ≤ ω2 · ω2.Then, we have n(ω2 · ω2) = 1, n(ω2 · ω1) = 2, n(ω2 · ω1 + ω) = 3. Also, we have j(ω2 · ω2) =

j(ω2 · ω1) = 1 and J(ω2 · ω2) = J(ω2 · ω1) = [0, ω2 · ω2).

If cf(J(ζ)+) ∈ κ, κ+, we denote by εζν : ν < cf(J(ζ)+) the increasing enumeration ofE(J(ζ)), i.e. εζν = ε

J(ζ)ν for ν < cf(J(ζ)+).

Now if ζ < δ, we define the basic orbit of ζ (with respect to I) as

o(ζ) =⋃(E(I(ζ,m)) ∩ ζ) : m < n(ζ). (1.18)

Note that this is the notion of orbit used in [84] in order to construct by forcing an LCS∗space X such that CS(X) = 〈κ〉η for any specific regular cardinal κ and any ordinal η < κ++.However, this notion of orbit can not be used to construct an LCS∗ space X such that CS(X) =〈κ〉κ+

_〈κ+〉. To check this point, assume on the contrary that such a space X can be constructedby forcing from the notion of a basic orbit. Then, since the basic orbit of κ+ is 0, we havethat if x, y are any two different elements of Iκ+(X) and U, V are basic neighbourhoods of x, yrespectively, then U ∩ V ⊂ I0(X). But then, we deduce that |I1(X)| = κ+.

However, we will show that a refinement of the notion of basic orbit can be used to proofTheorem 1.6.

If ζ < δ with cf ζ ≥ κ, we define the extended orbit of ζ by

o(ζ) = o(ζ) ∪ (E(J(ζ)) ∩ ζ). (1.19)

Consider the tree of intervals defined in Example-2.2. Then, we have o(ω2 ·ω1) = o(ω2 ·ω1) =ω2 · ξ : 0 ≤ ξ < ω1, o(ω2 · ω2) = 0, o(ω2 · ω2) = ω2 · ξ : 0 ≤ ξ < ω2.

Note that if ζ < δ, the basic orbit of ζ is a set of cardinality at most κ (see [84, Proposition1.3]). Then, it is easy to see that for any ζ < δ with cf ζ ≥ κ , the extended orbit of ζ is a cofinalsubset of ζ of cardinality cf ζ.

In order to define the desired notion of forcing, we need some preparations. The underlyingset of the desired space will be the union of a collection of blocks.

LetB = S ∪ 〈ζ, η〉 : ζ < δ, cf ζ ∈ κ, κ+, η < κ+. (1.20)

LetBS = δ × κ (1.21)

andBζ,η = 〈ζ, η〉 × [ζ, δ)× κ (1.22)

for 〈ζ, η〉 ∈ B \ S.Let

X =⋃BT : T ∈ B. (1.23)

The underlying set of our space will be X. We should produce a partition X = Y ∪∗⋃∗Yζ :

ζ ∈ Lδκ such that(1) Y is an open subspace of X with CS(Y ) = 〈κ〉δ ,

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(2) Y ∪ Yζ is an open subspace of X with CS(Y ∪ Yζ) = 〈κ〉ζ _〈κ+〉δ−ζ .We will have Y = BS , Yζ =

⋃Bζ,η : η < κ+ for ζ ∈ Lδκ.

Let

π : X −→ δ such that π(〈α, ν〉) = α,π(〈ζ, η, α, ν〉) = α.

(1.24)

Let

π− : X −→ δ such that π−(〈α, ν〉) = α,π−(〈ζ, η, α, ν〉) = ζ.

(1.25)

DefineπB : X −→ B by the formula x ∈ BπB(x). (1.26)

Define the block orbit function oB : B \ S −→[δ]≤κ as follows:

oB(〈ζ, η〉) =

o(ζ) if cf ζ = κ,o(ζ) ∪ εζν : ν < η if cf ζ = κ+.

(1.27)

That is, if cf ζ = κ+ thenoB(〈ζ, η〉) = o(ζ) ∩ εζη.

Finally we define the orbits of the elements of X as follows:

o* : X −→[δ]≤κ such that o*(〈α, ν〉) = o(α),

o*(〈ζ, η, α, ν〉) = oB(〈ζ, η〉) ∪ (o(α) \ ζ).(1.28)

Let Λ ∈ I and x, y ∈[X]2. We say that Λ isolates x from y if

(i) Λ− < π(x) < Λ+,(ii) Λ+ ≤ π(y) provided πB(x) = πB(y),(iii) Λ+ ≤ π−(y) provided πB(x) 6= πB(y).

Now, we define the poset P = 〈P,≤〉 as follows: 〈A,, i〉 ∈ P iff

(P1) A ∈[X]<κ.

(P2) is a partial order on A such that x y implies x = y or π(x) < π(y).(P3) Let x y.

(a) If πB(y) = 〈ζ, η〉 and ζ ≤ π(x) then πB(x) = πB(y).(b) If πB(y) = 〈ζ, η〉 and ζ > π(x) then πB(x) = S.(c) If πB(y) = S then πB(x) = S.

(P4) i :[A]2 −→ A ∪ undef such that for each x, y ∈

[A]2 we have

∀a ∈ A([a x ∧ a y] iff a ix, y).

(P5) ∀x, y ∈[A]2 if x and y are -incomparable but -compatible, then π(ix, y) ∈ o*(x) ∩

o*(y).(P6) Let x, y ∈ [A]2 with x y. Then:

(a) If πB(x) = S and Λ ∈ I isolates x from y, then there is z ∈ A such that x z y andπ(z) = Λ+.

(b) If πB(x) 6= S, π(x) 6= π−(x) and Λ ∈ I isolates x from y, then there is z ∈ A such thatx z y and π(z) = Λ+.

The ordering on P is the extension: 〈A,, i〉 ≤⟨A′,′, i′

⟩iff A′ ⊂ A, ′= ∩(A′ ×A′), and

i′ ⊂ i.

By using (P3), we obtain:

Claim 1.4.10. Assume that x, y, z and Λ are as in (P6). Then we have:(a) If πB(x) = πB(y), then πB(z) = πB(x) = πB(y).(b) If πB(x) 6= πB(y) and Λ+ < π−(y), then πB(z) = πB(x).(c) If πB(x) 6= πB(y) and Λ+ = π−(y), then πB(z) = πB(y).

Since κ<κ = κ implies (κ+)<κ = κ+, we have that the cardinality of P is κ+. Then, usingthe arguments of [84] it is enough to prove that Lemmas 1.4.11, 1.4.12 and 1.4.13 below hold.

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1.5. UNIVERSAL SPACES 27

Lemma 1.4.11 ([15]).P is κ-complete.

Lemma 1.4.12 ([15]).P satisfies the κ+-c.c.

Lemma 1.4.13 ([15]).Assume that p = 〈A,, i〉 ∈ P , x ∈ A, and α < π(x). Then there isp′ =

⟨A′,′, i′

⟩∈ P with p′ ≤ p and there is b ∈ A′ \ A with π(b) = α such that b ′ y iff x y

for y ∈ A.

The statement of Lemma 1.4.11 clearly holds; and the proof of lemma 1.4.13 is standard.However, the proof of of Lemma 1.4.12 is the real burden of [15]. Basically it is a very

complicated amalgamation argument which is 14 page long. The detailed proof can be found in[15].

1.5. Universal spaces


The notion of universal spaces was introduced in Definition 1.4.2. We conjecture that thereare always universal spaces, but we can prove just the following special cases:

Theorem 1.5.1 ([16]).(1) There is a Cω(ω1)-universal LCS∗ space.(2) (GCH) For every infinite cardinal λ and every ordinal δ < ω2, there is a Cλ(δ)-universal

LCS∗ space.

The proofs of the above statements are based on a carefull analysis of the construction ofLCS∗ spaces in [61] and in [14]. The details can be found in [16].

What about Cω(ω2)? Baumgartner and Shelah introduced the notion of ∆-functions in [25,Section 8]. In that paper they also proved that (a) the existence of a ∆-function is consistentwith ZFC + GCH, (b) if there is a ∆-function then 〈ω〉ω2

∈ Cω(ω2) holds in a “natural” c.c.cforcing extension of the ground model. “Natural” means that the elements of the posets are justfinite approximations of the locally compact right-separating neighbourhoods of the points of thedesired space. Building on their method, Bagaria, [21], proved that

Cω(ω2) ⊇ s ∈ ω2ω, ω1 : s(0) = ω (†)

is also consistent. More precisely, he showed that if there is a ∆-function andMAℵ2 holds (whichis a consistent assumption), then (†) above holds.

However, MAℵ2 implies 2ω0 ≥ ω3, and if 2ω0 = ωα, then the natural “upper bound” of Cω(ω2)is a much larger family of sequences:

Cω(ω2) ⊆ s ∈ ω2ων : ν ≤ α : s(0) = ω. (‡)

These results naturally raised the following questions.

Problem 1.Does 〈ω〉ω2∈ Cω(ω2) imply (†), or even

Cω(ω2) ⊇ s ∈ ω2ω, ω1, ω2 : s(0) = ω. (∗)Although these questions are still open we proved Theorem 1.5.10 claiming that if there is a

“natural” poset P such that 〈ω〉ω2∈ Cω(ω2) holds in V P , then there is a natural poset Q such

that (∗) holds in V Q. Moreover,

Theorem 1.5.2 ([16]).Con(ZFC) −→ Con(ZFC + 2ω = ω2 + there is a Cω(ω2)-universal LCS∗space witnessing that Cω(ω2) is as large as possible, i.e.

Cω(ω2) = s ∈ ω2ω, ω1, ω2 : s(0) = ω.Before proving Theorem 1.5.2 we need some preparation.Let T0 = 0 × ω, Tα = α × ω2 for 1 ≤ α < ω2, and

T =⋃Tα : α < ω2.

Let π : T → ω2 be the natural projection: π(〈α, ξ〉) = α.

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Definition 1.5.3.Define the poset P∗ = 〈P ∗,〉 as follows. The underlying set P ∗ consists oftriples p = 〈ap,≤p, ip〉 satisfying the following requirements:

(1) ap ∈[T]<ω,

(2) ≤p is a partial ordering on ap with the property that if x <p y then x ∈ ω2 × ω andπ(x) < π(y),

(3) ip :[ap]2 → [

ap]<ω is such that

(3.1) if x, y ∈[ap]2 then

(3.1.1) if x, y ∈ ω2 × ω and π(x) = π(y) then ipx, y = ∅,(3.1.2) if x <p y then ipx, y = x.

(3.2) if x, y ∈[ap]2 and z ∈ ap then(

(z ≤p x ∧ z ≤p y) iff ∃t ∈ ix, y z ≤p t).

Set p q iff ap ⊇ aq, ≤p aq =≤q and ip [aq]2 = iq.

Let P ∗ω = p ∈ P ∗ : ap ⊆ ω2 × ω and P∗ω = 〈P ∗ω ,〉.

Consider a function d :[ω2

]2 → [ω2

]≤ω. An element p ∈ P ∗ω is d-good iff

(?d) if x, y ∈[ap]2, π(x) < π(y) and x 6<p y then

π′′ipx, y ⊆ d(π(x), π(y)).

Let P ∗d be the family of d-good elements of P ∗ω and put P∗d = 〈P ∗d ,〉.Observation 1.5.4.Our poset P ∗d is just “the poset P defined from d” in [25, Section 7]. (Stip-ulation (?d) corresponds to (3.1.3), the other requirements have the same numbering here as in[25].)

A condition r ∈ P is an amalgamation of conditions p and q iff r ≺ p and r ≺ q, ar = ap∪aq,and ≤r is the partial ordering on ar generated by ≤p ∪ ≤q. Let Q ⊆ P ∗. The poset Q = 〈Q,≺〉has the amalgamation property iff every uncountable subset of Q contains two elements whichhave an amalgamation in Q.

Clearly the amalgamation property implies the countable chain condition.Baumgartner and Shelah proved, [25, Theorem 8.1], that if d :

[ω2

]2 → [ω2

]≤ω is a ∆-function then P∗d has the countable chain condition. Actually, they proved the following:

Proposition* 1.5.5. If d is a ∆-function then P∗d has the amalgamation property.

For a condition p ∈ P ∗ and x ∈ T \ ap define the condition q = p ] x, as follows. Letaq = ap ∪ x. Put u ≤q t iff u ≤p t or u = t = x. Let iqu, t = ipu, t unless u or t is x. Letiqu, x = ∅. Clearly q = p ] x ∈ P ∗.

Let P ′ ⊆ P ∗. The poset P ′ = 〈P ′,〉 has the density property D (or the density propertyDω) iff p ] x ∈ P for each p ∈ P and for each x ∈ (T ) \ ap (or for each x ∈ (ω2 × ω) \ ap,respectively).

For p ∈ P ∗, y ∈ ap and x ∈ (ω2 × ω) \ ap with π(x) < π(y) define the condition q = p ]y xas follows. Let aq = ap ∪ x. Put u ≤q t iff u ≤p t, or u = x and y ≤p t. Let iqu, t = ipu, tunless u or t, say t, is x. Let iqx, u = x if x ≤q u and iqx, u = ∅ otherwise.

Since x is a minimal element in ≤q we have q = p ]y x ∈ P ∗.

Let P ⊆ P ∗. The poset P = 〈P,≺〉 has the density property E iff p ]x y ∈ P for eachp ∈ P , x ∈ ap and y ∈ (ω2 × ω) \ ap with π(y) < π(x).

The following claim is straightforward from the definition of P∗d .

Proposition 1.5.6. For each function d :[ω2

]2 → [ω2

]≤ω the poset P∗d has the density propertiesDω and E.

Definition 1.5.7. (a) Let Q ⊆ P ∗ω . We say that the poset Q = 〈Q,〉 is a BS-poset iff Q has theamalgamation property, and the density properties Dω and E.

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1.5. UNIVERSAL SPACES 29

(b) Let P ⊆ P ∗. We say that the poset P = 〈P,〉 is a U-poset iff P has the amalgamationproperty, and the density properties D and E.

In [25, Section 9] Baumgartner and Shelah also proved that

Proposition 1.5.8. It is consistent that 2ω ≤ ω2 and there is a ∆-function d.

Putting together Propositions 1.5.5, 1.5.6 and 1.5.8 we obtain

Proposition 1.5.9. It is consistent that 2ω ≤ ω2 and there is a BS-poset Q = 〈Q,〉.As we will see, Theorem 1.5.2 follows almost immediately from the Proposition above and

from the next two theorems.

Theorem 1.5.10. If there is a BS-poset then there is a U-poset as well.

Theorem 1.5.11. If P is a U-poset then in V P there is an LCS space X such that CS(X) =〈ω〉_〈ω2〉ω2

∈ Cω(ω2), and for every s ∈ ω2ω, ω1, ω2 with s(0) = ω there is an open subspaceY ⊆ X with CS(Y ) = s.

Proof of Theorem 1.5.2 from 1.5.9, 1.5.10, and 1.5.11. By Proposition 1.5.9 and Theo-rem 1.5.10 we can assume that in the ground model we have 2ω ≤ ω2 and there is a U-posetP = 〈P,〉. We show that the model V P satisfies the requirements.

Since |P | = ω2, P satisfies c.c.c and 2ω ≤ ω2, we have (2ω)VP ≤ ((|P|ω)ω)V = ω2.

By Theorem 1.5.11, in V P there is an LCS space X such that CS(X) ∈ Cω(ω2) and

CS(Y ) : Y ⊆ X is open ⊇ s ∈ ω2ω, ω1, ω2 : s(0) = ω. (•)

Since |X| ≥ ω2 and | I0(X)| = ω, we have 2ω ≥ ω2 in V P . So 2ω = ω2 in V P .Thus

Cω(ω2) ⊆ s ∈ ω2ω, ω1, ω2 : s(0) = ω. ()

But (•) and () together yield that

Cω(ω2) = s ∈ ω2ω, ω1, ω2 : s(0) = ω,

and that X is a Cω(ω2)-universal space. 1.5.2

Proof of Theorem 1.5.11. Let G be a P-generic filter. Recall that T0 = 0×ω, Tα = α×ω2

for 1 ≤ α < ω2, and T =⋃Tα : α < ω2 Let

≤G=⋃≤p: p ∈ G

and for each x ∈ T putU(x) = z ∈ T : z ≤G x.

Let X = 〈T, τ〉 be the LCS∗ space generated by the family U(x) : x ∈ T. Density properties Dand E imply that Iα(X) = Tα for α < ω2.

Let s ∈ ω2ω, ω1, ω2 with s(0) = ω. Put

Y = 〈α, ξ〉 : α < ω2, ξ < s(α).

If x ∈ Iα(X) then U(x) \ x ⊆ α× ω. Hence

Y =⋃U(y) : y ∈ Y ,

therefore Y is open. Thus Iα(Y ) = Iα(X) ∩ Y , that is, Iα(Y ) = α × s(α). Thus the cardinalsequence of Y is exactly s. 1.5.11

Proof of Theorem 1.5.10. Fix an injective function ϕ : ω2 × ω21–1−→ ω2 × ω such that

(A) if π(x) < π(y) and x ∈ ω2 × ω then π(ϕ(x)) < π(ϕ(y)),(B) if x 6= y then π(ϕ(x)) = π(ϕ(y)) iff π(x) = π(y) and x, y ∈ ω2 × ω.

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“Lift” this ϕ to a function ϕ : P ∗ → P ∗ω in the natural way: for p = 〈ap,≤p, ip〉 ∈ P ∗ let ϕ(p) =⟨aϕ(p),≤ϕ(p), iϕ(p)

⟩, where aϕ(p) = ϕ′′ap, ϕ(x) ≤ϕ(p) ϕ(y) iff x ≤p y and iϕ(p)ϕ(x), ϕ(y) =

ϕ′′ipx, y.Let Q = 〈Q,〉 be a BS-poset. Take

P = p ∈ P ∗ : ϕ(p) ∈ Q

and P = 〈P,〉.We claim that P is a U-poset. For the details see [16].

1.5.10

1.6. A lifting theorem

(This section is based on [18] )

To simplify the formulation of the forthcoming results denote by T HIN (α) the statementthat there is an LCS∗ space with cardinal sequence 〈ω〉α.

Baumgartner and Shelah proved that if there is a ∆-function then T HIN (ω2) holds in anatural c.c.c forcing extension, where the meaning of "natural“ was explained in the previoussection. Building on their method, but using much more involved combinatorics, Martinez [84]proved that if there is a strong ∆-function, then for each δ < ω3 there is a c.c.c poset Pδ suchthat T HIN (δ) holds in V Pδ . These results naturally raised the following problem.

Problem 2.Does T HIN (ω2) imply T HIN (δ) for each δ < ω3?

Although this question remains still open we prove a “lifting theorem” (Theorem 1.6.21)claiming that if there is a natural poset Pω2 such that T HIN (ω2) holds in V Pω2 , then for eachδ < ω3 there is a natural poset Pδ such that T HIN (δ) holds in V Pδ : the posets used by Martinezcan be constructed directly from the poset applied by Baumgartner and Shelah without evenmentioning the ∆-function. Moreover, our lifting theorem works for each cardinal κ, not only forω2! Since there is no ∆-function on ω3 you can not expect to apply the method of Baumgartnerand Shelah to prove T HIN (ω3). However, if anybody can construct a “natural” c.c.c poset P suchthat T HIN (ω3) holds in V P then our theorem gives immediately the consistency of T HIN (α)for each α < ω4.

To formulate our statement more precisely we introduce some notation, so we postpone theformulation of our main result till theorem 1.6.15.

First we recall some definition and results.To formulate our result we neeed some preparation. Fix a cardinal κ ≥ ω and let π :

κ+ × ω −→ κ+ be the natural projection: π(〈α, n〉) = α.Define the poset P 0 =

⟨P 0,≺

⟩as follows. The underlying set P 0 consists of triples 〈a,≤, i〉

satisfying the following requirements:

(i) a ∈[κ+ × ω

]<ω,(ii) ≤ is a partial ordering on a,(iii) ∀x, y ∈

[a]2 if x ≤ y the π(x) < π(y),

(iv) i :[a]2 −→ P(a) is a function,

(v) ∀x, y ∈[a]2 if π(x) = π(y) then ix, y = ∅,

(vi) ∀x, y ∈[a]2 if x ≤ y then ix, y = x.

Write p = 〈ap,≤p, ip〉 for p ∈ P 0. Define the function hp : ap −→ P(ap) by the formulahp(x) = y ∈ ap : y ≤p x. For b ⊂ ap write hp[b] =

⋃hp(x) : x ∈ b.

Let p ≺ q iff aq ⊂ ap,≤q=≤p ∩(aq × aq),iq ⊂ ip.

Clearly ≺ is a partial ordering on P 0.

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1.6. A LIFTING THEOREM 31

Let

P ∗ =〈a,≤, i〉 ∈ P 0 : ∀x, y ∈

[a]2 ∀z ∈ a

(z ≤ x ∧ z ≤ y) iff ∃t ∈ ix, y z ≤ t.

Fact 1.6.1. For p ∈ P 0,

p ∈ P ∗ iff ∀x, y ∈[ap]2hp(x) ∩ hp(y) = hp[ipx, y].

The elements of P ∗ can be considered as the natural finite approximations of an LCS∗-orderon κ+ × ω and the witnessing function i.

Definition 1.6.2.Two condition p, q ∈ P 0 are twins iff (i)–(ii) below hold, where a = ap ∩ aq:(i) ≤p a =≤q a,(ii) ip

[a]2 = iq

[a]2.

Definition 1.6.3. Let p, q ∈ P 0 be twins. A condition r ∈ P 0 is an amalgamation of p and q iff(a) ar = ap ∪ aq(b) ≤r is the partial ordering on ar generated by ≤p ∪ ≤q,(c) ir ⊃ ip ∪ iq.Let

amalg(p, q) = r : r is an amalgamation of p and q.When we speak about amalgamations of two conditions we will always assume that these

conditions are twins.

Fact 1.6.4. If r ∈ P 0 is an amalgamation of p and q, then(1) ≤r ap =≤p,(2) r ≺ p and r ≺ q,(3) If x ∈ ap and y ∈ aq then x ≤r y iff there is z ∈ ap ∩ aq such that x ≤p z ≤q y.Fact 1.6.5. If r ∈ P 0 is an amalgamation of p and q, moreover p, q ∈ P ∗ then

∀x, y ∈[ap]2 ∪ [aq]2 hr(x) ∩ hr(y) = hr[ix, y].

Proof. See in [18].

For A ⊂ κ+ letP ∗A = p ∈ P ∗ : ap ⊂ A× ω.

Next we introduce three properties, (K+), DA1 and DA

2 , of posets 〈P,≺〉, where P ⊂ P ∗Afor some A ⊂ κ+. The first one is a strong version of property (K), the two others are densityrequirements.

Definition 1.6.6. Let P ⊂ P ∗. The poset P = 〈P,≺〉 has property (K+) iff

∀S ∈[P]ω1 ∃T ∈

[S]ω1 ∀p, q ∈

[T]2p and q have an amalgamation in P .

Definition 1.6.7. For a condition p ∈ P 0 and x ∈ (κ+ × ω) \ ap define q = p ] x ∈ P 0 asfollows:

• aq = ap ∪ x,• ≤q=≤p ∪〈x, x〉,• ip ⊂ iq,• iqx, y = ∅ for y ∈ ap.

Fact 1.6.8. p ] x ∈ P ∗A for each p ∈ P ∗A and x ∈ (A× ω) \ ap.

Definition 1.6.9. Let P ⊂ P ∗A. The poset P = 〈P,≺〉 has property DA1 iff

p ] x ∈ P for each p ∈ P and x ∈ (A× ω) \ ap.Definition 1.6.10. For p ∈ PA, x ∈ ap, y0, y1, . . . , yn−1 ∈ (A × ω) \ ap with π(y0) < π(y1) <. . . π(yn−1) < π(x) define the condition q = p ]x 〈y0, . . . , yn−1〉 ∈ P 0 as follows:

• aq = ap ∪ y0, . . . , yn−1,

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• ≤q=≤p ∪〈yi, yj : i ≤ j < n〉 ∪ 〈yi, z〉 : z ∈ ap, x ≤p z,• ip ⊂ iq,• iqyi, yj = ymin(i,j),

• iqyi, z =yi if x ≤p z∅ otherwise for z ∈ ap.

Fact 1.6.11. If p ∈ P ∗A, x ∈ ap, y0, y1, . . . , yn−1 ∈ (A×ω)\ap with π(y0) < π(y1) < . . . π(yn−1) <π(x), then q = p ]x 〈x0, . . . , xn−1〉 ∈ P ∗A.

Definition 1.6.12. Let P ⊂ P ∗A. The poset P = 〈P,≺〉 has property DA2 iff

∀α, β ∈ A, α < β, there is a finite set of ordinals LP (α, β) = α0, . . . , αn−1 ∈[A]<ω

such that α = α0 < α1 < . . . αn−1 < β and if p ∈ P , x ∈ ap with π(x) = β andxi ∈ (A× ω) \ ap with π(xi) = αi for i < n, then p ]x 〈y0, . . . yn−1〉 ∈ P .

Definition 1.6.13. Let A ⊂ κ+ and P ⊂ P ∗. The poset P = 〈P,≺〉 is A-nice iff P ⊂ P ∗A andP has properties (K+), (DA

1 ) and (DA2 ). For δ < κ+ let NAT (δ) be the statement that there is

δ-nice poset Pδ.

Proposition 1.6.14. If a poset P is δ-nice then P has property (K) and T HIN (δ) holds in V P .

Proof. By fact 1.6.4(2) property (K+) implies property (K). Let G ⊂ P be a generic filter. PutA =

⋃ap : p ∈ G, i =

⋃ip : p ∈ G and ≺=

⋃≺p: p ∈ G. Then A = δ × ω by (Dδ

1). ByLemma 1.8.6 it is enough to prove that 〈δ × ω,≺〉 is a 〈δ〉ω-poset (see definition 1.8.5).

Since A = δ×ω, 1.8.5(1) holds. The partial ordering ≺ satisfies 1.8.5(2) because every p ∈ Psatisfies (iii). The function i :

[δ × ω

]2 −→ [δ × ω

]<ω satisfies 1.8.5(3) because every element ofP is in P ∗. Finally 1.8.5(4) holds because (D2

δ) can be applied in a suitable density argument.Thus 〈δ × ω,≺〉 is an 〈δ〉ω-poset, hence there is an LCS∗ space with character sequence 〈δ〉ω byLemma 1.8.6. 1.6.14

After this preparation we are able to formulate the main lifting theorem.

Theorem 1.6.15.NAT (κ) implies NAT (δ) for each cardinal κ and ordinal δ < κ+.

First, in lemma 1.6.16 below, we show that our lifting theorem works downwards. AlthoughT HIN (κ) clearly implies T HIN (δ) for δ < κ we should prove NAT (δ) for δ < κ as well, becausewe will use the posets witnessing this to prove NAT (γ) for γ ≥ κ.

If p ∈ P 0 and I ⊂ κ+ let

p I =⟨ap ∩ (I × ω),≤p (I × ω), ip

[I × ω

]2⟩.

Observe that• p I ∈ P 0 iff ipx, y ⊂ I × ω for each x, y ∈

[ap ∩ (I × ω)

]2,• if p ∈ P ∗ and p I ∈ P 0 then p I ∈ P ∗.

Lemma 1.6.16 ([18]).NAT (κ) implies NAT (δ) for δ < κ.

Proof of theorem 1.6.15. Since we know the statement for δ < κ we prove the theorem byinduction on δ ≥ κ. When we constructed Pδ we will also have PA ⊂ P ∗A for each A ⊂ κ+ withorder type δ such that PA = 〈PA,≺〉 has properties (K+), (DA

1 ) and (DA2 ).

We will write LA(α, β) for LPA(α, β). Let LA(α, α) = ∅.Successor step:

Assume that Pδ is constructed. Then we can get Pδ+1 as follows.A p is in Pδ+1 iff

(i) p ∈ P ∗δ+1,(ii) p δ ∈ Pδ,(iii) ∀x, y ∈

[ap]2 if π(x) < δ and π(y) = δ then either ix, y = x (i.e. x ≤p y) or ix, y = ∅

(i.e. hp(x) ∩ hp(y) = ∅).The next lemma claims that Pδ+1 = 〈Pδ,≺〉 works:

Lemma 1.6.17 ([18]).Pδ+1 satisfies (K+), (Dδ+11 ). and (Dδ+1

2 ). 1.6.15

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1.6. A LIFTING THEOREM 33

The successor step is done.Limit step:

Assume that δ is limit ordinal, and PA is constructed for each A ⊂ κ+ with order type < δ.Fix a club C ⊂ δ, C = γζ : ζ < cf(δ). Let Iζ = [γζ , γζ+1) for ζ < cf(δ).Let ρ : δ −→ cf(δ) s.t. ρ(α) = ζ iff α ∈ Iζ .Let p ∈ Pδ iff

(δ1) p ∈ P ∗δ ,(δ2) p C ∈ PC ,(δ3) p Iζ ∈ PIζ for each ζ < cf(δ),(δ4) ∀x, y ∈ ap if x ≤p y, γζ < π(x) < γζ+1 ≤ π(y) then ∃u ∈ ap x ≤p u ≤p y and π(u) = γζ+1

(δ5) ∀x, y ∈ ap if x ≤p y, π(x) < γξ ≤ π(y) < γξ+1 then ∃v ∈ ap x ≤p v ≤p y and π(v) = γξ(δ6) ∀x, y ∈ ap, γζ ≤ π(x) < γζ+1 ≤ γξ ≤ π(y) < γξ+1, x 6≤p y then

ipx, y ⊂⋃ipu, v : u ≤p x, v ≤p y, π(u) = γζ , π(v) = γξ.

We show that Pδ = 〈Pδ,≺〉 works, i.e. it satisfies properties (K+), (Dδ1) and (Dδ

2).

Lemma 1.6.18.Pδ satisfies (K+).

Proof. Let pν : ν < ω1 ∈[Pδ]ω1 , pν = 〈aν ,≤ν , iν〉, hν = hpν . Let cν = η < cf(δ) : aν ∩ Iη 6=

∅. By thinning out the sequence pν : ν < ω1 we can assume that

(a) aν : ν ∈ ω1 forms a ∆-system with kernel d,(b) there is a partial ordering ≤d on d such that ≤ν d =≤d for each ν ∈ ω1,(c) cν : ν < ω1 forms a ∆-system with kernel c.(d) ∀η ∈ c ∃eη ∀ν ∈ ω1 aν ∩ (γη, γη + 1 × ω) = eη

(e) ∀η ∈ c ∀ν, µ ∈[ω1

]2 the conditions pν Iη and pµ Iη have an amalgamation rην,µ =⟨aην,µ,≤ην,µ, iην,µ

⟩in PIη .

(f) ∀ν, µ ∈[ω1

]2 the conditions pν C and pµ C have an amalgamation rCν,µ =⟨aCν,µ,≤Cν,µ, iCν,µ

⟩in PC .

(g) iνx, y = iµx, y for each x, y ∈[d]2 and ν, µ ∈

[ω1

]2.To ensure (g) fix x, y ∈

[d]2. If ρ(x) = ρ(y) = η then (g) holds by (e): iνx, y = iµx, y. If

π(x), π(y) ∈[C]2 then iνx, y = iµx, y

def= iCx, y by (f). If η = ρ(x) 6= ρ(y) = σ then by(δ6) we have

iνx, y ⊂⋃iνu, v : u ≤ν x, v ≤ν y, π(u) = γρ(x), π(v) = γρ(y) ⊂⋃

iCu, v : u ∈ eρ(x), v ∈ eρ(y),

i.e. iνx, y is a subset of a fixed finite set for each ν ∈ ω1. So, by thinning out our sequence wecan guarantee that (g) holds.

Claim 1.6.18.1. pν and pµ are twins for each ν, µ ∈[ω1

]2.Fix ν, µ ∈

[ω1

]2. Define r = 〈a,≤, i〉 ∈ P 0 as follows:

(r1) a = aν ∪ aµ,(r2) ≤ is the partial ordering on a generated by ≤ν ∪ ≤µ,

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(r3)

ix, y =

iνx, y if x, y ∈[aν]2,

iµx, y if x, y ∈[aµ]2,

iCν,µx, y if x, y ∈[C]2,

iην,µx, y if x, y ∈[Iη]2,

M(x, y) otherwise,where

M(x, y) =⋃iu, v : u, v ∈

[a]2, u ≤ x, v ≤ y, π(u) = γρ(x), π(v) = γρ(y).

Claim 1.6.18.2. r ∈ P ∗δ is an amalgamation of pν and pµ.

Proof. The proof is quite boring and technical. See in [18].

Hence Pδ satisfies (K+). 1.6.18

Lemma 1.6.19.Pδ satisfies (Dδ1).

Proof. Assume that p ∈ Pδ and z ∈ (δ×ω) \ ap. Let q = p] x. We need to show that q ∈ Pδ,i.e., q satisfies (δ1)–(δ6). For further details see [18].

Lemma 1.6.20. Pδ satisfies (Dδ2).

Proof. If α, β ∈[Iη]2 for some η then let Lδ(α, β) = LIη (α, β). Otherwise, if α ∈ Iη, β ∈ Iσ,

η < σ, then let α+ = min(C \ α+ 1) and put

Lδ(α, β) = α, α+ ∪ LC(α+, γσ) ∪ LIσ (γσ, β).

Enumerate LP (α, β) as α = α0 < α1 < · · · < αn−1 < β. Let p ∈ Pδ, z ∈ ap with π(z) = β andzi ∈ (δ × ω) \ ap with π(zi) = αi for i < n. Let q = p ]z 〈z0, . . . zn−1〉.

We should show that q ∈ Pδ, i.e. q satisfies (δ1)–(δ6). The details of this quite long argumentcan be found in [18].

Thus the limit step is done as well, which completes the inductive construction, so theorem1.6.15 is proved. 1.6.15

Now we can conclude the section.

Theorem 1.6.21. If there is a κ-nice poset P for some regular cardinal κ then there is a c.c.cposet Q such that T HIN (δ) holds in V Q for each δ < κ+.

Proof. Using theorem 1.6.15 we fix, for each δ < κ+, a δ-nice poset Pδ. Let Q be the finite-support product of Pδ : δ < κ+. Since every Pδ has property (K), so has Q.

Let G be a Q-generic filter and let δ < κ+ be arbitrary. Then Gδ = p(δ) : p ∈ G ∧ δ ∈ dom pis a Pδ-generic filter, hence T HIN (δ) holds in V [Gδ] winessed by some space Xδ by proposition1.6.14. Since V [Gδ] ⊂ V [G] the space Xδ witnesses T HIN (δ) in V [G].

1.7. Wide scattered spaces and morasses


By Baumgartner and Shelah, [25], it is relatively consistent with ZFC that 〈ω〉ω2∈ C(ω2).

Refining their argument, first Bagaria, [21], proved that ω2ω, ω1 ⊂ C(ω2) in some ZFC model,then we showed in [16]that 2ω = ω2 and ω2ω, ω1, ω2 ⊂ C(ω2) is also consistent.

For a long time ω2 was a mystique barrier in both height and width. In this section we canconstruct wider spaces.

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1.7. WIDE SCATTERED SPACES AND MORASSES 35

Theorem 1.7.1. If GCH holds and λ ≥ ω2 is a regular cardinal, then in some cardinal preservinggeneric extension 2ω = λ and every sequence s = 〈sα : α < ω2〉 of infinite cardinals with sα ≤ λis the cardinal sequence of some locally compact scattered space.

We will find the suitable generic extension in three steps:(I) The first extension adds a “strongly stationary strong (ω1, λ)-semimorass” to the ground

model (see Definition 1.7.2 and Theorem 1.7.4).(II) Using that strong semimorass the second extension adds a ∆(ω2 × λ)-function to the first

extension (see Definition 1.7.12 and Theorem 1.7.13).(III) Using the ∆(ω2 × λ)-function we add an “LCS∗ space with stem” to the second model

and we show that those2 space alone guarantees that every sequence s = 〈sα : α < ω2〉 ofinfinite cardinals with sα ≤ λ is the cardinal sequence of some locally compact scatteredspace (see Theorem 1.7.26).

Steps (I) and (II) are based on works of P. Koszmider, see [75] and [72].

1.7.1. Strong semimorasses. If ρ is a function and X is set, write ρ[X] = ρ(ξ) : ξ ∈ X.If X and Y are sets of ordinals with tp(X) = tp(Y ), denote the unique order preserving

bijection between X and Y by ρX,Y .For X ∈

[λ]ω and F ⊂

[λ]ω let F X = Y ∈ F : Y ( X.

If X, X1 and X2 are sets of ordinals, we writeX = X1 ⊕X2 iff tp(X1) = tp(X2), X = X1 ∪X2 and ρX1,X2 X1 ∩X2 = id;X = X1 X2 iff tp(X1) = tp(X2), X = X1 ∪X2 and X1 ∩X2 < X1 \X2 < X2 \X1;

andX = X1 ⊗X2 iff X = X1 ⊕X2 and X ∩ ω2 = (X1 ∩ ω2) (X2 ∩ ω2).

In [75] Koszmider introduced the notion of semimorasses and proved several properties con-cerning that structures. Unfortunately, in our proof we need structures with a bit strongerproperties.

Definition 1.7.2. Let ω1 ≤ λ be a cardinal. A family F ⊂[λ]ω is a strong (ω1, λ)-semimorass

iff(M1) 〈F ,⊆〉 is well-founded, (and so we have the rank function on F),(M2) F is locally small, i.e. |F X| ≤ ω for each X ∈ F .(M3) F is homogeneous, i.e. ∀X,Y ∈ F if rank(X) = rank(Y ) then tp(X) = tp(Y ) and F Y =

ρX,Y [Z] : Z ∈ F X.(M4) F is directed, i.e. ∀X,Y ∈ F (∃Z ∈ F) X ∪ Y ⊂ Z.(M5) F is strongly locally semidirected, i.e. ∀X ∈ F either (a) or (b) holds:

(a) F X is directed,(b) ∃X1, X2 ∈ F rank(X1) = rank(X2), X = X1 ⊗ X2, and F X = (F X1) ∪ (F

X2) ∪ X1, X2.(M6) F covers λ, i.e. ∪F = λ.If in (M5)(b) we weaken the assumption X = X1 ⊗X2 to X = X1 ⊕X2 then we obtain the defi-nition of an (ω1, λ)-semimorass (see [75, Definition 1]). Moreover, a strong (ω1, ω2)-semimorassis just Velleman’s simplified (ω1, ω2)-morass.

Definition 1.7.3.A family F ⊂[λ]ω is strongly stationary iff for each function c :

[F]<ω → [

λ]ω

there are stationary many X ∈ F such that X is c-closed, i.e. c(X∗) ⊂ X for each X∗ ∈[F X

]<ω.Theorem 1.7.4. If 2ω = ω1 < λ = λω1 then there is a σ-complete ω2-c.c. forcing notion P suchthat

V P |= “λω1 = λ and there is a strongly stationary strong (ω1, λ)-semimorass F . ”

We say that a family p ⊂[λ]ω is neat iff ∪p = ∪(p \ ∪p).

Proof of Theorem 1.7.4. Define P = 〈P,≤〉 as follows. LetP = p ⊂

[λ]ω : |p| ≤ ω,∪p ∈ p, p is neat and satisfies (M1)–(M5).

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Write supp(p) = ∪p for p ∈ P . Clearly supp(p) is the ⊂-largest element of p. Put

p ≤ q iff supp(q) ∈ p ∧ q = (p supp(q)) ∪ supp(q). (1.29)

P is σ-complete. Indeed, if p0 ≥ p1 ≥ p2 . . . then let

p = ∪n<ωpn ∪ ∪n<ω supp(pn).

Then p ∈ P and p ≤ pn for each n.

Definition 1.7.5.We say that two conditions p and p′ are twins iff(i) tp(supp(p)) = tp(supp(p′)),(ii) supp(p) ∪ supp(p′) = supp(p)⊗ supp(p′),(iii) p′ = ρsupp(p),supp(p′)[p].

Lemma 1.7.6. If p and p′ are twins then they have a common extension in P

Proof. Write D = supp(p) and D′ = supp(p′). Put r = p ∪ p′ ∪ D ∪D′. We show that r is acommon extension of p and p′.Claim: p = (r D) ∪ D and p′ = (r D′) ∪ D′.

Indeed, assume that X ∈ r D. Then X ∈ p or X ∈ p′. If X ∈ p′ then X ⊂ D′, and soX ⊂ D ∩D′. Since ρD,D′ D ∩D′ = id it follows that X = ρ−1

D,D′ [X] ∈ p. So r D ⊂ p, whichproves the Claim.

First we check that that r ∈ P . (M1) and (M2) are clear. Since supp(r) = supp(p)∪supp(p′),r is neat. r has the largest element supp(r) = D ∪ D′ ∈ r, and so (M4) also holds. In (M5)we have just one new instance X = supp(r). But in this case the choice X1 = D and X2 = D′

works. To check (M3) assume that X,Y ∈ r, rank(X) = rank(Y ). If X,Y ∈ p or X,Y ∈ p′ thenwe can apply that p and p′ satisfy (M3). So we can assume that X ∈ p \ p′ and Y ∈ p′ \ p. LetX ′ = ρD,D′ [X]. Then rank(X ′) = rank(X) = rank(Y ) and X ′, Y ∈ p′. Since p′ satisfies (M3), wehave tp(X ′) = tp(Y ), and so tp(X) = tp(Y ). Since ρX′,Y : p′ X ′ → p′ Y is an isomorphism,and ρX,Y = ρD,D′ ρX′,Y it follows that ρX,Y : p X → p′ Y is also an isomorphism. However: p X = r X and p′ Y = r Y by the Claim, and so ρX,Y : r X → r Y is also anisomorphism, which proves (M3).

Finally r ≤ p, p′ follows immediately from the Claim.

Lemma 1.7.7. P satisfies ω2-c.c.

Proof. Assume that rα : α < ω2 ⊂ P . Write Dα = supp(rα) for α < ω2. By standardargument we can find I ∈

[ω2

]ω2 such that(a) Dα : α ∈ I forms a ∆-system with kernel D, and tp(Dα) = tp(Dβ) for α, β ∈ I,(b) For α < β ∈ I we have D ∩ ω2 < Dα \D < Dβ \D,(c) ρDα,Dβ [Dα ∩ ω2] = Dβ ∩ ω2 and ρDα,Dβ D = id,(d) rβ = ρDα,Dβ [X] : X ∈ rα.Then for each α 6= β ∈ I the conditions rα and rβ are twins, so they are compatible by Lemma1.7.6.

Lemma 1.7.8. (a) ∀α ∈ ω2

Dα = p ∈ P : supp(p) ∩ (ω2 \ α) 6= ∅

is dense in P .(b) ∀β ∈ λ \ ω2

Eβ = p ∈ P : β ∈ supp(p)is dense in P .

Proof. (a) For each q ∈ P and α < ω2 there is q′ such that q and q′ are twins and supp(q′) ∩(ω2 \ α) 6= ∅. Then q and q′ has a common extension p ∈ Dα by Lemma 1.7.6.(b) For all q and β ∈ λ \ ω2 there is q′ such that q and q′ are twins and β ∈ supp(q′). Then thecommon extension p of q and q′ is in Eβ .

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Let G be a P -generic filter over V , and put F ′ = ∪G and F = ∪F ′. Then F ′ ⊂[λ]ω and so

F ⊂ λ. By the previous lemma, F ⊃ λ \ ω2 and |F ∩ ω2| = ω2. So F = ρF,λ[X] : X ∈ F ′ isa strong (ω1, λ)-semimorass. To complete the proof of 1.7.4 it is enough to prove the followinglemma.

Lemma 1.7.9. V P |= “F is strongly stationary”.

Proof. It is enough to prove that if

q C ⊂[F]ω is club and c :

[F ′]<ω → [

F]ω

then there are p ≤ q and C ∈[λ]ω such that p “C ∈ F ′ ∩ C is c-closed”.

First we need a claim.

Claim 1.7.9.1. If p A ∈[F]ω then ∃p′ ≤ p such that A ⊂ supp(p′).

Proof. If α ∈ A and p α ∈ F then there are p′ ≤ p and p′′ ∈ P such that α ∈ supp(p′′) andp′ p′′ ∈ F . Then p′ and p′′ have a common extension q, and then α ∈ supp(q) and q α ∈ F .

Since P is σ-complete and A is countable, we are done by a straightforward induction.

We will choose a decreasing sequence 〈pn : n < ω〉 ⊂ P and an increasing sequence 〈Cn : n < ω〉 ⊂[λ]ω as follows. Let C0 = ∅ and p0 = q. If pn and Cn are given, let Zn ⊃ Cn ∪ supp(pn) and

p′n ≤ pn s.t.p′n ∪c(X) : X ∈

[pn]<ω ⊂ Zn ∈ [F ]ω.

Let pn+1 ≤ p′n and Cn+1 ⊃ Zn ∪ Cn such that Cn+1 ⊂ supp(pn+1) and pn+1 Cn+1 ∈ C.Having constructed the sequence finally put C = ∪Cn : n < ω and p = ∪n<ωpn ∪ C.

Then p ∈ P , p ≤ q, C = supp(p). Since p “Cn ∈ C and C is club”, we have p “C ∈ C. Sincep c′′

[[pn]<ω] ⊂ supp(pn+1), we have p C is c-closed. Moreover p p ⊂ F ′, so p C ∈ F ′.

Putting these together we obtain that p and C have the desired properties, which proves thelemma.

Since (λω1)V [F ] ≤ ((|P |+ λ)ω1)V = (λω1)V = λ, the proof of Theorem 1.7.4 is complete.

Next we investigate some properties of strong semimorasses.

Lemma 1.7.10. Let F ⊂[λ]ω be a strongly stationary strong (ω1, λ)-semimorass.

(1) If X,Y ∈ F , rank(X) = rank(Y ), α ∈ X ∩ Y ∩ ω2, then X ∩ α = Y ∩ α.(2) If X,Y ∈ F , X ⊂ Y and rank(X) < α < rank(Y ) then there is Z ∈ F such that rank(Z) = α

and X ⊂ Z ⊂ Y .(3) If X ∈ F and rank(X) < α < ω1 then there is Z ∈ F such that rank(Z) = α and X ⊂ Z.(4) If X,Y ∈ F , rank(X) ≤ rank(Y ), and α ∈ X ∩ Y ∩ ω2, then X ∩ α ⊂ Y ∩ α.

Proof. (1) We prove the statement by induction on the minimal rank of Z ∈ F with Z ⊃ X ∪Y .If rank of Z is minimal, then clearly Z = Z1 ⊗ Z2 where X ⊂ Z1 and Y ⊂ Z2. Let X ′ =

ρZ1,Z2 [X] ∈ F Z2. Since α ∈ Z1∩Z2∩ω2, we have Z1∩α = Z2∩α and so ρZ1,Z2 (α+1) = id.Thus X ′ ∩ α = X ∩ α and α ∈ X ′. Since X ′, Y ∈ F Z2, α ∈ X ′ ∩ Y and rank(Z2) < rank(Z),by the inductive assumption we have X ′ ∩ α = Y ∩ α. Thus X ∩ α = Y ∩ α.(2) Easy by straightforward induction on rank(Y ).(3) By straightforward induction on α there is Y ∈ F such that X ⊂ Y and rank(Y ) ≥ α. Thenapply (2).(4) By (3) there is Y ′ ⊃ X such that rank(Y ) = rank(Y ′). Then apply (1) for Y and Y ′.

In [72] Koszmider proved several statements for Velleman’s simplified morasses. Here we needsimilar results for strong semimorasses. The following lemma corresponds to [72, Fact 2.6-Fact2.7].

Lemma 1.7.11. Let F ⊂[λ]ω be a strongly stationary strong (ω1, λ)-semimorass. Assume λω =

λ, fix an injective function c : F → λ, and consider the stationary set

F ′ = X ∈ F : c(X∗) ∈ X for each X∗ ∈ F X. (1.30)

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Assume that F ,F ′, c ∈M ≺ H(θ), |M | = ω, and M ∩ λ ∈ F ′. Then(1) F M ∩ λ ⊂M .(2) rank(M ∩ λ) = M ∩ ω1.(3) If Y ∈ F with rank(Y ) < δ = M ∩ ω1 then there is Z ∈M ∩ F such that (M ∩ λ) ∩ Y ⊂ Z,

and rank(Z) = rank(Y ).(4) If A ∈

[F]<ω then there is Z ∈ F ∩M such that

∪X ∩M : X ∈ A, rank(X) < M ∩ ω1 ⊂ Z. (1.31)

Proof. (1) Let X ∈ F M ∩ λ, i.e. X ∈ F and X ( M ∩ λ. Then X ( M ∩ λ ∈ F ′ impliesα = c(X) ∈M ∩ λ. But c, α ∈M and c is injective, so X = c−1α ∈M .(2) If X ( M ∩ λ, X ∈ F , then X ∈ M by (1) and so rank(X) ∈ M ∩ ω1. Thus rank(M ∩ λ) ≤M ∩ ω1.

Assume that α < M ∩ ω1. Then

M |= “∃X ∈ F rank(X) = α.′′

Thus there is X ∈M ∩ F such that rank(X) = α. Hence rank(M ∩ λ) ≥M ∩ ω1.(3) There is Y ′ ⊃ Y , Y ′ ∈ F and rank(Y ′) = rank(M ∩ λ). Let Z = ρY ′,M∩λ[Y ]. SinceY ∩ (M ∩ λ) ⊂ Y ′ ∩ (M ∩ λ) and ρY ′,M∩λ Y ′ ∩ (M ∩ λ) = id, we have Z ⊃ Y ∩ (M ∩ λ).(4) Just apply (3) and the fact that F is directed.

1.7.2. A ∆(ω2 × λ)-function. Let λ ≥ ω2 be an infinite cardinal and let π : ω2 × λ → ω2

be the natural projection: π(〈ξ, α〉) = ξ.

Definition 1.7.12. (1) Assume that f is a function such that dom(f) ⊂[ω2 × λ

]2 and fx, y ∈[π(x) ∩ π(y)

]<ω for each x, y ∈ dom(f). We say that two finite subsets d and d′ of ω2 × λ aregood for f provided

[d ∪ d′

]2 ⊂ dom(f) and ∀x ∈ d′ \ d ∀y ∈ d \ d′ ∀z ∈ d ∩ d′ ∩ (ω2 × ω)(S1) if π(z) < π(x), π(y) then π(z) ∈ fx, y,(S2) if π(z) < π(y) then fx, z ⊂ fx, y,(S3) if π(z) < π(x) then fy, z ⊂ fx, y.(2) A function f :

[ω2 × λ

]2 → [ω2

]<ω is a ∆(ω2 × λ)-function iff fx, y ⊂ min(π(x), π(y)) andfor each sequence dα : α < ω1 ⊂

[ω2 × λ

]<ω there are α 6= β such that dα and dβ are good forf .

Remark .The assumption |fx, y| < ω, instead of the usual |fx, y| ≤ ω, is not a misprint.

Theorem 1.7.13. If 2ω = ω1 < λ = λω1 and there is a strongly stationary strong (ω1, λ)-semimorass, then in some cardinal preserving generic extension λω1 = λ and there is a ∆(ω2×λ)-function.

Proof. To start with fix a strongly stationary strong (ω1, λ)-semimorass F ⊂[λ]ω. We can

assume thatω ⊂ X for each X ∈ F . (1.32)

Fix an injective function c : F → λ, and consider the stationary set

F ′ = X ∈ F : c(X∗) ∈ X for each X∗ ∈ F X. (1.33)

Definition 1.7.14.We define a poset P = 〈P,≤〉 as follows: P consists of triples p = 〈a, f,A〉,where a ∈

[ω2 × λ

]<ω, f :[a]2 → P(π[a]) with fs, t ⊂ min(π(s), π(t)), A ∈

[F]<ω such that

∀s, t ∈ a ∀X ∈ A if s, t ∈ X ×X then fs, t ⊂ X. (1.34)

Write p = 〈ap, fp,Ap〉 for p ∈ P . Put p ≤ q iff ap ⊃ aq, fp ⊃ fq and Ap ⊃ Aq.For p ∈ P let

supp(pν) = ∪aν ∪ ∪Aν .

Clearly supp(p) ∈[λ]≤ω.

If ρ is a function and x = 〈a, b〉 ∈ dom(ρ)2, let ρ(x) = 〈ρ(a), ρ(b)〉. We say p, q ∈ P are twinsiff

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(A) supp(p) and supp(q) have the same order type,(B) the unique <On-preserving bijection ρ between supp(p) and supp(q) gives an isomorphism

between p and q, i.e.(a) aq = ρ[ap],(b) ρ[X] : X ∈ Ap = Aq,(c) for each s, t ∈

[ap]2, ρ[fps, t] = fqρ(s), ρ(t).

Definition 1.7.15 ([75, Definition 22]). Let K ⊂[λ]ω. A poset P is K-proper iff for some large

enough regular cardinal θ if M is a countable elementary submodel of H(θ) with P ∈ M andM ∩ λ ∈ K then for each p ∈M ∩ P there is an (M,P )-generic q ≤ p.

Lemma 1.7.16 ([75, Fact 23]). If K ⊂[λ]ω is stationary and a poset P is K-proper, then forcing

with P preserves ω1.

Definition 1.7.17 ([75, Definition 24]).Assume that P is a poset, M ≺ H(θ), |M | = ω, q ∈ P ,and P, π1, . . . , πn ∈M . We say that the formula Φ(x, π1, . . . , πn) well-reflects q in M iff(1) H(θ) |= Φ(q, π1, . . . , πn),(2) if s ∈M ∩ P and M |= Φ(s, π1, . . . πn) then q and s are compatible in P .

Definition 1.7.18 ([75, Definition 25]).Assume that P is a poset, K ⊂[λ]ω. We say that P is

simply K-proper if the following holds: for some/each large enough regular cardinal θIF(i) M ≺ H(θ), |M | = ω,(ii) p ∈ P , P, p,K ∈M ,(iii) M ∩ λ ∈ K,THEN there is p0 ≤ p such that for each q ≤ p0 some formula Φ(x, π1, . . . , πn) well-reflects q inM .

By lemmas [75, Fact 23 and Lemma 26] we have

Lemma 1.7.19. If K ⊂[λ]ω is stationary and a poset P is simply K-proper, then forcing with

P preserves ω1.

To show that ω1 is preserved we prove the following lemma.

Lemma 1.7.20. P is simply F ′-proper.Actually we will prove some stronger statement. To formulate it we need some preparation.If M ≺ H(θ), p ∈ P ∩M , M ∩ λ ∈ F and q ∈ P let

pM = 〈ap, fp,Ap ∪ M ∩ λ〉and

q M = 〈aq ∩M,fq M,Aq ∩M〉 .

Lemma 1.7.21. (1) IfM ≺ H(θ) and p ∈ P∩M then pM ∈ P . (2) If q ≤ pM then q M ∈ P∩Mas well.

Proof. (1) We should only check (1.34) for pM . Assume that s, t ∈ ap and X ∈ Ap ∪ M ∩ λ.Since p ∈ P , we can assume X = M ∩ λ. However s, t ∈ M , and so fps, t ∈ M as well byp ∈M . Since |fps, t| ≤ ω, it follows fps, t ⊂M ∩ λ = X which was to be proved.(2) It is straightforward that q M ∈ P . To show q M ∈M we should check that fq M ∈M .So assume that s, t ∈ aq ∩M . Then s, t ∈ (M ∩ λ) × (M ∩ λ) and M ∩ λ ∈ ApM ⊂ Aq. So, by(1.34), fqs, t ⊂M ∩ λ. Since fqs, t is finite, we have fqs, t ∈M .

Lemma 1.7.22.Assume that M ≺ H(θ), |M | = ω, P,F ∈ M , p ∈ P ∩M , M ∩ λ ∈ F ′. Letδ = rank(M ∩ λ) = M ∩ ω1. Assume that Z ∈M ∩ F such that

Z ⊃ ∪X ∩M : X ∈ Aq, rank(X) < δ. (1.35)

Let Φ(x, Z, q M) be the following formula:

“ x ∈ P , x ≤ q M , (ax \ aqM ) ∩ (Z × Z) = ∅.” (1.36)

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Then(1) Φ(q, Z, q M) holds.(2) If s ∈M , M |= Φ(s, Z, q M) and

h : [as \ aqM , aq \ aqM ]→ P(π[as ∪ aq])

such that

hx, y ⊂ min(π(x), π(y)) ∩⋂X ∈ Aq : x, y ∈ X ×X, rank(X) ≥ δ, (1.37)

then r = 〈as ∪ aq, fs ∪ fq ∪ h,As ∪ Aq〉 ∈ P is a common extension of q and s.(3) Φ(x, Z, q M) well reflects q in M .

Proof. (1) Since q M ∈ P by Lemma 1.7.21(2), we have q ≤ q M by the definition of therelation ≤. Since Z ∈M , we have Z × Z ⊂M ×M ⊂M and aq \ aqM = aq \M .(2) To show that r ∈ P we need to check that r = 〈ar, fr,Ar〉 satisfies (1.34). So let x, y ∈ ar,X ∈ Ar.Case 1. x, y ∈ as and X ∈ As or x, y ∈ aq and X ∈ Aq.

Then everything is fine, because s, q ∈ P .

Case 2. x, y ∈[as]2, x ∈ as \ aq and X ∈ Aq.

If rank(X) < δ then (as \ aq) ∩ (X ×X) = ∅ by (1.35), so x /∈ X ×X. Thus (1.34) is void.If rank(X) ≥ δ then ν = min(π(x), π(y)) ∈ M ∩X, so M ∩ ν ⊂ X ∩ ν by Lemma 1.7.10(4).

Thus frx, y = fsx, y ∩ ν ⊂M ∩ ν ⊂ X.

Case 3. x, y ∈[aq]2, x ∈ aq \ as and X ∈ As.

Since (aq \ as) ∩M = ∅, it is not possible that X ∈ As. Then x ∈ aq \ as = aq \ aqM , sox /∈M . However X ⊂M and so x /∈ X ×X, so (1.34) is void.

Case 4. x ∈ aq \ as and y ∈ as \ aq.Then the assumption concerning h in (1.37) is stronger than (1.34). Indeed, if y ∈ as \ aq

then y /∈ Z × Z. So if y ∈ X ×X for some X ∈ Aq then rank(X) ≥ δ.

(3) Define the functionh : [as \ aqM , aq \ aqM ]→

[ω2

]<ωby hx, y = ∅. Then (1.37) holds, so s and q are comparable by (2), which was to be proved.

Proof of Lemma 1.7.20. We can apply Lemma 1.7.22 because by Lemma 1.7.11 we can pickZ ∈M ∩ F such that Z ⊃ ∪X ∩M : X ∈ Aq, rank(X) < δ.

Lemma 1.7.23. P satisfies ω2-c.c.

Proof. Let pν : ν < ω2 ⊂ P . Put pν = 〈aν , fν ,Aν〉. Recall that supp(pν) = ∪aν ∪ ∪Aν .Wecan assume that(i) supp(pν) : ν < ω2) forms a ∆-system with kernel D(ii) the conditions are pairwise twins witnessed by functions ρν,µ : supp(pν)→ supp(pµ).

Fix ν < µ < ω2. Define the function e as follows:

dom(e) = [aν \ aµ, aµ \ aν ] and es, t = ∅.

We claim thatr = 〈aν ∪ aµ, fν ∪ fµ ∪ e,Aν ∪ Aµ〉 (1.38)

is a common extension of pν and pµ. We need to show that r ∈ P . Since pν and pµ are twins,we should check only (1.34). So let t, s ∈ aν ∪ aµ and X ∈ Aν ∪ Aµ with s, t ∈ X × X. Wecan assume e.g. X ∈ Aν . Since X ⊂ supp(pν), we have s, t ∈ supp(pν) × supp(pν). Thensupp(pν)× supp(pν)∩ aµ ⊂ aν because supp(pν)× supp(pν)∩ aµ ⊂ D and aν , aµ are twins. Thuswe have s, t ∈ aν , and so we are done because pν satisfies (1.34).

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To complete the proof of Theorem 1.7.13 we claim that if G is a P -generic filter then thefunction

f = ∪fp : p ∈ G (1.39)is a ∆(ω2 × λ)-function.

Assume thatp dξ : ξ < ω1 ⊂

[ω2 × λ

]<ω.

We can assumep dξ : ξ < ω1 is a ∆-system with kernel d.

Assume M ≺ H(θ), |M | = ω, p,F ′⟨dξ : ξ < ω1

⟩∈M and X0 = M ∩ λ ∈ F ′. Let

pM = 〈ap, fp,Ap ∪ X0〉 .

Let q ≤ pM , ξ1 < ω1 and e1 ∈[ω2 × λ

]<ω that

q dξ1 = e1 ∧ (e1 \ d) ∩M = ∅ ∧ e1 ⊂ aq.Put δ = rank(M ∩ λ). By Lemma 1.7.11 we can pick Z ∈M ∩ F such that

Z ⊃ ∪X ∩M : X ∈ Aq, rank(X) < δ.Consider the following formula Ψ(x, ξ, e) with free variables x, ξ and e and parameters Z, q M,⟨dξ : ξ < ω1

⟩, d ∈M :

Φ(x, Z, q M) ∧ ξ ∈ ω1 ∧ (x dξ = e) ∧ e ⊂ ax ∧ (e \ d) ⊂ ax \ aqMwhere the formula Φ(x, Z, q M) was defined in (1.36) in Lemma 1.7.22. Then Ψ(q, ξ1, e1) holds.Thus ∃x∃ξ∃eΨ(x, ξ, e) also holds. Since the parameters are all in M , we have

M |= ∃x ∃ξ ∃e Ψ(x, ξ, e). (1.40)

Thus there are s, ξ2, e2 ∈M such that

Φ(s, Z, q M) ∧ (s dξ2 = e2) ∧ e2 ⊂ as ∧ (e2 \ d) ⊂ as \ aqM .Since e2 \ d ⊂ as \ aqM and (as \ aqM ) ∩ Z × Z = ∅ by Φ(s, Z, q M), we have

(e2 \ d) ∩ Z × Z = ∅.Define the function

h : [as \ aqM , aq \ aqM ]→ P(π[as ∪ aq])by the formula

hx, y = π[as ∪ aq] ∩min(π(x), π(y))∩ ⋂X ∈ Aq : x, y ∈ X ×X, rank(X) ≥ δ. (1.41)

So hx, y is as large as it is allowed by (1.37).Then, by Lemma 1.7.22, the condition r = 〈ar, fr,Ar〉, where ar = as ∪ aq, fr = fs ∪ fq ∪ h

and Ar = As ∪ Aq, is a common extension of q and s.

Lemma 1.7.24. e1 and e2 are good for fr.

Proof. We should check conditions (S1)-(S3).Assume that z ∈ e1 ∩ e2 ∩ (ω2 × ω), x ∈ e1 \ e2 ⊂ aq \ aqM , and y ∈ e2 \ e1 ⊂ as \ aqM .

Observe that z, y ∈M , and so π(z), π(y) ∈M as well.(S1): Assume that π(z) < π(x), π(y).

We should show that π(z) ∈ frx, y. However, frx, y was defined by (1.41). So we shouldshow that

if X ∈ Aq, rank(X) ≥ δ, x, y ∈ X ×X then π(z) ∈ X.Since π(y) ∈M ∩X ∩ ω2 and rank(M ∩ λ) ≤ rank(X) we have M ∩ π(y) ⊂ X ∩ π(y) by Lemma1.7.10(4). Since π(z) < π(y) and π(z) ∈M it follows that π(z) ∈ X.(S2): Assume that π(z) < π(y).

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We need to show that frx, z ⊂ frx, y. Since frx, z = fqx, z and frx, y was definedby (1.41) we should show that

if X ∈ Aq, rank(X) ≥ δ, x, y ∈ X ×X then fqx, z ⊂ X.

Since π(y) ∈M ∩X we have π(z) ∈M ∩ π(y) ⊂ X ∩ π(y) by Lemma 1.7.10(4).Since π(z) ∈ X, z ∈ ω2×ω and ω ⊂ X by (1.32), it follows that z ∈ X×X. Since x, z ∈ X×X

and X ∈ Aq, we have fqx, z ⊂ X by (1.34), which was to be proved.(S3): Assume that π(z) < π(x).

We need to show that fry, z ⊂ frx, y. Since fry, z = fsy, z and frx, y was definedby (1.41) we should show that

if X ∈ Aq, rank(X) ≥ δ, x, y ∈ X ×X then fsy, z ⊂ X.

Since z, y ∈M we have fsy, z ⊂M .Moreover y ∈ X × X, and so π(y) ∈ M ∩ X ∩ ω2, which implies M ∩ π(y) ⊂ X ∩ π(y) by

Lemma 1.7.10(4) .Thus fsy, z = fsy, z ∩ π(y) ⊂M ∩ π(y) ⊂ X ∩ π(y) ⊂ X, which was to be proved.

Since r dξ1 = e1 ∧ dξ2 = e2 ∧ f ⊃ fr, by Lemma 1.7.24 r “ dξ1 and dξ2 are good for f ”.So f is a ∆(ω2 × λ)-function in V [G].

Since |P | ≤ λ and so (λω1)V [G] ≤ ((|P |+ λ)ω1)V = (λω1)V = λ, the proof of Theorem 1.7.13is complete.

1.7.3. Space construction. Assume that X is a scattered space. We say that a subspaceY ⊂ X is a stem of X provided(i) ht(Y ) = ht(X),(ii) X \ Y is closed discrete in X.

Clearly (ii) holds iff every x ∈ X has a neighborhood Ux such that Ux \ x ⊂ Y .

Proposition 1.7.25.Assume that X is an LCS∗ space, Y ⊂ X is a stem, CS(X) = 〈κν : ν < µ〉and CS(Y ) = 〈λν : ν < µ〉. Then

CS(Z) : Y ⊂ Z ⊂ X = s ∈ µCard : λν ≤ s(ν) ≤ κν for each ν < µ. (1.42)

Proof. Assume that s ∈ µCard such that λν ≤ s(ν) ≤ κν for each ν < µ. For ν < µ pickZν ∈ [Iν(X)]s(ν) with Zν ⊃ Iν(Y ). Put Z =

⋃Zν : ν < µ. Since Y ⊂ Z and Y is a stem, we

have Iν(Z) = Zν for ν < µ, and so CS(Z) = s.

Theorem 1.7.26. If there is a ∆(ω2×λ)-function, then there is a c.c.c poset P such that in V Pthere is an LCS space X with stem Y such that CS(X) = 〈λ〉ω2

and CS(Y ) = 〈ω〉ω2.

Corollary 1.7.27. If there is a ∆(ω2 × λ)-function, then there is a c.c.c poset P such that inV P every sequence s = 〈sα : α < ω2〉 of infinite cardinals with sα ≤ λ is the cardinal sequence ofsome locally compact scattered space.

Proof of Theorem 1.7.26. Instead of constructing the topological space directly, we actuallyproduce a certain “graded poset” which guarantees the existence of the desired locally compactscattered space. We use the ideas from [21] to formulate the properties of our required poset.

Definition 1.7.28.Given two sequences t = 〈κα : α < δ〉 and s = 〈λα : α < δ〉 of infinite cardinalswith λα ≤ κα, we say that a poset 〈T,≺〉 is a t-poset with an s-stem iff the following conditionsare satisfied:(T1) T =

⋃Tα : α < δ where Tα = α × κα for each α < δ. Let Sα = α × λα, and

S =⋃Sα : α < δ.

(T2) For each s ∈ Tα and t ∈ Tβ , if s ≺ t then α < β and s ∈ Sα.(T3) For every s, t ∈

[T]2 there is a finite subset is, t of S such that for each u ∈ T :

(u s ∧ u t) iff u v for some v ∈ is, t.(T4) For α < β < δ, if t ∈ Tβ then the set s ∈ Sα : s ≺ t is infinite.

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Lemma 1.7.29. If there is a t-poset with an s-stem then there is an LCS∗ space X with stem Ysuch that CS(X) = t and CS(Y ) = s.

Indeed, if T = 〈T,≺〉 is an s-poset, we write UT (x) = y ∈ T : y x for x ∈ T , and wedenote by XT the topological space on T whose subbase is the family

UT (x), T \ UT (x) : x ∈ T, (1.43)

then XT is our desired LCS∗ space with stem.So, to prove Theorem 1.7.26 it will be enough to show that a 〈λ〉ω2

-poset with an 〈ω〉ω2-stem

may exist.We follow the ideas of [25] to construct P . Fix a ∆(ω2×λ)-function f :

[ω2 × λ

]2 → [ω2

]<ω.Definition 1.7.30.Define the poset P = 〈P,〉 as follows. The underlying set P consists oftriples p = 〈ap,≤p, ip〉 satisfying the following requirements:

(1) ap ∈[ω2 × λ

]<ω,(2) ≤p is a partial ordering on ap with the property that if x <p y then x ∈ ω2 × ω and

π(x) < π(y),(3) ip :

[ap]2 → [

ap]<ω is such that

(3.1) if x, y ∈[ap]2 then

(3.1.1) if x, y ∈ ω2 × ω and π(x) = π(y) then ipx, y = ∅,(3.1.2) if x <p y then ipx, y = x,(3.1.3) if x and y are <p-incomparable, then

ipx, y ⊂ fx, y × ω.(3.2) if x, y ∈

[ap]2 and z ∈ ap then(

(z ≤p x ∧ z ≤p y) iff ∃t ∈ ipx, y z ≤p t).

Set p q iff ap ⊇ aq, ≤p aq =≤q and ip [aq]2 = iq.

Lemma 1.7.31. P satisfies ω1-c.c..

Proof. Let pν : ν < ω1 ⊂ P , pv = 〈aν ,≤ν , iν〉. By thinning out our sequence we can assumethat(i) aν : ν < ω1 forms a ∆-system with kernel a′.(ii) iν

[a′]2 = i.

(iii) ≤ν a′ × a′ =≤.(iv) for each ν < µ < ω1 there is a bijection ρν,µ : aν → aµ such that

(a) ρν,µ a′ = id(b) π(x) ≤ π(y) iff π(ρν,µ(x)) ≤ π(ρν,µ(y)),(c) x ≤ν y iff ρν,µ(x) ≤µ ρν,µ(y),(d) x ∈ ω2 × ω iff ρν,µ(x) ∈ ω2 × ω,(e) ρν,µ[iνx, y] = iµρν,µ(x), ρν,µ(y).

Now it follows from condition (3.1) and condition (iv) that if ν < µ < ω2 and x, y ∈[a′]2 then

iνx, y = iµx, y.Since f is a ∆(ω2 × λ)-function there is ν < µ < ω1 such that aν and aµ are good for f , i.e.

(S1)–(S3) hold. Define r = 〈a,≤, i〉 as follows:(a) a = aν ∪ aµ,(b) x ≤ y iff x ≤ν y or x ≤µ y or there is s ∈ aν ∩ aµ such that x ≤ν s ≤µ y or x ≤µ s ≤ν y,(c) i ⊃ iν ∪ iµ,(d) for x ∈ aν \ aµ and y ∈ aµ \ aν , if x and y are ≤-incomparable then

ix, y =(fx, y × ω

)∩ t ∈ a : t ≤ x ∧ t ≤ y. (1.44)

(e) for x, y ∈[a]2 with x < y, ix, y = x.

We claim that r ∈ P .By the construction, we have ≤ aν × aν =≤ν and ≤ aµ × aµ =≤µ.

Claim: ≤ is a partial order.

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We should check only the transitivity. Assume x ≤ y ≤ z. If x ≤ν y ≤ν z or x ≤µ y ≤µ zthen we are done. Assume that x ≤ν u ≤µ y ≤µ z for some u ∈ aν ∩ aµ. Then x ≤ν u ≤µ z sox ≤ z.

If x ≤ν u ≤µ y ≤µ t ≤ν z for some u, t ∈ aν ∩ aµ, then u ≤µ t, which implies u ≤ν t. Thusx ≤ν u ≤ν t ≤ν z and so x ≤ν z, and hence x ≤ z.

The other cases are similar to these ones.

(3.1.3) holds by the construction of i.To show that p is a condition we should finally check (3.2). Let x, y ∈ a be ≤-incomparable

elements. It is clear that if u ≤ t for some t ∈ ix, y then u ≤ x and ≤ y. So we should checkthat(∗) if z ≤ x and z ≤ y then there is t ∈ ix, y such that z ≤ t.

If x, y, z ∈ aν or x, y, z ∈ aµ then it is clear because pν , pµ ∈ P .Case 1. x, y ∈ aν and z ∈ aµ \ aν .Subcase 1.1 x, y ∈ aν \ aµ.

There are x′, y′ ∈ aν ∩ aµ such that z ≤µ x′ ≤ν x and z ≤µ y′ ≤ν y. Then there ist′ ∈ iµx′, y′ such that z ≤µ t′. Then t′ ∈ aν ∩ aµ, so t′ ≤ν x, y. Thus there is t ∈ iνx, y suchthat t′ ≤ν t, and so z ≤ t. Since ix, y = iνx, y, we are done.Subcase 1.2 x ∈ aν \ aµ and y ∈ aν ∩ aµ

Put y′ = y, then proceed as in Subcase 1.1.Case 2. x, z ∈ aν \ aµ and y ∈ aµ \ aν .

Then z ≤ν y′ ≤µ y for some y′ ∈ aν ∩ aµ. Then there is t ∈ iνx, y′ such that z ≤ν t.Clearly t ≤ x, y. We show that t ∈ ix, y.

If t = y′ then t ≤ x, y and π(t) ∈ fx, y by (S1). Thus t ∈ ix, y.Assume that t <ν y′. Then π(t) ∈ fx, y′ ⊂ fx, y by (S2), because y′ ∈ aν ∩ aµ and

π(y′) < π(y). Thus t ∈ ix, y by (1.44).

Assume that G is a P-generic filter. We claim that if we take

= ∪≤p: p ∈ G.

then 〈ω2 × λ,〉 is a 〈λ〉ω2-poset with an 〈ω〉ω2

-stem. By standard density arguments, is apartial order on ω2 × λ which satisfies (T4). Moreover, every p ∈ P satisfies (2), so (T2) alsoholds. Finally the function

i =⋃ip : p ∈ G

witnesses (T3) because every p ∈ P satisfies (3.2).

1.8. Spaces constructed from strongly unbounded function


By using the combinatorial notion of the new ∆ property (NDP) of a function, it was provedby Roitman that the existence of an LCS∗ spaces with cardinal sequence 〈ω〉ω1

_〈ω2〉 is consistentwith ZFC (see [94] and [92]). (Actually, she considered superatomic Boolean algebras, but inthis thesis we mainly use the language of topology) It is worth to mention that [94] was the firstpaper in which such a special function was used to guarantee the chain condition of a certainposet. Roitman’s result was generalized in [66], where for every infinite regular cardinal κ, it wasproved that the existence of an LCS∗ space with cardinal sequence 〈κ〉κ+

_〈κ++〉 is consistentwith ZFC. Then, our aim here is to prove the following stronger result.

Theorem 1.8.1.Assume that κ, λ are infinite cardinals such that κ+++ ≤ λ, κ<κ = κ and2κ = κ+. Then for each ordinal η with κ+ ≤ η < κ++ and cf(η) = κ+, in some cardinal-preserving generic extension there is an LCS space with cardinal sequence 〈κ〉η _〈λ〉.

Corollary 1.8.2. 〈ω〉ω1

_〈ω3〉 ∈ C(ω + 1) is consistent with ZFC. 〈ω1〉ω2

_〈ω4〉 ∈ C(ω1 + 1) isalso consistent.

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1.8. SPACES CONSTRUCTED FROM STRONGLY UNBOUNDED FUNCTION 45

In order to prove Theorem 1.8.1, we shall use the main result of [75]. Assume that κ, λ areinfinite cardinals such that κ is regular and κ < λ. We say that a function F : [λ]2 → κ+ is aκ+-strongly unbounded function on λ iff for every ordinal δ < κ+, every cardinal ν < κ and everyfamily A ⊆ [λ]ν of pairwise disjoint sets with |A| = κ+, there are different a, b ∈ A such thatFα, β > δ for every α ∈ a and β ∈ b.Theorem* 1.8.3 (Koszmider, [75].). If κ, λ are infinite cardinals such that κ+++ ≤ λ, κ<κ = κand 2κ = κ+, then there is a κ-closed and cardinal-preserving partial order that forces the existenceof a κ+- strongly unbounded function on λ.

So, in order to prove Theorem 1.8.1 it is enough to show the following result.

Theorem 1.8.4.Assume that κ, λ are infinite cardinals with κ+++ ≤ λ and κ<κ = κ, and η isan ordinal with κ+ ≤ η < κ++ and cf(η) = κ+. Assume that there is a κ+- strongly unboundedfunction on λ. Then, there is a cardinal-preserving partial order that forces the existence of anLCS∗ space with cardinal sequence 〈κ〉η _〈λ〉.

In [66], [84], [94] and in many other papers, the authors proved the existence of certainLCS∗ space in such a way that instead of constructing the spaces directly, they actually producedcertain “graded posets” which guaranteed the existence of the wanted LCS∗ space. From theseconstructions, Bagaria, [21], extracted the following notion and proved the Lemma 1.8.6 belowwhich was implicitly used in many earlier papers.

Definition 1.8.5 ([21]).Given a sequence s = 〈κα : α < δ〉 of infinite cardinals, we say that aposet 〈T,≺〉 is an s-poset iff the following conditions are satisfied:(1) T =

⋃Tα : α < δ where Tα = α × κα for each α < δ.

(2) For each s ∈ Tα and t ∈ Tβ , if s ≺ t then α < β.(3) For every s, t ∈

[T]2 there is a finite subset is, t of T such that for each u ∈ T :

(u s ∧ u t) iff u v for some v ∈ is, t.(4) For α < β < δ, if t ∈ Tβ then the set s ∈ Tα : s ≺ t is infinite.Lemma 1.8.6 ([21, Lemma 1]). If there is an s-poset then there is an LCS∗ space with cardinalsequence s.

Actually, if T = 〈T,≺〉 is an s-poset, we write UT (x) = y ∈ T : y x for x ∈ T , and wedenote by XT the topological space on T whose subbase is the family

UT (x), T \ UT (x) : x ∈ T, (1.45)

then XT is a locally compact, Hausdorff, scattered space whose cardinal sequence is s.So, to prove Theorem 1.8.4 it will be enough to show that 〈κ〉η _〈λ〉-posets may exist for κ, η

and λ as above.

The organization of this section is as follows. In Subsection 2, we shall prove Theorem 1.8.4for the special case in which κ = ω and λ ≥ ω3, generalizing in this way the result proved byRoitman in [94]. In Subsection 1.8.2, we shall define the combinatorial notions that make theproof of Theorem 1.8.4 work. And in Section 1.8.3, we shall present the proof of Theorem 1.8.4.

1.8.1. Generalization of Roitman’s Theorem. In this subsection, our aim is to provethe following result.

Theorem 1.8.7. Let λ be a cardinal with λ ≥ ω3. Assume that there is an ω1-strongly unboundedfunction on λ. Then, in some cardinal-preserving generic extension for each ordinal η withω1 ≤ η < ω2 and cf(η) = ω1 there is an LCS∗ space with cardinal sequence 〈ω〉η _〈λ〉.

The theorem above is a bit stronger than Theorem 1.8.4 for κ = ω, because the genericextension does not depend on η. However, as we will see, its proof is much simpler than the proofof the general case.

By Lemma 1.8.6, it is enough to construct a c.c.c. poset P such that in V P for each η < ω2

with cf(η) = ω1 there is an 〈ω〉η _〈λ〉-poset.

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For η = ω1 it is straightforward to obtain a suitable P: all we need is to plug Kosmider’sstrongly unbounded function into the original argument of Roitman. For ω1 < η < ω2 this simpleapproach does not work, but we can use the “stepping-up” method of Er-rhaimini and Veličkovicfrom [41]. Using this method, it will be enough to construct a single 〈ω〉ω1

_〈λ〉-poset (with someextra properties) to obtain 〈ω〉η _〈λ〉-posets for each η < ω2 with cf(η) = ω1.

To start with, we adapt the notion of a skeleton introduced in [41] to the cardinal sequenceswe are considering.

Definition 1.8.8.Assume that T = 〈T,≺〉 is an s-poset such that s is a cardinal sequence of theform 〈κ〉µ _〈λ〉 where κ, λ are infinite cardinals with κ < λ and µ is a non-zero ordinal. Let i bethe infimum function associated with T . Then for γ < µ we say that Tγ , the γth-level of T , is abone level iff the following holds:

(1) is, t = ∅ for every s, t ∈ Tγ with s 6= t.(2) If x ∈ Tγ+1 and y ≺ x then there is a z ∈ Tγ with y z ≺ x.We say that T is a µ-skeleton iff Tγ is a bone level of T for each γ < µ.

The next statement can be proved by a straightforward modification of the proof of [41,Theorem 2.8].

Theorem 1.8.9. Let κ, λ be infinite cardinals. If there is a 〈κ〉κ+_〈λ〉-poset which is a κ+-

skeleton, then for each η < κ++ with cf(η) = κ+ there is a 〈κ〉η _〈λ〉-poset.

So, to get Theorem 1.8.7 it is enough to prove the following result.

Theorem 1.8.10. Let λ be a cardinal with λ ≥ ω3. Assume that there is an ω1- strongly un-bounded function on λ. Then, in some c.c.c. generic extension there is an 〈ω〉ω1

_〈λ〉-poset whichis an ω1-skeleton.

Let F : [λ]2 → ω1 be an ω1- strongly unbounded function on λ. In order to prove Theorem1.8.10, we shall define a c.c.c. forcing notion P = 〈P,≤〉 that adjoins an s-poset T = 〈T,〉 whichis an ω1-skeleton, where s is the cardinal sequence 〈ω〉ω1

_〈λ〉.So, the underlying set of the required s-poset is the set T =

⋃Tα : α ≤ ω1 where Tα =

α × ω for α < ω1 and Tω1 = ω1 × λ. If s = (α, ν) ∈ T , we write π(s) = α and ξ(s) = ν.

Then, we define the poset P = 〈P,≤〉 as follows. We say that p = 〈X,, i〉 ∈ P iff thefollowing conditions hold:

(P1) X is a finite subset of T .(P2) is a partial order on X such that s ≺ t implies π(s) < π(t).(P3) i : [X]2 → [X]<ω is an infimum function, that is, a function such that for every s, t ∈ [X]2

we have:∀x ∈ X([x s ∧ x t] iff x v for some v ∈ is, t).

(P4) If s, t ∈ X ∩ Tω1 and v ∈ is, t, then π(v) ∈ Fξ(s), ξ(t).(P5) If s, t ∈ X with π(s) = π(t) < ω1, then is, t = ∅.(P6) If s, t ∈ X, s ≺ t and π(t) = α+ 1, then there is a u ∈ X such that s u ≺ t and π(u) = α.

Now, we define ≤ as follows: 〈X ′,′, i′〉 ≤ 〈X,, i〉 iff X ⊆ X ′, =′ ∩(X ×X) and i ⊆ i′.

We will need condition (P4) in order to show that P is c.c.c.

Lemma 1.8.11.Assume that p = 〈X,, i〉 ∈ P , t ∈ X, α < π(t) and n < ω. Then, there is ap′ = 〈X ′,′, i′〉 ∈ P with p′ ≤ p and there is an s ∈ X ′ \ X with π(s) = α and ξ(s) > n suchthat, for every x ∈ X, s ′ x iff t ′ x.

Proof. Let L = α ∪ ξ : α < ξ < π(t) ∧ ∃j < ω ξ + j = π(t). Let α = α0, . . . , α` be theincreasing enumeration of L. Since X is finite, we can pick an sj ∈ Tαj \X with ξ(sj) > n forj ≤ `. Let X ′ = X ∪ sj : j ≤ ` and let

≺′=≺ ∪(sj , y) : j ≤ l, t y ∪ (sj , sk) : j < k ≤ `.

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Now, we put i′x, y = ix, y if x, y ∈ X, i′sj , y = sj if t y, i′sj , sk = smin(j,k), andi′sj , y = ∅ otherwise. Clearly, 〈X ′,′, i′〉 is as required.

Lemma 1.8.12. If P preserves cardinals, then P adjoins an 〈ω〉ω1

_〈λ〉-poset which is an ω1-skeleton.

Proof. Let G be a P-generic filter. We put p = 〈Xp,p, ip〉 for p ∈ G. By Lemma 10 andstandard density arguments, we have

T =⋃Xp : p ∈ G, (1.46)

and taking=

⋃p: p ∈ G, (1.47)

the poset 〈T,〉 is an 〈ω〉ω1

_〈λ〉-poset. Especially, Lemma 1.8.11 ensures that 〈T,〉 satisfies(4) in Definition 1.8.5. Properties (P5) and (P6) guarantee that 〈T,〉 is an ω1-skeleton.

Now, we prove the key lemma for showing that P adjoins the required poset.

Lemma 1.8.13.P is c.c.c.

Proof. Assume that R = 〈rν : ν < ω1〉 ⊆ P with rν 6= rµ for ν < µ < ω1. For ν < ω1, writerν = 〈Xν ,ν , iν〉 and put Lν = π[Xν ]. By the ∆-System Lemma, we may suppose that the setXν : ν < ω1 forms a ∆-system with root X∗. By thinning out R again if necessary, we mayassume that Lν : ν < ω1 forms a ∆-system with root L∗ in such a way that Xν ∩Tα = Xµ ∩Tαfor every α ∈ L∗ \ω1 and ν < µ < ω1. Without loss of generality, we may assume that ω1 ∈ L∗.Since β \ α is a countable set for α, β ∈ L∗ with α < β < ω1, we may suppose that L∗ \ ω1is an initial segment of Lν for every ν < ω1. Of course, this may require a further thinning outof R. Now, we put Zν = Xν ∩ Tω1 for ν < ω1. Without loss of generality, we may assume thatthe domains of the forcing conditions of R have the same size and that there is a natural numbern > 0 with |Zν \X∗| = |Zµ \X∗| = n for ν < µ < ω1. We consider in Tω1 the well-order inducedby λ. Then, by thinning out R again if necessary, we may assume that for every ν, µ ∈ [ω1]2

there is an order-preserving bijection h = hν,µ : Lν → Lµ with h L∗ = L∗ that lifts to anisomorphism of Xν with Xµ satisfying the following:

(A) For every α ∈ Lν \ ω1, h(α, ξ) = (h(α), ξ).(B) h is the identity on X∗.(C) For every i < n, if x is the ith-element in Zν \X∗ and y is the ith-element in Zµ \X∗, then

h(x) = y.(D) For every x, y ∈ Xν , x ν y iff h(x) µ h(y).(E) For every x, y ∈ [Xν ]2, h[iνx, y] = iµh(x), h(y).

Now, we deduce from condition (P4) and the fact that R is uncountable that if x, y ∈ [X∗]2

then iνx, y ⊆ X∗ for every ν < ω1. So if x, y ∈ [X∗]2, then iνx, y = iµx, y for ν < µ < ω1.

Let δ = max(L∗ \ ω1). Since F is an ω1-strongly unbounded function on λ, there areordinals ν, µ with ν < µ < ω1 such that if we put a = ξ ∈ λ : (ω1, ξ) ∈ Zν \ X∗ anda′ = ξ ∈ λ : (ω1, ξ) ∈ Zµ \ X∗, then Fξ, ξ′ > δ for every ξ ∈ a and every ξ′ ∈ a′. Ourpurpose is to prove that rν and rµ are compatible in P. We put p = rν and q = rµ. And wewrite p = 〈Xp,p, ip〉 and q = 〈Xq,q, iq〉. Then, we define the extension r = 〈Xr,r, ir〉 of pand q as follows. We put Xr = Xp ∪ Xq. We define r=p ∪ q. Note that r is a partialorder on Xr, because L∗ \ ω1 is an initial segment of π[Xp] and π[Xq] . Now, we define theinfimum function ir. Assume that x, y ∈ [Xr]2. We put irx, y = ipx, y if x, y ∈ Xp, andirx, y = iqx, y if x, y ∈ Xq. Suppose that x ∈ Xp \Xq and y ∈ Xq \Xp. Note that x, y arenot comparable in 〈Xr,r〉 and there is no u ∈ (Xp ∪Xq) \X∗ such that u r x, y. Then, wedefine irx, y = u ∈ X∗ : u ≺r x, y. It is easy to check that r ∈ P , and so r ≤ p, q.

After finishing the proof of Theorem 1.8.4 for κ = ω, try to prove it for κ = ω1. So, assumethat 2ω = ω1, ω4 ≤ λ, and there is an ω2-strongly unbounded function on λ. We want to find〈ω1〉η _〈λ〉-posets for each ordinal η < ω3 with cf(η) = ω2 in some cardinal-preserving generic

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extension. Since the “stepping-up” method of Er-rhaimini and Veličkovic worked for κ = ω, itis natural to try to apply Theorem 1.8.9 for the case κ = ω1. That is, we can try to find acardinal-preserving generic extension that contains an 〈ω1〉ω2

_〈λ〉-poset which is an ω2-skeleton.For this, first we should consider the forcing construction given in [66, Section 4] to add an〈ω1〉ω2

_〈ω3〉-poset, and then try to extend this construction to add the required ω2-skeleton.However, the construction from [66] is σ-complete and requires that CH holds in the groundmodel. Then, the following results show that the forcing construction of an 〈ω1〉ω2

_〈λ〉-posetwhich is an ω2-skeleton is quite hopeless, at least by using the standard forcing from [66].

If X is the topological space associated with a skeleton and x ∈ X, we denote by t(x,X) thetightness of x in X.

Proposition 1.8.14.Assume that T = 〈T,≺〉 is a µ-skeleton, α < µ and x ∈ Iα+1(XT ). Then,t(x,XT ) = ω.

Proof. Assume that A ⊆ T and x ∈ A′. We can assume that a ≺ x for each a ∈ A.Let

U = u ∈ Iα(XT ) : u ≺ x ∧ ∃au ∈ A au u. (1.48)Since y ≺ x iff y u for some u ≺ x with u ∈ Iα(XT ), the set U is infinite.

Pick V ∈[U]ω, and put B = av : v ∈ V . We claim that x ∈ B′. Indeed, if y ≺ x then

there is a u ∈ Iα(XT ) such that y u ≺ x. So |b ∈ B : b y| ≤ 1. Hence y /∈ B′. However,B has an accumulation point because B ⊆ UT (x) and UT (x) is compact in XT . So, B shouldconverge to x.

Corollary 1.8.15. If T is a µ-skeleton, then µ ≤ |I0(XT )|ω. Especially, under CH an 〈ω1〉ω2

_〈λ〉-poset can not be an ω2-skeleton.

Thus, we are unable to use Theorem 1.8.9 to prove Theorem 1.8.4 even for κ = ω1. Insteadof this stepping-up method, in the next two sections we will construct 〈ω1〉η _〈λ〉-posets directlyusing the method of orbits from [84]. This method was used to construct by forcing 〈ω1〉η-posetsfor ω2 ≤ η < ω3. It is not difficult to get an 〈ω1〉ω2

-poset by means of countable “approximations”of the required poset. However, for ω2 ≤ η < ω3 we need the notion of orbit and a much moreinvolved forcing to obtain 〈ω1〉η-posets (see [84]).

1.8.2. Combinatorial notions. In this section, we define the combinatorial notions thatwill be used in the proof of Theorem 1.8.4.

If α, β are ordinals with α ≤ β let

[α, β) = γ : α ≤ γ < β. (1.49)

We say that I is an ordinal interval iff there are ordinals α and β with α ≤ β and I = [α, β).Then, we write I− = α and I+ = β.

Assume that I = [α, β) is an ordinal interval. If β is a limit ordinal, let E(I) = εIν : ν <cf(β) be a cofinal closed subset of I having order type cf (β) with α = εI0, and then put

E(I) = [εIν , εIν+1) : ν < cf (β). (1.50)

If β = β′ + 1 is a successor ordinal, put E(I) = α, β′ andE(I) = [α, β′), β′. (1.51)

Now, for an infinite cardinal κ and an ordinal η with κ+ ≤ η < κ++ and cf(η) = κ+, wedefine Iη =

⋃In : n < ω where:

I0 = [0, η) and In+1 =⋃E(I) : I ∈ In. (1.52)

Note that Iη is a cofinal tree of intervals in the sense defined in [84]. So, the followingconditions are satisfied:(i) For every I, J ∈ Iη, I ⊆ J or J ⊆ I or I ∩ J = ∅.(ii) If I, J are different elements of Iη with I ⊆ J and J+ is a limit, then I+ < J+ .(iii) In partitions [0, η) for each n < ω.

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(iv) In+1 refines In for each n < ω.(v) For every α < η there is an I ∈ Iη such that I− = α.

Then, for each α < η and n < ω we define I(α, n) as the unique interval I ∈ In such thatα ∈ I. And for each α < η we define n(α) as the least natural number n such that there is aninterval I ∈ In with I− = α. So if n(α) = k, then for every m ≥ k we have I(α,m)− = α.

Assume that α < η. If m < n(α), we define om(α) = E(I(α,m)) ∩ α. Then, we define theorbit of α (with respect to Iη) as

o(α) =⋃om(α) : m < n(α). (1.53)

For basic facts on orbits and trees of intervals, we refer the reader to [84, Section 1]. Inparticular, we have |o(α)| ≤ κ for every α < η.

We write E([0, η)) = εν : ν < κ+.Claim 1.8.15.1. o(εν) = εζ : ζ < ν for ν < κ+.

Proof. Clearly I(εν , 0) = [0, η) and I(εν , 1) = [εν , εν+1). So n(εν) = 1. Thus o(εν) = o0(εν) =E(I(εν , 0)) ∩ εν = E([0, η)) ∩ εν = εζ : ζ < ν.

For α < β < η letj(α, β) = maxj : I(α, j) = I(β, j), (1.54)

and putJ(α, β) = I(α, j(α, β) + 1). (1.55)

For α < η letJ(α, η) = I(α, 1). (1.56)

Claim 1.8.15.2. If εζ ≤ α < εζ+1 ≤ β ≤ η, then J(α, β) = [εζ , εζ+1).

Proof. For β = η, J(α, β) = I(α, 1) = [εζ , εζ+1).Now assume that β < η. Since I(α, 0) = I(β, 0) = [0, η), but I(α, 1) = [εζ , εζ+1) and I(β, 1) =

[εξ, εξ+1) for some εξ with εζ+1 ≤ εξ, we have j(α, β) = 0 and so J(α, β) = [εζ , εζ+1).

1.8.3. Proof of the Main Theorem 1.8.4. Suppose that κ, λ are infinite cardinals withκ+++ ≤ λ and κ<κ = κ, η is an ordinal with κ+ ≤ η < κ++ and cf(η) = κ+, and there is aκ+-strongly unbounded function on λ. We will use a refinement of the arguments given in [84]and [66, Section 4].

First, we define the underlying set of our construction. For every ordinal α < η, we putTα = α × κ. And we put Tη = η × λ. We define T =

⋃Tα : α ≤ η. Let T<η = T \ Tη. If

s = (α, ν) ∈ T , we write π(s) = α and ξ(s) = ν.We put I = Iη. Also, we define E = E([0, η)) = εν : ν < κ+. Since there is a κ+-strongly

unbounded function on λ and cf(η) = κ+ there is a function F :[λ]2 → E such that

(?) For every ordinal γ < η and every family A ⊆ [λ]<κ of pairwise disjoint sets with|A| = κ+, there are different a, b ∈ A such that Fα, β > γ for every α ∈ a and β ∈ b.

Let Λ ∈ I and s, t ∈ [T ]2 with π(s) < π(t). We say that Λ isolates s from t iff Λ− < π(s) <Λ+ and Λ+ ≤ π(t).

Now we define the poset P = 〈P,≤〉 as follows. We say that p = 〈X,, i〉 ∈ P iff the followingconditions hold:(P1) X ∈ [T ]<κ.(P2) is a partial order on X such that s ≺ t implies π(s) < π(t).(P3) i :

[X]2 → X ∪ undef is an infimum function, that is, a function such that for every

s, t ∈[X]2 we have:

∀x ∈ X([x s ∧ x t] iff x is, t).

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(P4) If s, t ∈ X are compatible but not comparable in 〈X,〉, v = is, t and π(s) = α1,π(t) = α2 and π(v) = β, we have:(a) If α1, α2 < η, then β ∈ o(α1) ∩ o(α2).(b) If α1 < η and α2 = η, then β ∈ o(α1) ∩ E.(c) If α1 = η and α2 < η, then β ∈ o(α2) ∩ E.(d) If α1 = α2 = η, then β ∈ Fξ(s), ξ(t) ∩ E.

(P5) If s, t ∈ X with s t and Λ = J(π(s), π(t)) isolates s from t, then there is a u ∈ X suchthat s u t and π(u) = Λ+.

Now, we define ≤ as follows:⟨X ′,′, i′

⟩≤ 〈X,, i〉 iff X ⊆ X ′, =′ ∩(X ×X) and i ⊆ i′.

Lemma 1.8.16.Assume that p = 〈X,, i〉 ∈ P , t ∈ X, α < π(t) and ν < κ. Then, there is ap′ =

⟨X ′,′, i′

⟩∈ P with p′ ≤ p and there is an s ∈ X ′ \ X with π(s) = α and ξ(s) > ν such

that, for every x ∈ X, s ′ x iff t ′ x.

Proof. Since |X| < κ, we can take an s ∈ Tα \X with ξ(s) > ν. Let I0, . . . , In be the list ofall the intervals in I that isolate s from t in such a way that I+

0 > I+1 > · · · > I+

n . Put γi = I+i

for i ≤ n. We take points ci ∈ T \X with π(ci) = γi for i ≤ n. Let X ′ = X ∪ s ∪ ci : i ≤ nand let

≺′=≺ ∪〈s, ci〉 : i ≤ n ∪ 〈s, y〉 : t y ∪ 〈cj , ci〉 : i < j∪ 〈ci, y〉 : i ≤ n, t y.

Note that, for z ∈ X ′ and y ∈ s ∪ ci : i ≤ n, either z and y are comparable or they areincompatible with respect to ′. So, the definition of i′ is clear.

Finally, observe that p′ satisfies (P5) because if x ≺′ y with x ∈ s ∪ ci : i ≤ n, y ∈ X ′and J(π(x), π(y)) isolates x from y then either J(π(x), π(y)) = Ik for some 0 ≤ k ≤ n orJ(π(x), π(y)) = J(π(t), π(y)). But if J(π(x), π(y)) = Ik, then ck witnesses (P5) for x and y; andif J(π(x), π(y)) = J(π(t), π(y)), we are done by condition (P5) for p.

For p ∈ P we write p = 〈Xp,p, ip〉, Yp = Xp ∩ T<η and Zp = Xp ∩ Tη.Lemma 1.8.17. If P preserves cardinals, then forcing with P adjoins an LCS∗ space with cardinalsequence 〈κ〉η _〈λ〉.

Proof. Let G be a P-generic filter. Then

T =⋃Xp : p ∈ G, (1.57)

and taking=

⋃p: p ∈ G (1.58)

the poset 〈T,〉 is a 〈κ〉η _〈λ〉-poset. Especially, Lemma 1.8.16 guarantees that 〈T,≺〉 satisfies(4) from Definition 1.8.5. So, by Lemma 1.8.6, in V [G] there is an LCS∗ space with cardinalsequence 〈κ〉η _〈λ〉.

To complete our proof we should check that forcing with P preserves cardinals. It is straight-forward that P is κ-closed. The burden of our proof is to verify the following statement, whichcompletes the proof of Theorem 1.8.4.

Lemma 1.8.18.P has the κ+-chain condition.

Define the subposet Pη = 〈Pη,≤η〉 of P as follows:

Pη = p ∈ P : Xp ⊆ η × κ, (1.59)

and let ≤η=≤ Pη. The poset Pη was defined in [84, Definition 2.1], and it was proved thatPη satisfies the κ+-chain condition. In [84, Lemmas 2.5 and 2.6] it was shown that every set

R ∈[Pη]κ+

has a linked subset of size κ+. Actually, a stronger statement was proved, and we willuse that statement to prove Lemma 1.8.18. However, before doing so, we need some preparation.

Definition 1.8.19. Suppose that g : A → B is a bijection, where A,B ∈[T]<κ. We say that g

is adequate iff the following conditions hold:

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(1) g[A ∩ T<η] = B ∩ T<η and g[A ∩ Tη] = B ∩ Tη.(2) For every s, t ∈ A, π(s) < π(t) iff π(g(s)) < π(g(t)).(3) For every s = 〈α, ν〉 ∈ A ∩ T<η, g(α, ν) = (β, ζ) implies ν = ζ.(4) For every s, t ∈ A ∩ Tη, ξ(s) < ξ(t) iff ξ(g(s)) < ξ(g(t)).

For A,B ⊆ T<η, this definition is just [84, Definition 2.2].

Definition 1.8.20.A set Z ⊆ P is separated iff the following conditions are satisfied:

(1) Xp : p ∈ Z forms a ∆-system with root X.(2) For each α < η, either Xp ∩ Tα = X ∩ Tα for every p ∈ Z, or there is at most one p ∈ Z such

that Xp ∩ Tα 6= ∅.(3) For every p, q ∈ Z there is an adequate bijection hp,q : Xp → Xq which satisfies the following:

(a) For any s ∈ X, hp,q(s) = s.(b) If s, t ∈ Xp , then s ≺p t iff hp,q(s) ≺q hp,q(t).(c) If s, t ∈ Xp, then hp,q(ips, t) = iqhp,q(s), hp,q(t).For Z ⊆ Pη, this definition is just [84, Definition 2.3].

Lemma 1.8.21.Assume that Z ∈[P]κ+

is separated and X is the root of the ∆-system Xp :p ∈ Z. If s, t are compatible but not comparable in p ∈ Z and s ∈ X ∩ T<η, then ips, t ∈ X.

Proof. Assume that s, t are compatible but not comparable in p ∈ Z and s ∈ X ∩ T<η. Assumethat ips, t 6∈ X. Then since

iqs, hp,q(t) : q ∈ Z = hp,q(ips, t) : q ∈ Z, (1.60)

the elements of iqs, hp,q(t) : q ∈ Z are all different. But this is impossible, because π(iqs, hp,q(t)) ∈o(s) for all q ∈ Z and |o(s)| ≤ κ.

In [84, Lemmas 2.5 and 2.6], as we explain in the Appendix of this section, actually thefollowing statement was proved.

Proposition 1.8.22. For each subset R ∈[Pη]κ+

there is a separated subset Z ∈[R]κ+

and anordinal γ < η such that every p, q ∈ Z have a common extension r ∈ Pη such that the followingholds:

(R1) sup π[Xr \ (Xp ∪Xq)] < γ.(R2) (a) y ≺r s iff y ≺r hp,q(s) for each s ∈ Xp and y ∈ Xr \ (Xp ∪Xq),

(b) s ≺r y iff hp,q(s) ≺r y for each s ∈ Xp and y ∈ Xr \ (Xp ∪Xq),(c) if s ≺r y for s ∈ Xp ∪Xq and y ∈ Xr \ (Xp ∪Xq), then there is a w ∈ Xp ∩Xq with

s r w ≺r y,(d) for s ∈ Xp \Xq and t ∈ Xq \Xp,

s ≺r t iff ∃u ∈ Xp ∩Xq such that s ≺p u ≺q t, (1.61)t ≺r s iff ∃u ∈ Xp ∩Xq such that t ≺q u ≺p s.

After this preparation, we are ready to prove Lemma 1.8.18.

Proof of Lemma 1.8.18. We will argue in the following way. Assume that R = 〈rν : ν < κ+〉 ⊆P , where rν = 〈Xν ,ν , iν〉. For each ν < κ+ we will “push down” rν into Pη, more precisely, wewill construct an isomorphic copy r′ν ∈ Pη of rν . Using Proposition 1.8.22 we can find a separatedsubfamily r′ν : ν ∈ K of size κ+ and an ordinal γ < η such that for each ν, µ ∈ K with ν 6= µthere is a condition r′ν,µ ∈ Pη such that r′ν,µ ≤η r′ν , r′µ and (R1)–(R2) hold, especially

supπ[X ′ν,µ \ (X ′ν ∪X ′µ)] < γ. (1.62)

Let X be the root of Xν : ν < κ+, Y = X \ Tη and γ0 = max(γ, sup π[Y ]). Since F isκ+-strongly unbounded, there are ν, µ ∈ K with ν < µ such that

∀s ∈ (Xν \Xµ) ∩ Tη ∀t ∈ (Xµ \Xν) ∩ Tη Fξ(s), ξ(t) > γ0. (1.63)

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Then we will be able to “pull back” r′ = r′ν,µ into P to get a condition r = rν,µ which is a commonextension of rν and rµ. Let us remark that r will not be an isomorphic copy of r′, rather r willbe a “homomorphic image“ of r′.

Now we carry out our plan.Since κ<κ = κ, by thinning out our sequence we can assume that R itself is a separated set.

So Xr : r ∈ R forms a ∆-system with kernel X. We write Y = X ∩ T<η and Z = X ∩ Tη.Recall that E = E([0, η)) = εζ : ζ < κ+ is a closed unbounded subset of η.Fix ν < κ+. Write Yν = Xν ∩ T<η and Zν = Xν ∩ Tη. Pick a limit ordinal ζ(ν) < κ+ such

that:(i) sup(π[Yν ]) < εζ(ν),(ii) ζ(µ) < ζ(ν) for µ < ν.

Let θ = tp(ξ[Zν ]) and α = εζ(ν). We put Z ′ν = 〈α, ξ〉 : ξ < θ. Clearly, Z ′ν ⊆ Tεζ(ν) andtp(ξ[Z ′ν ]) = tp(ξ[Zν ]). We consider in Z ′ν and Zν the well-orderings induced by κ and λ respec-tively. Put X ′ν = Yν ∪ Z ′ν , and let gν : X ′ν → Xν be the natural bijection, i.e. gν Yν = id andgν(s) = t if for some ξ < tp(ξ[Zν ]) s is the ξ-element in Z ′ν and t is the ξ-element in Zν .

Let Z ′ν = g−1ν Z. We define the condition r′ν =

⟨X ′ν ,′ν , i′ν

⟩∈ Pη as follows: for s, t ∈ X ′ν

with s 6= t we put

s ≺′ν t iff gν(s) ≺ν gν(t), (1.64)and

i′νs, t = iνgν(s), gν(t). (1.65)

Claim 1.8.22.1. r′ν ∈ Pη.

Proof. (P1), (P2) and (P3) are clear because gν is an isomorphism between r′ν =⟨X ′ν ,′ν , i′ν

⟩and rν = 〈Xν ,ν , iν〉, moreover π(s) < π(t) iff π(gν(s)) < π(gν(t)).(P4) Since X ′ν ⊆ T<η we should check just (a). So assume that s′, t′ ∈ X ′ν are compatible butnot comparable in 〈X ′ν ,≤′ν〉 and v′ = i′νs′, t′. Put s = gν(s′), t = gν(t′). Since gν Yν = id, wecan assume that s′, t′ /∈

[Yν]2, e.g. s′ ∈ Z ′ν and so s ∈ Zν .

First observe that v′ ∈ Yν , so v′ = gν(v′).If t′ ∈ Yν , then t′ = gν(t′), and v′ = iνs, t′. By applying (P4)(c) in rν for s and t′ we obtain

π(v′) ∈ E ∩ o(π(t′)) ⊆ E ∩ εζ(ν) ∩ o(π(t′)) = o(π(s′)) ∩ o(π(t′)) (1.66)

because o(π(s′)) = E ∩ εζ(ν) by Claim 1.8.15.1.If t′ ∈ Z ′ν , then t = gν(t′) ∈ Zν ⊆ Tη. Since v′ = i′νs′, t′ = iνs, t, applying (P4)(d) in rν

for s and t we obtain

π(v′) ∈ Fξ(s), ξ(t) ∩ E ∩ εζ(ν) ⊆ E ∩ εζ(ν) = o(π(s′)) ∩ o(π(t′))

because o(π(s′)) = o(π(t′)) = E ∩ εζ(ν) by Claim 1.8.15.1.(P5) Assume that s′, t′ ∈ X ′ν , s′ ≺′ν t′ and Λ = J(π(s′), π(t′)) isolates s′ from t′. Then s′ ∈ Yν ,so gν(s′) = s′. Since gν Yν = id, we can assume that s′, t′ /∈

[Yν]2, i.e. t′ ∈ Z ′ν .

Write t = gν(t′). Since π(t′) = εζ(ν) ∈ E , by Claim 1.8.15.2, J(π(s′), π(t′)) = J(π(s′), π(t)) =[εζ , εζ+1) = I(π(s′), 1), where εζ ≤ π(s′) < εζ+1. Applying (P5) in rν for s′ and t, we obtain av ∈ Yν such that π(v) = Λ+ and s′ ≺ν v ≺ν t. Then gν(v) = v, so s′ ≺′ν v ≺′ν t′, which was to beproved.

Now applying Proposition 23 to the family r′ν : ν < κ+, there are K ∈[κ+]κ+

and γ < ηsuch that r′ν : ν ∈ K is separated and for every ν, µ ∈ K with ν 6= µ there is a commonextension r′ ∈ Pη of r′ν and r′µ such that (R1)-(R2) hold. Let γ0 = max(γ, sup π[Y ]). Recall thatY is the root of the ∆-system Yν : ν ∈ κ+. For ν < µ < κ+ we denote by h′ν,µ the adequatebijection hr′ν ,r′µ .

Since F satisfies (?), there are ν, µ ∈ K with ν 6= µ such that for each s ∈ (Zν \ Zµ) andt ∈ (Zµ \ Zν) we have

Fξ(s), ξ(t) > γ0. (1.67)

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We show that the conditions rν and rµ have a common extension r = 〈X,, i〉 ∈ P .Consider a condition r′ =

⟨X ′,′, i′

⟩which is a common extension of r′ν and r′µ and satisfies

(R1)–(R2). We define the condition r = 〈X,, i〉 as follows. Let

X = (X ′ \ (Z ′ν ∪ Z ′µ)) ∪ (Zν ∪ Zµ). (1.68)

Write U = X ′ \ (Z ′ν ∪ Z ′µ) = X \ (Zν ∪ Zµ) and V = X ′ \ (X ′ν ∪X ′µ). Clearly, V ⊆ U . We definethe function h : X ′ → X as follows:

h = gν ∪ gµ ∪ (id U). (1.69)

Then h is well-defined, h is onto, h X ′ \ (Z ′ν ∪ Z ′µ) is injective, and h[Z ′ν ] = h[Z ′µ] = Z.Now, if s, t ∈ X we put

s ≺ t iff there is a t′ ∈ X ′ with h(t′) = t and s ≺′ t′. (1.70)

Finally, we define the meet function i on[X]2 as follows:

is, t = max≺′i′s′, t′ : h(s′) = s and h(t′) = t. (1.71)

We will prove in the following claim that the definition of the function i is meaningful. Then, theproof of Lemma 1.8.18 will be complete as soon as we verify that r ∈ P and r ≤ rν , rµ.Claim 1.8.22.2. i is well-defined by (1.71), moreover i ⊇ iν ∪ iµ.

Proof. We need to verify that the maximum in (1.71) does exist when we define is, t. So,suppose that s, t ∈ [X]2.

If s, t ∈[X \ Z

]2 then there is exactly one pair (s′, t′) such that h(s′) = s and h(t′) = t, andhence there is no problem in (1.71). So if s, t ∈

[Xν

]2 then is, t = i′s′, t′ = iνs, t by theconstruction of r′ν . If s, t ∈

[Xµ

]2 proceeding similarly we obtain is, t = i′s′, t′ = iµs, t.So we can assume that e.g. s ∈ Z. Then h−1(s) = s′, s′′ for some s′ ∈ Z ′ν and s′′ ∈ Z ′µ.First assume that t /∈ Z, so there is exactly one t′ ∈ X ′ with h(t′) = t. We distinguish the

following cases.

Case 1. t ∈ V .Note that since t ∈ V , t = t′. We show that i′s′, t = i′s′′, t.Let v = i′s′, t. Assume that v ∈ X ′ν ∪X ′µ. Then, by (R2)(c), v ≺′ t and t ∈ V imply that

there is a w ∈ Y = X ′ν ∩X ′µ such that v ′ w ≺′ t. Thus v = i′s′, w and i′s′, w = i′νs′, w =iνs, w ∈ Y by Lemma 1.8.21 for w ∈ Y . Clearly, v ≺′ t, s′′. Hence v ′ i′s′′, t.

Now assume that v ∈ V . Then v ≺′ s′ implies v ≺′ h′ν,µ(s′) = s′′ by (R2)(a). So v ≺′ t, s′′,thus i′s′, t ′ i′s′′, t.

So, in both cases i′s′, t ′ i′s′′, t. But s′ and s′′ are symmetrical, hence i′s′′, t ′i′s′, t, and so we are done.

Case 2. t ∈ Xν \ Z.We show that in this case i′s′′, t′ ′ i′s′, t′.Let v = i′s′′, t′. If v ∈ V , then v ≺′ s′′ and h′ν,µ(s′) = s′′ imply v ≺′ s′ by (R2)(a). Thus

v ′ s′, t′, and so v ′ i′s′, t′.Now assume that v ∈ X ′ν ∪ X ′µ. If v ∈ Y = X ′ν ∩ X ′µ, then v ≺′ s′, so v ≺′ i′s′, t′. We

show that it is not possible that v /∈ Y . For this, assume that v ∈ (X ′ν ∪X ′µ) \ Y . Without lossof generality, we may suppose that v ∈ X ′ν \X ′µ. Then, by (R2)(d), there is a w ∈ Y such thatv ≺′ w ≺′ s′′. Thus v = i′w, t′ = i′νw, t′ ∈ Y by Lemma 1.8.21.

Moreover, s, t ∈[Xν

]2 and is, t = i′s′, t′ = iνs, t because gν(s′) = h(s′) = s andgν(t′) = h(t′) = t.

Case 3. t ∈ Xµ \ Z.Proceeding as in Case 2, we can show that i′s′, t′ ′ i′s′′, t′ = iµs, t.Finally, assume that t ∈ Z. Then h−1(t) = t′, t′′ for some t′ ∈ Z ′ν and t′′ ∈ Z ′µ.

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Note that by Cases (2) and (3),

i′s′′, t′ ′ i′s′, t′ and i′s′, t′′ ′ i′s′′, t′′.Since i′s′, t′ = iνs, t = iµs, t = i′s′′, t′′ by the construction of r′ν and r′µ, we have

i′s′, t′ = i′s′′, t′′ = max≺′

(i′s′, t′, i′s′′, t′, i′s′, t′′, i′s′′, t′′). (1.72)

Moreover, in this case s, t ∈[Xν

]2 ∩ [Xµ

]2 and we have just proved that is, t = iνs, t =iµs, t.

By Claim 1.8.22.2 above, r is well-defined. Since i ⊇ iν ∪ iµ, it is easy to check that if r ∈ Pthen r ≤ rν , rµ. So, the following claim completes the verification of the chain condition.

Claim 1.8.22.3. r ∈ P .

Proof. (P1) and (P2) are clear.(P3) Assume that s, t ∈ [X]2. Without loss of generality, we may assume that s, t are compatiblebut not comparable in 〈X,〉. Note that by (1.70), (1.71) and condition (P3) for r′, we haveis, t ≺ s, t. So, we have to show that if v ≺ s, t then v is, t.

Assume that v ≺ s, t. Then, v ∈ U and there are s′, t′ ∈ X ′ such that h(s′) = s, h(t′) = tand v ≺′ s′, t′. By (P3) for r′, v ′ i′s′, t′. Now as v, i′s′, t′, is, t ∈ U and h U = id, weinfer from (1.71) that v ′ i′s′, t′ ′ is, t and hence v is, t.(P4) Assume that s, t ∈ X are compatible but not comparable in 〈X,〉. Let v = is, t.(a) In this case π(s), π(t) < η. Then s, t ∈ X \ (Zν ∪ Zµ) = U , so h(s) = s and h(t) = t. Thusis, t = i′s, t. Hence, it follows from condition (P4)(a) for r′ that π(is, t) ∈ o(s) ∩ o(t).(b) In this case π(s) < η and π(t) = η. Then s ∈ X \ (Zν ∪ Zµ) = U and t ∈ Zν ∪ Zµ.

By (1.71) and Claim 1.8.22.2, there is a t∗ ∈ Z ′ν∪Z ′µ such that h(t∗) = t and is, t = i′s, t∗.Now, applying (P4)(a) for r′, we infer that π(v) ∈ o(s) ∩ o(t∗). Since π(t∗) ∈ E, we have

o(t∗) ⊆ E by Claim 1.8.15.1. Then we deduce that π(v) ∈ o(s) ∩ E, which was to be proved.(c) The same as (b).

(d) In this case π(s) = π(t) = η. If s, t ∈[Zν]2 then is, t = iνs, t, and by (P4)(d) for rν ,

we deduce that π(is, t) ∈ Fξ(s), ξ(t) ∩ E. A parallel argument works if s, t ∈ Zµ.So we can assume that s ∈ Zν \ Zµ and t ∈ Zµ \ Zν . Note that there are a unique s′ ∈ Z ′ν

with h(s′) = s and a unique t′ ∈ Z ′µ with h(t′) = t. Then, v = is, t = i′s′, t′ ∈ U . Henceeither v ∈ V , or v ∈ Xν ∪Xµ and in this case there is a w ∈ Xν ∩Xµ with v ′ w by (R2)(d).

In both cases π(v) ≤ γ0. Note that, applying (P4)(a) in r′ for s′, t′ and v = i′s′, t′, weobtain π(v) ∈ o(s′) ∩ o(t′). Since π(s′), π(t′) ∈ E we have o(s′) ∪ o(t′) ⊆ E by Claim 1.8.15.1.Thus π(v) ∈ E. And since π(v) ≤ γ0, we have π(v) ∈ Fξ(s), ξ(t) ∩E, which was to be proved.(P5) Assume that s, t ∈ X, s ≺ t and Λ = J(π(s), π(t)) isolates s from t. Then s /∈ Tη, soh(s) = s.

If t /∈ Tη then h(t) = t, so we are done because r′ satisfies (P5).Assume that t ∈ Tη. As s ≺ t, there is a t′ ∈ Tεζ(ν) ∪ Tεζ(µ) such that h(t′) = t and s ≺′ t′.

Since π(t′) ∈ E, by Claim 1.8.15.2 we have J(π(s), π(t′)) = I(π(s), 1) = J(π(s), π(t)). Applying(P5) in r′ for s and t′, we obtain a v ∈ X ′ such that s ≺′ v ′ t′ and π(v′) = Λ+. But asζ(ν), ζ(µ) are limit ordinals, we have v ≺′ t′, and hence v ∈ X ′ \ (Z ′ν ∪ Z ′µ) = U . Then h(v) = v,so s ≺ v ≺ t, which was to be proved.

Hence we have proved that P satisfies the κ+-chain condition, which completes the proof ofTheorem 1.8.4.

1.8.4. Appendix. We explain in detail how Proposition 23 was proved in [84].Assume that Z ⊆ Pη is a separated set. Let X be the root of Xp : p ∈ Z. For every n ∈ ω

and every I ∈ In with cf(I+) = κ+, we define ξ(I) = the least ordinal γ such that εIγ ⊇ π[X]∩ Iand we put γ(I) = εIξ(I)+κ. Now for every α < η, if there is an n < ω and an interval I ∈ Inwith cf(I+) = κ+ such that α ∈ I and γ(I) ≤ α, we consider the least natural number k withthis property and write I(α) = I(α, k). Otherwise, we write I(α) = α. Then we say that Z is

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1.9. REGULAR AND ZERO-DIMENSIONAL SPACES 55

pairwise equivalent iff for every p, q ∈ Z and every s ∈ Xp, I(π(s)) = I(π(hp,q(s))). In [84], thefollowing two lemmas were proved:

Lemma 1.8.23 ([84, Lemma 2.5]). Every set in[Pη]κ+

has a pairwise equivalent subset of sizeκ+.

Lemma 1.8.24 ([84, Lemma 2.6]).A pairwise equivalent set Z ⊆ Pη of size κ+ is linked.

To get Proposition 23 we explain that the proof of [84, Lemma 2.6] actually gives the followingstatement:

If Z ⊆ Pη is a pairwise equivalent set of size κ+, then there is an ordinal γ < η such thatevery p, q ∈ Z have a common extension r ∈ Pη satisfying (R1)-(R2).

As above, we denote by X the root of Xp : p ∈ Z. Assume that p, q ∈ Z with p 6= q.First observe that the ordering ≺r is defined in [84, Definition 2.4]. For this, adequate bijectionsg1 : Xr \ (Xp ∪Xq) → Xp \ X and g2 : Xr \ (Xp ∪Xq) → Xq \ X are considered in such a waythat g2 = hp,q g1. Then since g2 = hp,q g1, [84, Definition 2.4](b) and (c) imply (R2)(a) and[84, Definition 2.4](d) and (f) imply (R2)(b). Also, (R2)(c) follows directly from [84, Definition2.4](d) and (f), and (R2)(d) is just [84, Definition 2.4](e) and (g). So, we have verified (R2).

To check (R1), i.e. to get the right γ we need a bit more work. Let

J = I(π(s)) : s ∈ Xp (1.73)

where p ∈ Z. Since Z is pairwise equivalent, J does not depend on the choice of p ∈ Z. Forevery I ∈ Iη with cf(I+) = κ+ we can choose a set D(I) ∈ [E(I)∩ γ(I)]κ unbounded in γ(I). Weclaim that

γ = sup(⋃D(I) : I ∈ J ) + 1 (1.74)

works.First observe that γ < η, because cf(η) = κ+, |J | < κ and |D(I)| = κ for any I ∈ J .Now assume that p, q ∈ Z with p 6= q. Write Lp = π[Xp], Lq = π[Xq] and L = π[X].

Let αξ : ξ < δ and α′ξ : ξ < δ be the strictly increasing enumerations of Lp \ L and Lq \ Lrespectively. In the proof of [84, Lemma 2.6], for each ξ < δ an element βξ ∈ D(I(αξ)) = D(I(α′ξ))was chosen, and then a condition r ≤η p, q was constructed in such a way that Xr = Xp ∪Xq ∪Ywhere Y ∩ (Xp ∪Xq) = ∅ and π[Y ] = βξ : ξ < δ. Then since βξ : ξ < δ ⊆

⋃D(I) : I ∈ J ,

we infer that

supπ[Xr \ (Xp ∪Xq)] = supπ[Y ] < γ, (1.75)which was to be proved.

1.9. Regular and zero-dimensional spaces


Since the classes of the regular, of the zero-dimensional, and of the locally compact scatteredspaces are different, it was a natural question what is the relationship between their cardinalsequences. Actually, Juhász raised the question whether there is a regular space with cardinalsequence 〈ω〉2ω in ZFC. In [25, Theorem 10.1] the authors answered his question positively.

Theorem* 1.9.1. For each α < (2ω)+ there is a regular (even zero-dimensional) scattered spacewith cardinal sequence 〈ω〉α.

In [10] we succeeded in giving a complete characterization of the cardinal sequences of bothT3 and zero-dimensional T2 scattered spaces. Although the classes of the regular and of thezero-dimensional scattered spaces are different, it will turn out that they yield the same class ofcardinal sequences.

For any regular, scattered space X we have |X| ≤ 2|I0(X)|, hence for such a space X itscardinal sequence s satisfies length(s) < (2|I0(X)|)+ and s(α) ≤ 2s(β) whenever β < α. We shallshow below that these properties of a sequence s actually characterize the cardinal sequences ofregular scattered spaces.

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In [25], for each γ < (2ω)+, a 0-dimensional, scattered space of height γ and width ω wasconstructed. The next lemma generalizes that construction.

For an infinite cardinal κ, let Sκ be the following family of sequences of cardinals:

Sκ =〈κα : α < δ〉 : δ < (2κ)+, κ0 = κ and κ ≤ κα ≤ 2κ for each α < δ

.

Lemma 1.9.2. For any infinite cardinal κ and s ∈ Sκ there is a 0-dimensional scattered space Xwith CS(X) = s.

Proof. Let s = 〈κα : α < δ〉 ∈ Sκ. Write X =⋃α × κα : α < δ

. Since |X| ≤ 2κ we can fix

an independent family Fx : x ∈ X ⊂[κ]κ.

The underlying set of our space is X and and the topology τ on X is given by declaring foreach x = 〈α, ξ〉 ∈ X the set

Ux = x ∪ (α× Fx)to be clopen, i.e. Ux, X \ Ux : x ∈ X is a subbase for τ .

The space X is clearly 0-dimensional and T2.

Claim 1.9.2.1. If x = 〈β, ξ〉 ∈ U ∈ τ and α < β then U ∩ (α × κα) is infinite.

Proof of the claim. We can find disjoint sets A,B ∈[X \ x

]<ω such that

x ∈ Ux ∩⋂y∈A

Uy \⋃z∈B

Uz ⊂ U.

Observe that if 〈γ, ξ〉 ∈ A then β < γ. Thus

U ∩ (α × κα) ⊃ α ×( ⋂y∈A∪x

Fy \⋃z∈B

Fz

),

and the set on the right side is infinite because Fx : x ∈ X was chosen to be independent.

To complete our proof, by induction on α < κ, we verify that Iα(X) = α × κα, henceCS(X) = s. Assume that this is true for ν < α. If x ∈ α × κα then

Ux ∩(X \

⋃ν<α

Iν(X))

= x,

hence α × κα ⊂ Iα(X). On the other hand, if x = 〈β, ξ〉 ∈ X with β > α and U ∈ τ is aneighbourhood of x, then, by the claim above, U ∩ (α × κα) is infinite, hence x is not isolatedin X \

⋃ν<α Iν(X), i.e., x /∈ Iα(X). Thus Iα(X) = α × κα.

Theorem 1.9.3. For any sequence s of cardinals the following statements are equivalent:(1) s = CS(X) for some regular scattered space X,(2) s = CS(X) for some 0-dimensional scattered space X,(3) for some natural number m there are infinite cardinals κ0 > κ1 > · · · > κm−1 and for all

i < m sequences si ∈ Sκi such that s = s0_s1

_ . . ._sm−1 or s = s0_s1

_ . . ._sm−1_ 〈n〉

for some natural number n > 0.

Proof.(1)=⇒ (3)By induction on j we choose ordinals νj < ht(X) and cardinals κj such that ν0 = 0 and κ0 =| I0(X)|, moreover, for j > 0 with κj−1 infinite

νj = minν ≤ ht(X) : | Iν(X)| < κj−1,

and κj = | Iνj (X)|. We stop when κm is finite. For each j < m let δj = νj+1

.− νj . Then

the sequence sj =⟨| Iνj+δ(X)| : δ < δj

⟩is in Sκj . Thus CS(X) = s0

_s1_ . . ._sm−1 provided

κm = 0 (i.e. Iνm(X) = ∅) and CS(X) = s0_s1

_ . . ._sm−1_ 〈κm〉 when 0 < κm < ω.

(3)=⇒ (2)First we prove this implication for sequences s of the form s0

_s1_ . . ._sm−1 by induction on m.

If s ∈ Sκo then the statement is just lemma 1.9.2

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1.9. REGULAR AND ZERO-DIMENSIONAL SPACES 57

Assume now that s = s0_s1

_ . . ._sm−1, where κ0 > κ1 > · · · > κm−1 and si ∈ Sκi fori < m.

According to lemma 1.9.2 there is a 0-dimensional space Y with cardinal sequence sm−1.Using the inductive assumption we can also fix pairwise disjoint 0-dimensional topological spacesXy,n for 〈y, n〉 ∈ I0(Y )× ω, each having the cardinal sequence s′ = s0

_s1_ . . ._sm−2. We then

define the space Z = 〈Z, τ〉 as follows. Let

Z = Y ∪⋃Xy,n : y ∈ I0(Y ), n < ω.

A set U ⊂ Z is in τ iff(i) U ∩ Y is open in Y ,(ii) U ∩Xy,n is open in Xy,n for each 〈y, n〉 ∈ I0(Y )× ω,(iii) if y ∈ I0(Y ) ∩ U then there is m < ω such that

⋃Xy,n : m < n < ω ⊂ U .

If U is a clopen subset of Y and n < ω then it is easy to check that

Z(U, n) = U ∪⋃Xy,m : y ∈ I0(Y ) ∩ U, n < m < ω

is clopen in Z. Hence

B = Z(U, n) : U ⊂ Y is clopen, n < ω∪T : T is a clopen subset of some Xy,n

is a clopen base of Z and so Z is 0-dimensional.Let δ′ = length(s′) and δ = length(s).

Claim 1.9.3.1. Iα(Z) =⋃Iα(Xy,n) : 〈y, n〉 ∈ I0(Y )× ω for α < δ′.

Proof of the claim 1.9.3.1. Since Xy,n is an open subspace of Z it follows that Iα(Xy,n) ⊂Iα(Z). On the other hand,

Y ⊂⋃Iα(Xy,n) : 〈y, n〉 ∈ I0(Y )× ω

Z

,

hence Y ∩ Iα(Z) = ∅.

Since, by claim 1.9.3.1,Z \

⋃α<δ′

Iα(Z) = Y,

it follows that for δ′ ≤ α < δ we have

Iα(Z) = Iα.−δ′(Y ). (∗)

Thus Z =⋃α<δ Iα(Z), hence Z is a scattered space of height δ.

If α < δ′ then, by claim 1.9.3.1,

| Iα(Z)| = | I0(y)| · ω · s′(α) = κm−1 · ω · s′(α) = s′(α) = s(α).

If δ′ ≤ α < δ then, by (∗), | Iα(Z)| = | Iα.−δ′(Y )| = sm−1(α

.− δ′) = s(α), consequently CS(Z) = s.

Thus we proved the statement for sequences of the form s0_ . . ._sm−1.

If s = s0_ . . ._sm−1

_ 〈n〉 then writing s′ = s0_ . . ._sm−1 we can first find pairwise disjoint

0-dimensional scattered spaces Xi,m, 〈i,m〉 ∈ n× ω each having cardinal sequence s′. Let

Z = xi : i < n ∪⋃X〈i,m〉 : i < n,m < ω.

Declare a set U ⊂ Z open iff(i) U ∩Xi,m is open in Xi,m for each 〈i,m〉 ∈ n× ω,(ii) if xi ∈ U then there is ni < ω such that

⋃Xi,m : ni < m < ω ⊂ U .

Then Z is 0-dimensional, and

Iα(Z) = ⋃

Iα(Xi,m) : i < n,m < ω if α < length(s′),xi : i < n if α = length(s′).

Hence again Z is a scattered space with CS(Z) = s.

(2)=⇒ (1) is trivial.

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1.10. Initially ω1-compact spaces


Improving a result of M. Rabus we force a normal, locally compact, 0-dimensional, Frechet-Urysohn, initially ω1-compact and non-compact space X of size ω2 having the following property:for every open (or closed) set A in X we have |A| ≤ ω1 or |X \A| ≤ ω1.

1.10.1. Introduction. E. van Douwen and, independently, A. Dow [34] have observed thatunder CH an initially ω1-compact T3 space of countable tightness is compact. (A space X isinitially κ-compact if any open cover of X of size ≤ κ has a finite subcover, or equivalently anysubset of X of size ≤ κ has a complete accumulation point). Naturally, the question arose whetherCH is needed here, i.e. whether the same is provable just in ZFC. The question became evenmore intriguing when in [23] , D. Fremlin and P. Nyikos proved the same result from PFA. Quiterecently, A. V. Arhangel’skiıhas devoted the paper [20] to this problem, in which he has raisedmany related problems as well.

In [90] M. Rabus has answered the question of van Douwen and Dow in the negative. Heconstructed by forcing a Boolean algebra B such that the Stone space St(B) includes a coun-terexample X of size ω2 to the van Douwen–Dow question, in fact St(B) is the one point com-pactification of X, hence X is also locally compact. The forcing used by Rabus is closely relatedto the one due to J. Baumgartner and S. Shelah in [25], which had been used to construct athin very tall superatomic Boolean algebra. In particular, Rabus makes use of a ∆-function fwith some extra properties that are satisfied if f is obtained by the original, rather sophisticatedforcing argument of Shelah from [25].

In this section we give an alternative forcing construction of counterexamples to the vanDouwen–Dow question, which we think is simpler, more direct and more intuitive than the one in[90]. First of all, we directly force a topology τf on ω2 that yields an example from a ∆-function(with no extra properties) in the ground model which also satisfies CH. There is a wide variety ofsuch ground models since they are easily obtained when one forces a ∆-function or because ω1

implies the existence of a ∆-function (cf. [25]).Both in [25] and [90] the main use of the ∆-function f is to suitably restrict the partial order

of finite approximations to a structure on ω2 so as to become c.c.c. This we do as well, but inthe proof of the countable compactness of τf we also need the following simple result that yieldsan additional property of ∆-functions provided CH also holds.

Lemma 1.10.1 ([11]).Assume that CH holds, f is a ∆-function, cα : α < ω2 are pairwisedisjoint finite subsets of ω2 and B ∈

[ω2

]ω. Then for each n ∈ ω there are distinct ordinalsα0, α1, . . . , αn−1 ∈ ω2 such that

B ⊂⋂f(ξ, η) : ξ ∈ cαi , η ∈ cαj , i < j < n.

The following, even simpler, result about arbitrary functions f :[ω2

]2 −→ [ω2

]≤ω withfα, β ⊂ α ∩ β for α, β ∈

[ω2

]2 will also be needed.

Lemma 1.10.2. If f is a function as above then for each K,K ′ ∈[ω2

]≤ω there is a countable setclf (K,K ′) ⊂ ω2 such that(a) K ⊂ clf (K,K ′), supK = sup clf (K,K ′),(b) ∀ξ ∈ clf (K,K ′) ∀η ∈ clf (K,K ′) ∪K ′ (fξ, η ⊂ clf (K,K ′)).

The topology τf that we will construct on ω2 is right separated (in the natural order of ω2)and is also locally compact and 0-dimensional. Thus for each α ∈ ω2 one can fix a compact(hence closed) and open neighbourhood H(α) of α such that max H(α) = α. Conversely, if wecan fix for each α ∈ ω2 such a right-separating compact open neighbourhood H(α) then thefamily H(α) : α < ω2 determines the whole topology τ on ω2. In fact, using the notationU(α, b) = H(α) \

⋃H(β) : β ∈ b, it is easy to check that for each α ∈ ω2 the family Bα =

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1.10. INITIALLY ω1-COMPACT SPACES 59

U(α, b) : b ∈[α]<ω is a τ -neighbourhood base of α. Therefore, our notion of forcing consists

of finite approximations to a family H = H(α) : α < ω2 like above.Now, if H = H(α) : α < ω2 is as required and β < α < ω2 then either (i) β ∈ H(α)

or (ii) β /∈ H(α). If (i) holds then H(β) \ H(α), if (ii) holds then H(β) ∩ H(α) is a compactopen subset of β, hence there is a finite subset of β, call it iα, β, such that this set is coveredby H[iα, β] =

⋃H(γ) : γ ∈ iα, β It may come as a surprise, but the existence of such a

function i is also sufficient to insure that the collection H be as required. More precisely, we havethe following result.

Definition 1.10.3. If H = H(α) : α ∈ ω2 is a family of subsets of ω2 such that max H(α) = αfor each α ∈ ω2 then we denote by τH the topology on ω2 generated by H∪ ω2 \H : H ∈ H asa subbase.

Clearly, τH is a 0-dimensional, Hausdorff and right separated topology in which the elementsof H are clopen.

Theorem 1.10.4 ([11]).Assume that H is as in definition 1.10.3 above and there is a functioni :[ω2

]2 −→ [ω2

]<ω satisfying iα, β ⊂ α ∩ β for each α, β ∈[ω2

]2 such that if β < α thenβ ∈ H(α) implies H(β) \ H(α) ⊂ H[iα, β] and β /∈ H(α) implies H(β) ∩ H(α) ⊂ H[iα, β].Then each H(α) is compact in the topology τH, hence τH is locally compact.

It is now very natural to try to force a generic 0-dimensional, locally compact and rightseparated topology on ω2 by finite approximations (or pieces of information) of H and i. As wasalready mentioned, the ∆-function f comes into the picture when one wants to make this forcingc.c.c. The technical details of this are done in section 1.10.2.

We call the family H coherent if β ∈ H(α) implies H(β) ⊂ H(α). Clearly, this makes thingseasier because then H(β) \ H(α) = ∅, hence there is no problem covering it, the requirement oni is only that if β /∈ H(α) and β < α then H(β) ∩ H(α) ⊂ H[iα, β]. The original forcing ofBaumgartner and Shelah from [25] (when translated to scattered, i.e. right separated, locallycompact spaces rather than superatomic Boolean algebras) actually produced such a coherentfamily H. This is interesting because if H is coherent and τH is separable, which we have almostautomatically if H is obtained generically, then τH is also countably tight!

Theorem 1.10.5. If there is a coherent family H of right separating compact open sets for aseparable topology τ on ω2 then t(〈ω2, τ〉) = ω.

Proof. Let X = 〈ω2, τ〉. Then for each α ∈ ω2 we have t(α,X) = t(α,H(α)), hence it sufficesto prove t(α,H(α)) = ω. If we had t(α,H(α)) = ω1 then we could find A ⊂ H(α) such thatα ∈ A but α /∈ B for every countable B ⊂ A. Next we can find a strictly increasing sequenceS = xν : ν < ω1 in A such that xν > supFν , where Fν = xµ : µ < ν. This construction canbe carried out as supFν < α because α /∈ Fν , Fν is compact and every initial segment of H(α)is open and so one of them should cover Fν . Now for each ν < ω1 there is a finite subset bν ofFν such that Fν ⊂ H[bν ]. If µ < ν then bµ ⊂ Fµ ⊂ Fν ⊂ H[bν ] implies that for each β ∈ bµ wehave H(β) ⊂ H[bν ] by the coherence of H, hence H[bµ] ⊂ H[bν ]. But xµ ∈ H[bν ] \ H[bµ] for eachµ < ν ∈ ω1, hence the H[bν ]’s yield a strictly increasing ω1-sequence of clopen sets in a separablespace, which is a contradiction completing the proof.

Ironically, this general result that gives countable tightness so easily cannot be used in ourconstruction because we had to abandon the coherency of H in our effort to insure countablecompactness (implied by the initial ω1-compactness) of τH.

We mentioned above that our examples, by genericity, are separable. But this is not acoincidence. It is well-known and very easy to prove that if X is an initially ω1-compact spacethen t(X) ≤ ω implies that X has no uncountable free sequence. (Moreover, if X is T3 theconverse of this is also true.) Hence the following easy, but perhaps not widely known, resultimmediately implies that any non-compact, initially ω1-compact space of countable tightnesscontains a countable subset whose closure is not compact. Thus if there is a counterexample tothe van Douwen–Dow question then there is also a separable one.

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Lemma 1.10.6 ([11]). If Y is a non-compact topological space, then for some ordinal µ the spaceY contains a free sequence yξ : ξ < µ ⊂ Y with non-compact closure.

Note that under CH the weight of a separable T3 space is ≤ ω1, and an initially ω1-compactspace of weight ≤ ω1 is compact, hence the CH result of van Douwen and Dow is a trivialconsequence of 1.10.6. Arhangel’skii raised the question, [20, problem 3], whether in this CHcan be weakened to 2ω < 2ω1? We shall answer this question in the negative: theorem 1.10.34implies that the existence of a counterexample to the van Douwen–Dow question is consistentwith practically any cardinal arithmetic that violates CH.

In [20, problem 17] Arhangel’skii asked if it is provable in ZFC that an initially ω1-compactsubspace of a T3 space of countable tightness is always closed. (Clearly this is so under CH orPFA, or in general if the answer to the van Douwen–Dow question is “yes”.) In view of our nextresult both Rabus’ and our spaces give a negative answer to this question. More generally wehave the following result.

Theorem 1.10.7. If X is a locally compact counterexample to van Douwen–Dow then the one-point compactification αX = X ∪p of X also has countable tightness. On the other hand, X isan initially ω1-compact non-closed subset of αX.

Proof. Let A ⊂ X be such that p ∈ A ( i.e. AX

is not compact). By lemma 1.10.6 and ourpreceding remark then there is a countable set S ⊂ A

Xsuch that S

Xis not compact. But by

t(X) = ω then there is a countable T ⊂ A for which S ⊂ TX, hence T

Xis non-compact as well,

so p ∈ T . Consequently we have t(p, αX) = ω and so t(αX) = ω.

1.10.2. The forcing construction. The following notation will be used in the definitionof the poset Pf . Given a function h and a ⊂ dom(h) we write h[a] = ∪h(ξ) : ξ ∈ a. Givennon-empty sets x and y of ordinals with supx 6= sup y let

x ∗ y =

x ∩ y if supx /∈ y and sup y /∈ x,x \ y if supx ∈ y,y \ x if sup y ∈ x.

Definition 1.10.8. For each function f :[ω2

]2 −→ [ω2

]≤ω satisfying f(α, β) ⊂ α ∩ β for anyα, β ∈

[ω2

]2 we define a poset Pf = 〈Pf ,≤〉 as follows. The underlying set of Pf is the familyof triples p = 〈a, h, i〉 for which

(i) a ∈[ω2

]<ω, h : a −→ P(a) and i :[a]2 −→ P(a) are functions,

(ii) maxh(ξ) = ξ for each ξ ∈ a ,(iii) i(ξ, η) ⊂ f(ξ, η) for each ξ, η ∈

[a]2,

(iv) h(ξ) ∗ h(η) ⊂ h[i(ξ, η)] for each ξ, η ∈[a]2.

We will often write p = 〈ap, hp, ip〉 for p ∈ Pf .For p, q ∈ Pf let p ≤ q if ap ⊃ aq, hp(ξ) ∩ aq = hq(ξ) for ξ ∈ aq, and ip ⊃ iq. If p ∈ Pf , α ∈ ap,b ⊂ ap ∩ α, let us write up(α, b) = hp(α) \ hp[b].Lemma 1.10.9. For each α < ω2 the set Dα = p ∈ Pf : α ∈ ap is dense in Pf .

Definition 1.10.10. If G is a Pf -generic filter over V , in V [G] we can define the topological spaceXf [G] = Xf = 〈ω2, τf 〉 as follows. For α ∈ ω2 put H(α) =

⋃hp(α) : p ∈ G ∧ α ∈ ap, let

H = H(α) : α < ω2 and let τf = τH as defined in 1.10.3, that is, τf is the topology on ω2

generated by H ∪ ω2 \H : H ∈ H as a subbase.

If G is a Pf -generic filter over V then by lemma 1.10.9 we have⋃ap : p ∈ G = ω2, and

for each α < ω2 max H(α) = α and H(α) is clopen in Xf . Thus Xf is 0-dimensional and rightseparated. Of course, neither f nor i is needed for this. As was explained in section 1.10.1, weneed f to be a ∆-function in order to make Pf c.c.c (which insures that no cardinal is collapsed),and the function i is used to make Xf also locally compact.

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Theorem 1.10.11. If CH holds and f is a ∆-function, then Pf satisfies the c.c.c and V Pf |=“Xf = 〈ω2, τf 〉 is a 0-dimensional, right separated, locally compact space having the followingproperties:

(i) t(Xf ) = ω,(ii) ∀A ∈ [ω2]ω1 ∃α ∈ ω2 |A ∩H(α)| = ω1

(iii) ∀A ∈ [ω2]ω (A is compact or |ω2 \A| ≤ ω1).”Consequently, in V Pf , Xf is a locally compact, normal, countably tight, initially ω1-compact butnon-compact space.

Proof of theorem 1.10.11. To show that Pf satisfies c.c.c we will proceed in the following way.We first formulate when two conditions p and p′ from Pf are called good twins (definition 1.10.12),then we construct the amalgamation r = p+ p′ of p and p′ (definition 1.10.13) and show that r isa common extension of p and p′ in Pf . Finally we prove in lemma 1.10.15 that every uncountablefamily of conditions contains a couple of elements which are good twins.

Definition 1.10.12. Let p = 〈a, h, i〉 and p′ = 〈a′, h′, i′〉 be from Pf . We say that p and p′ aregood twins provided(1) p and p′ are twins, i.e., |a| = |a′| and the natural order-preserving bijection e = ep,p′ between

a and a′ is an isomorphism between p and p′:(i) h′(e(ξ)) = e′′h(ξ) for each ξ ∈ a,(ii) i′(e(ξ), e(η)) = e′′i(ξ, η) for each ξ, η ∈

[a]2 ,

(iii) e(ξ) = ξ for each ξ ∈ a ∩ a′,(2) i(ξ, η) = i′(ξ, η) for each ξ, η ∈

[a ∩ a′

]2,(3) a and a′ are good for f .

Let us remark that, in view of (ii) and (iii), condition 2 can be replaced by “i(ξ, η) ⊂ a ∩ a′

for each ξ, η ∈[a ∩ a′

]2”.Definition 1.10.13. If p = 〈a, h, i〉 and p′ = 〈a′, h′, i′〉 are good twins we define the amalgamationr = 〈b, g, j〉 of p and p′ as follows:

Let b = a ∪ a′. For ξ ∈ h[a ∩ a′] ∪ h′[a ∩ a′] defineδξ = minδ ∈ a ∩ a′ : ξ ∈ h(δ) ∪ h′(δ).

Now, for any ξ ∈ b let

g(ξ) =

h(ξ) ∪ h′(ξ) if ξ ∈ a ∩ a′,h(ξ) ∪ η ∈ a′ \ a : δη ∈ h(ξ) if ξ ∈ a \ a′,h′(ξ) ∪ η ∈ a \ a′ : δη ∈ h′(ξ) if ξ ∈ a′ \ a.

(•)

Finally for ξ, η ∈[b]2 let

j(ξ, η) =

i(ξ, η) if ξ, η ∈ a,i′(ξ, η) if ξ, η ∈ a′,f(ξ, η) ∩ b otherwise.

(••)

(Observe that j is well-defined because 1.10.12.(2) holds.)We will write r = p+ p′ for the amalgamation of p and p′.

Lemma 1.10.14. If p and p′ are good twins then their amalgamation, r = p + p′, is a commonextension of p and p′ in Pf .

Proof. First we prove two claims.

Claim 1.10.14.1. Let η ∈ a and δ ∈ a ∩ a′. Then η ∈ h(δ) if and only if δη is defined andδη ∈ h(δ). (Clearly, we also have a symmetric version of this statement for η ∈ a′.)

Proof of claim 1.10.14.1. Assume first η ∈ h(δ). Then δη is defined and clearly δη ∈ h(δ) ifδη = δ. So assume δη 6= δ. Since i(δη, δ) ⊂ a ∩ a′ and max i(δη, δ) < δη we have η /∈ h[i(δη, δ)] bythe choice of δη. Thus from p ∈ Pf we have

η /∈ h(δη) ∗ h(δ). (†)

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Then h(δη) ∗ h(δ) 6= h(δη) ∩ h(δ) by (†). Since η ∈ h(δ) implies δη < δ, we actually haveh(δη) ∗ h(δ) = h(δη) \ h(δ). Thus δη ∈ h(δ) by the definition of the operation ∗.

On the other hand, if δη ∈ h(δ), then either δη = δ or h(δη) ∗ h(δ) = h(δη) \ h(δ). Thusη ∈ h(δ) because in the latter case again η /∈ h[i(δη, δ)], hence (†) holds.

Claim 1.10.14.2. If ξ ∈ a ∩ a′ theng(ξ) = h(ξ) ∪ η ∈ a′ \ a : δη ∈ h(ξ) = h′(ξ) ∪ η ∈ a \ a′ : δη ∈ h′(ξ).

Proof of claim 1.10.14.2. Conditions 1.10.12.1(i) and (iii) imply h(ξ) ∩ a ∩ a′ = h′(ξ) ∩ a ∩ a′and so

g(ξ) = h(ξ) ∪ h′(ξ) = h(ξ) ∪ ((a′ \ a) ∩ h′(ξ)).By claim 1.10.14.1 we have

(a′ \ a) ∩ h′(ξ) = η ∈ a′ \ a : δη ∈ h′(ξ).But by 1.10.12.(1) we have δη ∈ h(ξ) iff δη ∈ h′(ξ), hence it follows that

g(ξ) = h(ξ) ∪ η ∈ a′ \ a : δη ∈ h(ξ).The second equality follows analogously.

Next we check r ∈ Pf . Conditions 1.10.8.(i)–(iii) for r are clear by the construction. So weshould verify 1.10.8.(iv).

Let ξ 6= η ∈ b and α ∈ g(ξ) ∗ g(η). We need to show that α ∈ g [j(ξ, η)]. We will distinguishseveral cases.

Case 5. ξ, η ∈ a ( or ξ, η ∈ a′ ).

Since g(ξ) ∩ a = h(ξ) and g(η) ∩ a = h(η) we have (g(ξ) ∗ g(η)) ∩ a = h(ξ) ∗ h(η) by thedefinition of operation ∗. Thus (g(ξ) ∗ g(η)) ∩ a ⊂ h[i(ξ, η)] = g [j(ξ, η))] ∩ a ⊂ g [j(ξ, η)]. So wecan assume that α ∈ a′ \ a. We know that δα is defined because α ∈ g(ξ) ∪ g(η) is also satisfied.Since α ∈ g(ξ) iff δα ∈ h(ξ) and α ∈ g(η) iff δα ∈ h(η) by (•) it follows that δα ∈ h(ξ) ∗ h(η).Thus there is ν ∈ i(ξ, η) such that δα ∈ h(ν). But i(ξ, η) = j(ξ, η) and α ∈ g(ν) by (•). Thusα ∈ g [j(ξ, η)].

Case 6. ξ ∈ a \ a′ and η ∈ a′ \ a.We can assume that α ∈ a, since the α ∈ a′ case is done symmetrically.

Subcase 6.1. g(ξ) ∗ g(η) = g(η) \ g(ξ).

Then α ∈ g(η) and η ∈ g(ξ) so δα and δη are both defined and δα ∈ h′(η), δη ∈ h(ξ)hold, hence α ≤ δα < η < δη < ξ. But a and a′ are good for f , so by 1.1.10(a) we haveδα ∈ f(η, ξ) ∩ b = j(ξ, η). Thus α ∈ h(δα) ⊂ g(δα) ⊂ g [j(ξ, η)] which was to be proved.

Subcase 6.2. g(ξ) ∗ g(η) = g(ξ) ∩ g(η) or g(ξ) ∗ g(η) = g(ξ) \ g(η).

Since now α ∈ g(ξ) ∗ g(η) ⊂ g(ξ), by the definition of the operation ∗ we have

|α, ξ ∩ g(η)| = 1. (1.76)

Thus, by the definition of g(η), δ∗ = minδ ∈ a ∩ a′ : α ∈ h(δ) ∨ ξ ∈ h(δ) is well-defined andδ∗ < η. If δ∗ < ξ then α ∈ h(δ∗) and by 1.1.10(a) we have δ∗ ∈ f(ξ, η) ∩ b = j(ξ, η) for a and a′are good for f , and so α ∈ g [j(ξ, η)].

Thus we can assume ξ < δ∗. We know that δ∗ = δα or δ∗ = δξ by the choice of δ∗, butδα = δξ is impossible by (1.76). Thus

|α, ξ ∩ h(δ∗)| = 1. (1.77)

Since α ∈ g(ξ) implies α ∈ h(ξ) and we have ξ < δ∗, (1.77) implies α ∈ h(ξ) ∗ h(δ∗) and soα ∈ h[i(ξ, δ∗)]. But i(ξ, δ∗) ⊂ f(ξ, δ∗) ⊂ f(ξ, η) because a and a′ are good for f , so 1.1.10(b) or(c) may be applied. Consequently, we have i(ξ, δ∗) ⊂ j(ξ, η) by (••). Hence α ∈ g [j(ξ, η)] whichwas to be proved.

Since we investigated all the cases it follows that r satisfies 1.10.8.(iv), that is, r ∈ Pf . Sincer ≤ p, q are clear from the construction, the lemma is proved.

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Lemma 1.10.15. Every uncountable family F of conditions in Pf contains a couple of elementswhich are good twins. Consequently, Pf satisfies c.c.c.

Proof. By standard counting arguments F contains an uncountable subfamily F ′ such that everypair p 6= p′ ∈ F ′ satisfies 1.10.12.(1)-(2). But f is a ∆-function, so there are p 6= p′ ∈ F ′ suchthat ap and ap

′are good for f , i.e. p and p′ satisfies 1.10.12.(3), too. In other words, p and p′

are good twins and so r = p+ p′ is a common extension of p and p′ in Pf .

Let G be a Pf -generic filter over V . As in definition 1.10.10, let H(α) =⋃hp(α) : p ∈ G∧α ∈

ap for α ∈ ω2, and let τf be the topology on ω2 generated by H(α) : α ∈ ω2∪ ω2 \H(α) : α ∈ω2 as a subbase. Put i =

⋃ip : p ∈ G.

Since Xf is generated by a clopen subbase and max(H(α)) = α for each α ∈ ω2 by 1.10.8(ii),it follows that Xf is 0-dimensional and right separated in its natural well-order.

The following proposition is clear by 1.10.8.(iv) and by the definition of H and i.

Proposition .H(α) ∗ H(β) ⊂ H[i(α, β)] for α < β < ω2. So by 1.10.4 every H(α) is a compactopen set in Xf .

Definition 1.10.16. For α ∈ ω2 and b ∈[α]<ω let

U(α, b) = H(α) \H[b]

and letBα = U(α, b) : b ∈

[α]<ω.

By theorem 1.10.4 every H(α) is compact and Bα is a neighborhood base of α in Xf . ThusXf is locally compact and the neighbourhood base Bα of α consists of compact open sets.

Unfortunately, the family H = H(α) : α < ω2 is not coherent, so we can’t apply theorem1.10.5 to prove that Xf is countably tight. It will however follow from the following result.

Lemma 1.10.17. In V Pf , if a sequence zζ : ζ < ω1 ⊂ H(β) converges to β, then there is someξ < ω1 such that β ∈ zζ : ζ < ξ.

Proof. Assume on the contrary that for each ξ < ω1 we can find a finite subset bξ ⊂ β such thatzζ : ζ < ξ ∩U(β, bξ) = ∅, that is, zζ : ζ < ξ ⊂ H[bξ].

Fix now a condition p ∈ Pf which forces the above described situation and decides the valueof β. Then, for each ξ < ω1 we can choose a condition pξ ≤ p which decides the value of zξ andbξ. We can assume that apξ : ξ < ω1 forms a ∆-system with kernel D, zξ ∈ apξ \D and thatzξ < zη for ξ < η < ω1.

Claim .Assume that ξ < η < ω1, pξ and pη are good twins and r = pξ + pη. Then r “zξ ∈H[D ∩ β]”.

Proof of the claim. Indeed, zξ ∈ hr[bη] because r ≤ pξ, pη. Since zξ ∈ apξ \ apη , (•) andclaim 1.10.14.2 imply that zξ ∈ hr[bη] holds if and only if δzξ ∈ D = apξ ∩ apη is defined andδzξ ∈ hpη [bη]. Since bη ⊂ β, we also have δzξ < β and so zξ ∈ hr[D ∩ β].

Applying lemma 1.10.15 to appropriate final segments of pξ : ξ < ω1 we can choose, by inductionon µ < ω1, pairwise different ordinals µ < ξµ < ηµ < ω1 with ηµ < ξν if µ < ν such that pξµ andpηµ are good twins. Let rµ = pξµ + pηµ . Since Pf satisfies c.c.c there is a condition q ≤ p suchthat q “ |µ ∈ ω1 : rµ ∈ G| = ω1”. Thus, by the claim q “ |zξ : ξ < ω1 ∩H[D ∩ β]| = ω1”, i.e.the neighbourhood U(β,D ∩ β) of β misses uncountably many of the points zξ which contradictsthat β is the limit of this sequence.

Corollary 1.10.18. t(Xf ) = ω.

Proof. Assume on the contrary that α ∈ ω2 and t(α,Xf ) = t(α,H(α)) = ω1. Then there is anω-closed set Y ⊂ H(α) that is not closed. Since the subspace H(α) is compact and right separatedand so it is pseudo-radial , for some regular cardinal κ there is a sequence zξ : ξ < κ ⊂ Y whichconverges to some point β ∈ H(α)\Y . Since Y is ω-closed and |Y | ≤ |H(α)| = ω1 we have κ = ω1.By lemma 1.10.17 there is some ξ < ω1 with β ∈ zζ : ζ < ξ ⊂ Y contradicting β /∈ Y .

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Lemma 1.10.19. In V Pf , for each uncountable A ⊂ Xf there is β ∈ ω2 such that |A∩H(β)| = ω1.

Proof. Assume that p “A = αξ : ξ < ω1 ∈[ω2

]ω1 ”. For each ξ < ω1 pick pξ ≤ p andαξ ∈ ω2 such that pξ αξ = αξ. Since Pf satisfies c.c.c we can assume that the αξ are pairwisedifferent. Let supαξ : ξ < ω1 < β < ω2. Now for each ξ < ω1 define the condition qξ ≤ pξ bythe stipulations aqξ = apξ ∪ β, hqξ(β) = aqξ and iqξ(ν, β) = ∅ for ν ∈ apξ . Then qξ ∈ Pf andqξ αξ ∈ H(β). But Pf satisfies c.c.c, so there is q ≤ p such that q “|ξ ∈ ω1 : qξ ∈ G| = ω1”.Thus q “|A ∩H(β)| = ω1.”

Since every H(β) is compact, lemma 1.10.19 above clearly implies that Xf is ω1-compact, i.e.every subset S ⊂ Xf of size ω1 has a complete accumulation point.

Now we start to work on (iii): in V Pf the closure of any countable subset Y of Xf is eithercompact or it contains a final segment of ω2. If Y is also in the ground model, then actually thesecond alternative occurs and this follows easily from the next lemma.

Lemma 1.10.20. If p ∈ Pf , β ∈ ap, b ⊂ ap ∩ β, α ∈ β \ ap, then there is a condition q ≤ p suchthat α ∈ uq(β, b).

Proof. Define the condition q ≤ p by the following stipulations: aq = ap ∪ α, hq(α) = α,

hq(ν) =hp(ν) ∪ α if β ∈ hp(ν)hp(ν) if β /∈ hp(ν)

for ν ∈ ap, and let iq ⊃ ip and iq(α, ν) = ∅ for ν ∈ ap.To show q ∈ Pf we need to check only 1.10.8(iv). Assume that α ∈ hq(ν)∗hq(µ). Then by the

construction of q we have β ∈ hp(ν) ∗ hp(µ). Thus there is ξ ∈ ip(ν, µ) with β ∈ hp(ξ). But thenα ∈ hq(ξ), so by iq(ν, µ) = ip(ν, µ) we have α ∈ hq[iq(ν, µ)]. In view of hp(ν)∗hp(µ) ⊂ hp[ip(ν, µ)]we are done. Thus q ∈ Pf , q ≤ p and clearly α ∈ uq(β, b), so we are done.

This lemma yields the following corollary.

Corollary 1.10.21. If Z ∈[ω2

]ω ∩ V and β ∈ ω2 \ supZ then β ∈ Z.

Proof. Let U(β, b) be a neighbourhood of β, b ∈[β]<ω. Since p “U(β, b) ⊃ up(β, b)” for each

p ∈ Pf and the setDβ,b,Z = q ∈ Pf : uq(β, b) ∩ Z 6= ∅

is dense in Pf by the previous lemma, it follows that U(β, b) intersects Z. Consequently β ∈ Z.

The space Xf is right separated, i.e. scattered, so we can consider its Cantor-Bendixsonhierarchy. According to corollary 1.10.21 for each α < ω2 the set Aα = [ωα, ωα + ω) is a denseset of isolated points in Xfd(ω2 \ ωα). Thus the αth Cantor-Bendixson level of Xf is just Aα.Therefore Xf is a thin very tall, locally compact scattered space in the sense of [93]. Let usemphasize that CH was not needed to get this result, hence we have also given an alternativeproof of the main result of [25].

Now we continue to work on proving property 1.10.11(iii) of Xf .Given p ∈ Pf and b ⊂ ap let pdb =

⟨b, h, ipd

[b]2⟩ where h is the function with dom(h) = b

and h(ξ) = hp(ξ) ∩ b for ξ ∈ b. Let us remark that pdb in not necessarily in Pf . In fact, pdb ∈ Pfif and only if ip(ξ, η) ⊂ b for each ξ 6= η ∈ b. Especially, if b is an initial segment of ap, thenpdb ∈ Pf . The order ≤ of Pf can be extended in a natural way to the restrictions of conditions:if p and q are in Pf , b ⊂ ap, c ⊂ aq, define pdb ≤ qdc iff b ⊃ c, hp(ξ)∩ c = hq(ξ)∩ c for each ξ ∈ c,and ipd

[c]2 = iqd

[c]2. Clearly if pdb ∈ Pf and qdc ∈ Pf then the two definitions of ≤ coincide.

Definition 1.10.22. Let p, p′ ∈ Pf with ap = ap′. We write p ≺ p′ if for each α ∈ ap and

b ⊂ ap ∩ α we have up(α, b) ⊂ up′(α, b).The following technical result will play a crucial role in the proof of 1.10.11(iii). Part (c) in

it will enable us to “insert” certain things in H(γ0) in a non-trivial way. But there is a price wehave to pay for this: this is the point where the coherency of the H(α) has to be abandoned. Part(d) will be needed in section 1.10.3.

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Lemma 1.10.23.Assume that s = 〈as, hs, is〉 ∈ Pf , as = S ∪ E ∪ F , Q ⊂ S, S <OnE<On F ,E = γi : i < k, γ0 < γ1 < · · · < γk−1, F = γi,0, γi,1 : i < k, moreover(i) ∀i < k hs(γi,0) ∩ hs(γi,1) = hs[Q ∪ E],(ii) ∀i < k ∀ξ ∈ S f(ξ, γi) = f(ξ, γi,0) = f(ξ, γi,1).Then there is a condition r = 〈ar, hr, ir〉 with ar = S ∪ E such that(a) r ≤ sdS,(b) r ≤ sd(Q ∪ E),(c) S \ hs[Q ∪ E] ⊂ hr(γ0),(d) sd(S ∪ E) ≺ r.

Proof. Let ar = S ∪ E and write C = S \ hs[Q ∪ E]. For ξ ∈ ar we set

hr(ξ) =hs(ξ) ∪ C if ξ = γi and γ0 ∈ hs(γi),hs(ξ) otherwise.

For ξ 6= η ∈ ar we let

ir(ξ, η) =is(ξ, η) if ξ, η ∈ Q ∪ E or ξ, η ∈ S,f(ξ, η) ∩ ar otherwise.

Finally let r = 〈ar, hr, ir〉.We claim that r satisfies the requirements of the lemma. a, b and c are clear from the

definition of r, once we establish that r ∈ Pf . To see that it suffices to check only 1.10.8.(iv)because the other requirements are clear from the construction of r. So let ξ < η ∈ ar. We haveto show hr(ξ) ∗ hr(η) ⊂ hr[ir(ξ, γi)].

If ξ, η ∈ S, then hr(ξ) ∗ hr(η) ⊂ hr[ir(ξ, η)] holds because rdS = sdS ∈ Pf .So we can assume that η = γi for some i < k.

Case 1. ξ ∈ S.Subcase 1.1. ξ /∈ hr(γi), hence hr(ξ) ∗ hr(γi) = hr(ξ) ∩ hr(γi).

In this case we also have hs(ξ) ∗ hs(γi) = hs(ξ) ∩ hs(γi and so

hs(ξ) ∩ hs(γi) ⊂ hs[is(ξ, γi)] ⊂ hr[ir(ξ, γi)] (1.78)

for ir(ξ, γi) ⊃ is(ξ, γi, ). If γ0 /∈ hs(γi) then hr(γi) = hs(γi) and since hr(ξ) = hs(ξ) we havehr(ξ) ∗ hr(γi) = hs(ξ) ∩ hs(γi) ⊂ hr[ir(ξ, γi)] by (1.78). Assume now that γ0 ∈ hs(γi). Thushr(γi) = hs(γi) ∪ C, and so ξ /∈ C, that is ξ ∈ hs[Q ∪E]. Then hr(ξ) ∗ hr(γi) = hr(ξ) ∩ hr(γi) =(hs(ξ)∩hs(γi))∪ (hs(ξ)∩C). By (1.78) above it is enough to show that hs(ξ)∩C ⊂ hr[ir(ξ, γi)].Since hs(ξ) ∩ C = ∅ for ξ ∈ Q we can assume that ξ /∈ Q. By (i) hs[Q ∪ E] = hs(γi,0) ∩ hs(γi,1)and so

hs(ξ) ∩ C = hs(ξ) \ hs[Q ∪ E] = (hs(ξ) \ hs(γi,0)) ∪ (hs(ξ) \ hs(γi,1)). (1.79)Since ξ ∈ hs[Q ∪ E] ⊂ hs(γi,j) for j = 0, 1 we have

hs(ξ) \ hs(γi,j) = hs(ξ) ∗ hs(γi,j) ⊂ hs[is(ξ, γi,j)]. (1.80)

By (ii), is(ξ, γi,j) ⊂ f(ξ, γi,j)∩as = f(ξ, γi)∩as. Since ξ /∈ Q it follows that f(ξ, γi)∩as = ir(ξ, γi).Thus from (1.80) we obtain

hs(ξ) \ hs(γi,j) ⊂ hs[ir(ξ, γi)] = hr[ir(ξ, γi)]. (1.81)

Putting (1.79) and (1.81) together we get hs(ξ) ∩ C ⊂ hr[ir(ξ, γi)] which was to be proved.

Subcase 1.2. ξ ∈ hr(γi), hence hr(ξ) ∗ hr(γi) = hr(ξ) \ hr(γi).If ξ ∈ hs(γi) then since hs(ξ) = hr(ξ) we have hr(ξ)∗hr(γi) = hr(ξ)\hr(γi) ⊂ hs(ξ)\hs(γi) ⊂

hs[is(ξ, γi)] ⊂ hr[ir(ξ, γi)] and we are done.So we can assume that ξ /∈ hs(γi) and so hr(γi) 6= hs(γi). By the construction of r, we have

γ0 ∈ hs(γi), hr(γi) = hs(γi) ∪ C and so ξ ∈ C, i.e ξ /∈ hs[Q ∪ E]. By (i) we can assume thatξ /∈ hs(γi,0). So s ∈ Pf implies

hs(ξ) ∩ hs(γi,0) = hs(ξ) ∗ hs(γi,0) ⊂ hs[is(ξ, γi,0)] (1.82)

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We havehr(ξ) ∗ hr(γi) = hs(ξ) \ (hs(γi) ∪ C) ⊂ hs(ξ) \ C (1.83)

and applying (i) again

hs(ξ) \ C = hs(ξ) ∩ hs[Q ∪ E] ⊂ hs(ξ) ∩ hs(γi,0). (1.84)

By (ii), is(ξ, γi,0) ⊂ f(ξ, γi,0) ∩ as = f(ξ, γi) ∩ as. Since ξ /∈ Q ⊂ hs[Q ∪ E] it follows thatf(ξ, γi) ∩ as = ir(ξ, γi) and so (1.82)–(1.84) together yield

hr(ξ) ∗ hr(γi) ⊂ hr[ir(ξ, γi)] (1.85)

which was to be proved.

Case 2. ξ = γj for some j < i.

Since is(γj , γi) = ir(γj , γi) we have

hs(γj) ∗ hs(γi) ⊂ hr[ir(γi, γj)]. (1.86)

It is easy to check that

hr(γj) ∗ hr(γi) =hs(γj) ∗ hs(γi) if γ0 /∈ hs(γj) ∗ hs(γi),hs(γj) ∗ hs(γi) ∪ C if γ0 ∈ hs(γj) ∗ hs(γi).

(1.87)

So we are done if γ0 /∈ hs(γj) ∗ hs(γi). Assume γ0 ∈ hs(γj) ∗ hs(γi). Then there is γl ∈ is(γj , γi)with γ0 ∈ hs(γl). Thus, by the construction of r we have

C ⊂ hr(γl) ⊂ hr[ir(γj , γi)] (1.88)

But (1.87) and (1.88) together imply what we wanted.Thus we proved r ∈ Pf .Clearly r satisfies 1.10.23.(a)–(c). To check 1.10.23.(d) write s′ = sd(S∪E) and let α ∈ S∪E

and b ⊂ (S ∪E)∩α. We need to show that us′(α, b) ⊂ ur(α, b). Since S ∪E is an initial segment

of as, we have us′(α, b) = us(α, b). If α ∈ S, then also us

′(α, b) = ur(α, b), so we can assume

α ∈ E.Let ξ ∈ us′(α, b) = us(α, b). Then ξ ∈ hs(α) ⊂ hs[E], and hence ξ /∈ C. But hr[b]\hs[b] ⊂ C,

more precisely, it is empty or just C. Since ξ ∈ us(α, b), it follows that ξ /∈ hs[b] and so ξ /∈ hr[b]because ξ /∈ C. Thus ξ ∈ ur(α, b). Hence r satisfies (d).

The lemma is proved.

Lemma 1.10.24. In V Pf , if Y ⊂ ω2 is countable, then either Y is compact or |ω2 \ Y | ≤ ω1.

Proof. Assume that 1Pf “Y = yn : n ∈ ω ⊂ ω2”. For each n ∈ ω fix a maximal antichainCn ⊂ Pf such that for each p ∈ Cn there is α ∈ ap with p “ yn = α”. Let A =

⋃ap : p ∈

⋃n<ω

Cn.

Since every Cn is countable by c.c.c we have |A| = ω.Assume also that 1Pf “Y is not compact”, that is, Y can not be covered by finitely many

H(δ) in V Pf .Let

I = δ < ω2 : ∃p ∈ Pf p “δ /∈ Y ”.

Clearly 1Pf ω2 \ I ⊂ Y . Since ω2 \ I is in the ground model, by corollary 1.10.21 it is enoughto show that ω2 \ I is infinite. Actually we will prove much more:

Claim . I is not stationary in ω2.

Assume on the contrary that I is stationary. Let us fix, for each δ ∈ I, a condition pδ ∈ Pfand a finite set Dδ ∈

[δ]<ω such that pδ “Y ∩ U(δ,Dδ) = ∅”. For each δ ∈ I let Qδ = apδ ∩ δ

and Eδ = apδ \ δ. We can assume that Dδ ⊂ Qδ and supA < δ for each δ ∈ I.Let Bδ = clf (A ∪Qδ, Eδ) for δ ∈ I (see 1.10.2). For each δ ∈ I the set Bδ is countable with

sup(Bδ) = sup(A∪Qδ), so we can apply Fodor’s pressing down lemma and CH to get a stationaryset J ⊂ I and a countable set B ⊂ ω2 such that Bδ = B for each δ ∈ J .

By thinning out J and with a further use of CH we can assume that for a fixed k ∈ ω wehave

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(1) Eδ = γδi : i < k for δ ∈ J , γδ0 < γδ1 < · · · < γδk−1,(2) f(ξ, γδi ) = f(ξ, γδ

′

i ) for each ξ ∈ B, δ, δ′ ∈ J and i < k.Let δ = min J , D = Dδ, E = Eδ, p = pδ, Q = Qδ. By lemma 1.10.1 there are ordinals δj ∈ J

with δ < δ0 < δ1 < · · · < δ2k−1 such that

B ∪ E ⊂⋂f(ξ, η) : ξ ∈ Eδi , η ∈ Eδj , i < j < 2k. (?)

For i < k and j < 2 let γi = γδi and γi,j = γδ2i+ji . Let F = γi,j : i < k, j < 2.

We know that ap = Q ∪ E. Define the condition q ∈ Pf by the following stipulations:(i) aq = ap ∪ F and q ≤ p,(ii) hq(γi,j) = γi,j ∪ ap for 〈i, j〉 ∈ k × 2(iii) iq(γi0,j0 , γi1,j1) = ap for 〈i0, j0〉 6= 〈i1, j1〉 ∈ k × 2,(iv) iq(ξ, γi,j) = ∅ for ξ ∈ ap and 〈i, j〉 ∈ k × 2.Since ap ⊂ B ∪ E, (?) implies that q ∈ Pf .

Since 1Pf “H[Q∪E] 6⊃ Y ”, there is a condition t ≤ q, a natural number n and an ordinal αsuch that t “α = yn” but α ∈ at \ ht[Q ∪ E]. Since Cn is a maximal antichain we can assumethat t ≤ v for some v ∈ Cn.

Let s = td(B ∪ E ∪ F ). Then s ∈ Pf because for each pair ξ < η ∈ as if ξ ∈ B thenit(ξ, η) ⊂ f(ξ, η) ⊂ B and so it(ξ, η) ⊂ as, and if ξ, η ∈ E∪F then it(ξ, η) = iq(ξ, η) ⊂ Q∪E ⊂ as.Moreover s ≤ v because av ⊂ B. Thus s “α = yn” and α /∈ hs[Q ∪ E]. Let S = as ∩B.

Since is(γi,0, γi,1) = iq(γi,0, γi,1) = Q ∪ E, and γi,j /∈ hs(γi,1−j) we have

hs(γi,0) ∩ hs(γi,1) = hs(γi,0) ∗ hs(γi,1) ⊂ hs[Q ∪ E]. (1.89)

Moreover, if ξ ∈ Q ∪ E and j < 2 then ξ ∈ hq(γi,j) ⊂ hs(γi,j) and is(ξ, γi,j) = ∅, consequentlyhs(ξ) \ hs(γi,j) = hs(ξ) ∗ hs(γi,j) ⊂ hs[is(ξ, γi,j)] = ∅. (1.90)

Putting (1.89) and (1.90) together it follows that hs(γi,0) ∩ hs(γi,1) = hs[Q ∪ E].Thus we can apply 1.10.23 to get a condition r such that r ≤ sd(Q ∪ E) = p, r ≤ sdS ≤ v

and α ∈ S \ hs[Q ∪ E] ⊂ hs(γ0) = hs(δ). Since D ⊂ Q, we have α ∈ hs(δ) \ hs[D].Thus

r α ∈ Y ∩ (H(δ) \H[D]).On the other hand r Y ∩ (H(δ) \ H[D]) = ∅ because r ≤ p. With this contradiction the claimis proved and this completes the proof of the lemma.

Clearly, lemma 1.10.24 implies that Xf is countably compact.

Corollary 1.10.25. If F ⊂ X is closed (or open), then either |F | ≤ ω1 or |X \ F | ≤ ω1.

Proof. If |F | = ω2 then F is not compact, so by lemma 1.10.6 F contains a free sequence Ywith non-compact closure. But F is initially ω1-compact and countably tight, so Y is countable.Consequently, we have |ω2 \ Y | ≤ ω1 by lemma 1.10.24 and so |X \ F | ≤ ω1.

Corollary 1.10.26.Xf is normal and z(Xf ) = hd(Xf ) ≤ ω1.

Proof. To show that Xf is normal let F0 and F1 be disjoint closed subsets of Xf . Since at leastone of them is compact by lemma 1.10.24 they can be separated by open subsets of Xf becauseXf is T3.

Concerning the hereditarily density of Xf it follows easily from corollary 1.10.25 that Xf doesnot contain a discrete subspace of size ω2. But Xf is right separated, so all the left separatedsubspaces of Xf are of size ≤ ω1, that is , z(X) ≤ ω1.

Thus theorem 1.10.11 is proved.

We know that the space Xf is not automatically hereditarily separable, so the followingquestion of Arhangel’skiı, [20, problem 5], remains unanswered: Is it true in ZFC that everyhereditarily separable, initially ω1-compact space is compact?

As we have seen our space Xf is normal. However, we don’t know whether Xf is or can bemade hereditarily normal, i.e. T5. This raises the following problem.

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Problem 3. Is it provable in ZFC that every T5, countably tight, initially ω1-compact space iscompact?

1.10.3. Making Xf Frechet-Urysohn. In [20, problem 12] Arhangel’skiıasks if it is prov-able in ZFC that a normal, first countable initially ω1-compact space is necessarily compact.We could not completely answer this question, but in this subsection we show that the Frechet-Urysohn property (which is sort of half-way between countable tightness and first countability)in not enough to get compactness.

To achieve that we want to find a further extension of the model V Pf in which Xf becomesFrechet-Urysohn but its other properties are preserved, for example, Xf remains initially ω1-compact and normal. Since Xf is countably tight and χ(Xf ) ≤ ω1 it is a natural idea to makeXf Frechet-Urysohn by constructing a generic extension of V Pf in which Xf remains countablytight and p > ω1, i.e. MAω1(σ-centered) holds.

The standard c.c.c poset P which forces p > ω1 is obtained by a suitable finite supportiteration of length 2ω1 . During this iteration in the αth step we choose a non-principal filterF ⊂ P(ω) generated by at most ω1 elements and we add a new subset A of ω to the αth

intermediate model so that A is almost contained in every element of F , i.e. A \ F is finite foreach F ∈ F . It is well-known and easy to see that P has property K. Thus, by theorem 1.10.27below, Xf remains countably tight in V Pf∗R and so indeed Xf becomes Frechet-Urysohn inthat model. Moreover, theorem 1.10.28 implies that the ω1-compactness of Xf is also preserved.Unfortunately, we could not prove that forcing with P preserves the countable compactness ofXf .

So, instead of aiming at p > ω1 we will consider only those filters during the iteration whichare needed in proving the Frechet-Urysohn property of Xf . As we will see, we can handle thesefilters in such a way that our iterated forcing R preserves not only the countable compactness ofXf but also property (iii) from theorem 1.10.11: in V Pf∗R the closure of any countable subset ofXf is either compact or contains a final segment of ω2. Of course, this will insure the preservationof the normality of Xf as well.

We start with the two easy theorems, promised above, about the preservation of countabletightness and ω1-compactness of Xf under certain c.c.c forcings.

Theorem 1.10.27. If the topological space X is right separated, compact, countably tight and theposet R has property K then forcing with R preserves the countable tightness of X.

Proof. First we recall that X remains compact (and clearly right separated) in any extension ofthe ground model by [60, lemma 7]. Since F(X) = t(X) for compact spaces, assume indirectlythat 1R “zξ : ξ < ω1 ⊂ X is a free sequence”. For every ξ < ω1 we have that 1R “zζ : ζ < ξand zζ : ξ ≤ ζ < ω1 are disjoint compact sets” and X is T3, so we can fix a condition pξ ∈ P ,open sets Uξ and Vξ from the ground model and a point zξ ∈ X such that Uξ ∩ Vξ = ∅ and

pξ “ zζ : ζ < ξ ⊂ Uξ, zζ : ξ ≤ ζ < ω1 ⊂ Vξ and zξ = zξ.”

Since R has property K, there is an uncountable set I ⊂ ω1 such that the conditions pξ :ξ ∈ I are pairwise compatible.

We claim that the sequence zξ : ξ ∈ I is an uncountable free sequence in the ground modelwhich contradicts F(X) = t(X) = ω. Indeed let ξ ∈ I. If ζ ∈ I ∩ ξ, then pξ and pζ has a commonextension q in P and we have

q “ zζ = zζ and zη : η < ξ ⊂ Uξ.”

Hence zζ ∈ Uξ. Similarly for ζ ∈ I \ ξ we have zζ ∈ Vξ. Therefore Uξ and Vξ separatezζ : ζ ∈ I ∩ ξ and zζ : ζ ∈ I \ ξ which implies that zξ : ξ ∈ I is really free.

Theorem 1.10.28. Forcing with a c.c.c poset R over V Pf preserves property 1.10.11.(ii) of thespace Xf , i.e. for each uncountable A ⊂ Xf there is β ∈ ω2 such that A ∩H(β) is uncountable.

Proof. We work in V Pf . Assume that r R “A = αξ : ξ < ω1 ∈[Xf

]ω1”. For each ξ < ω1

pick a condition rξ ≤ r from R which decides the value of αξ, rξ R “αξ = αξ”. Since R satisfiesc.c.c, αξ : ξ ∈ ω1 is uncountable, hence as Xf has property (ii) in V Pf , for some β < ω2 the set

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I = H(β) ∩ αξ : ξ < ω1 is also uncountable. Since R satisfies c.c.c there is a condition q ≤ r inR such that q R “|ξ ∈ I : rξ ∈ G| = ω1”, where G is the R-generic filter over V Pf . Thus q R“|A ∩H(β)| = ω1” which was to be proved.

Of course, theorem 1.10.28 implies that forcing with any c.c.c poset R preserves the ω1-compactness of Xf . It is much harder to find a property of a poset R which guarantees thatforcing with R over V Pf preserves the countable compactness of Xf . We will proceed in thefollowing way. In definition 1.10.29 we formulate when a poset R is called nice (over Pf ), andthen in theorem 1.10.30 we show that forcing with a nice poset preserves not only the countablecompactness of Xf , but also property 1.10.11(iii): the closure of any countable subset of Xf iseither compact or contains a final segment of ω2. Finally, in definitions 1.10.31 and 1.10.32 wedescribe a class of finite support iterated forcings, which by theorem 1.10.33 are nice and haveproperty K , and then in theorem 1.10.34 we show that forcing with a suitable member of thisclass makes Xf Frechet-Urysohn.

Definition 1.10.29. Let R be a name for a poset in V Pf . We say that R is nice (over Pf ) if thereis a dense subset D of the iteration Pf ∗ R with the following property: If 〈p0, r0〉 , 〈p1, r1〉 ∈ D,p, p′ ∈ Pf are such that p ≤ p0, p1, p′ ≤ p0, p1, and

p “r0 and r1 are compatible in R”,

moreover we have either p ≤ p′ or p ≺ p′ (see definition 1.10.22) then we also have

p′ “r0 and r1 are compatible in R”.

Theorem 1.10.30. If CH holds in V , f is a ∆-function and R is a Pf -name for a c.c.c posetwhich is nice over Pf , then 1.10.11.(iii) is preserved by forcing with R, i.e.

V Pf∗R |= ∀Y ∈[Xf

]ω( Y is compact or |Xf \ Y | ≤ ω1 ).

Proof. The proof is quite long and technical. See in [11].

Definition 1.10.31.Assume that A ∈[Xf

]ω and α ∈ A. We define the poset Q(A,α) as follows.Its underlying set is

[A]<ω × [α]<ω . If 〈s, C〉 and 〈s′C ′〉 are conditions, let 〈s, C〉 ≤ 〈s′C ′〉 if

and only if s ⊃ s′, C ⊃ C ′ and s \ s′ ⊂ U(α,C ′). For q ∈ Q(A,α) write q = 〈sq, Cq〉 andsupp(q) = sq ∪ Cq.

It is well-known and easy to see that if G is a Q(A,α)-generic filter, then S =⋃sq : q ∈ G

is a sequence from A which converges to α, i.e. every open neighbourhood of α contains all butfinitely many points of S.

Clearly Q(A,α) is σ-centered and well-met. In fact, if p0 = 〈s0, C0〉 and p1 = 〈s1, C1〉 arecompatible, then p0 ∧ p1 = 〈s0 ∪ s1, C0 ∪ C1〉.Definition 1.10.32.A finite support iterated forcing 〈Rξ : ξ ≤ κ〉 over V Pf is called an FU-iteration if for each ξ < κ we have

1Pf∗Rξ Rξ+1 = Rξ ∗Q(A, α) for some A ∈[Xf

]ω and α ∈ A.Since every Q(A,α) is σ-centered, it is clear that any FU-iteration is c.c.c, in fact it even has

property K. The really important, and much less trivial, property of them is given in our nextresult.

Theorem 1.10.33.Any FU-iteration is nice over Pf .

Proof. This proof is rather long and technical. See in [11].

Now we are ready to obtain the main result of this section.

Theorem 1.10.34.Assume that CH holds in the ground model V , there is a ∆-function f andλ is a cardinal such that ω1 < λ = λω. Then in V Pf there is an FU-iteration Rλ of lengthλ such that in V Pf∗Rλ the space Xf is Frechet-Urysohn and satisfies 1.10.11(i)–(iii), moreoverV Pf∗Rλ |=“2κ = (λκ)V for each cardinal κ ≥ ω”.

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Proof. It is standard to proof together the pieces we obtained so far. For details, see [11].

Theorem 1.10.34 answers a question raised by Arhangel’skiı, [20, problem 3], in the negative:CH can not be weakened to 2ω < 2ω1 in the theorem of van Douwen and Dow. In fact, if we havea ZFC model V in which CH holds and λ = λω ≤ 2ω1 , then we can find a cardinal preservinggeneric extension W of V which contains a Frechet-Urysohn and normal counterexample to thevan Douwen–Dow question, (2ω)W = λ, moreover (2κ)W = (2κ)V for each κ ≥ ω1.

Let us finish by formulating the following higher cardinal version of the van Douwen–Dowproblem:

Problem 4. Is it provable in ZFC that an initially ω2-compact T3 space of countable tightness iscompact ?

The main problem is trying to use out approach that worked for ω2 (instead of ω3) here isthat no ∆-function may exist for ω3! This problem has come up already in the efforts trying tolift the result of [25] from ω2 to ω3.

1.11. First countable, initially ω1-compact spaces


Improving the results from the previous section, we force a first countable, normal, locallycompact, initially ω1-compact and non-compact space X = 〈ω2 × 2ω, τ〉.

Actually, Alan Dow conjectured that applying the method of [74] (that "turns" a compactspace into a first countable one) to the space of Rabus in [90] yields an ω1-compact but non-compact first countable space. How one can carry out such a construction was outlined by thesecond author in the preprint [71]. However, [71] only sketches some arguments as the languageadopted there, which follows that of [90], does not seem to allow direct combinatorial controlover the space which is forced. This explains why the second author hesitated to publish [71].

One missing element of [71] was a language similar to that of [11] which allows workingwith the points of the forced space in a direct combinatorial way. In this section we combine theapproach of [11] with the ideas of [71] to obtain directly an ω1-compact but non-compact firstcountable space. Consequently, our proofs follow much more closely the arguments of [11] thanthose of [90] or their analogues in [71].

As before, we again use a ∆-function to make our forcing CCC but we need both CH and a∆-function with some extra properties to obtain first countability.

It is immediate from the countable compactness of X that its one-point compactification X∗is not first countable. In fact, one can show that the character of the point at infinity ∗ in X∗ isω2. As X is initially ω1-compact, this means that every (transfinite) sequence converging from Xto ∗ must be of type cofinal with ω2. Since X is first countable, this trivially implies that there isno non-trivial converging sequence of type ω1 in X∗. In other words: the convergence spectrumof the compactum X∗ omits ω1. As far as we know, this is the first and only (consistent) exampleof this sort.

1.11.1. A general construction. First we introduce a general method to construct locallycompact, zero-dimensional spaces. This generalizes the method for the construction of locallycompact right-separated (i.e. scattered) spaces that was described in [11].

Definition 1.11.1. Let ϑ be an ordinal, X be a 0-dimensional space, and fix a clopen subbase(i.e. a subbase consisting of clopen sets) S of X such that X ∈ S and

S ∈ S \ X implies (X \ S) ∈ S. (1.91)

Let K : ϑ× S −→ P(ϑ) be a function satisfying

K(δ, S) ⊂ K(δ,X) ⊂ δ, (1.92)

and for any δ ∈ ϑ and S ∈ S set

U(δ, S) = (δ × S) ∪ (K(δ, S)×X). (1.93)

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1.11. FIRST COUNTABLE, INITIALLY ω1-COMPACT SPACES 71

We shall denote by τK the topology on ϑ×X generated by the family

UK = U(δ, S) , (ϑ×X) \ U(δ, S) : δ < ϑ, S ∈ S (1.94)

as a subbase. Write XK = 〈ϑ×X, τK〉.If a is a set of ordinals and s is an arbitrary set we write

a⊗ a⊗ s = 〈ζ, ξ, σ〉 : ζ, ξ ∈ a, ζ < ξ, σ ∈ s. (1.95)

Theorem 1.11.2. (1) Assume that ϑ, X, S and K are as in definition 1.11.1 above. Then thespace XK = 〈ϑ×X, τK〉 is 0-dimensional and Hausdorff and the subspace α×X is homeomor-phic to X for each α < ϑ.(2) Assume, in addition, that X is compact and(K1) if S ∩ S′ = ∅ then K(δ, S) ∩K(δ, S′) = ∅,(K2) if X = ∪S ′ for some S ′ ∈

[S]<ω then

K(δ,X) = ∪K(δ, S) : S ∈ S ′,

(K3) there is a function i with dom(i) = ϑ ⊗ ϑ ⊗ S such that for each 〈δ, δ′, S〉 ∈ ϑ ⊗ ϑ ⊗ S wehave(i1) i(δ, δ′, S) ∈

[δ]<ω and

(i2) K(δ,X) ∗K(δ′, S) ⊂ ∪K(ν,X) : ν ∈ i(δ, δ′, S),where

K(δ,X) ∗K(δ′, S) =

K(δ,X) ∩K(δ′, S) if δ /∈ K(δ′, S),K(δ,X) \K(δ′, S) if δ ∈ K(δ′, S). (1.96)

Then all members of UK are compact, hence XK is locally compact.

Proof. The argument is quite long, but more or less standard. See the details in [8].

To describe a natural base of the space XK, we fix some more notation. For δ < ϑ, S ′ ∈[S]<ω

and F ∈[δ]<ω we shall write

B(δ,S ′, F ) = ∩U(δ, S) : S ∈ S ′ \ U [F ].

For a point x ∈ X we set S(x) = S ∈ S : x ∈ S, moreover we put

B(δ, x) = B(δ,S ′, F ) : S ′ ∈[S(x)

]<ω, F ∈

[δ]<ω. (1.97)

Lemma 1.11.3.Assume that ϑ, X, S and K are as in part (2) of the previous theorem 1.11.2.Then for each δ < ϑ and x ∈ X the family B(δ, x) forms a neighbourhood base of the point 〈δ, x〉in XK.

Proof. See in [8].

As we already mentioned above, our construction of the locally compact spaces XK generalizesthe construction of locally compact right-separated spaces given in [11]. In fact, the latter is thespecial case when X is a singleton space (and S is the only possible subbase X). We mayactually say that in the space XK the compact open sets U(δ) right separate the copies δ ×Xof X rather than the points.

Actually, a locally compact, right separated, and initially ω1-compact but non-compact spacecannot be first countable. (Indeed, this is because the scattered height of such a space mustexceed ω1.) So the transition to a more complicated procedure is necessary if we want to keepour example locally compact.

We now present a much more interesting example of our general construction, where X willbe the Cantor set C and S will be a natural subbase. For technical reasons, we put C = 2N

instead of 2ω, where N = ω\0.The clopen subbase S of C is the one that determines the product topology and is defined as

follows. If n > 0 and ε < 2 then let [n, ε] = f ∈ C : f(n) = ε. We then put

S = [n, ε] : n > 0, i < 2 ∪ C.

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Then S satisfies 1.11.1.(1.91), moreover if S ′ ⊂ S \ C covers C = 2N then there is n ∈ N suchthat both [n, 0], [n, 1] ∈ S ′.

In order to apply our general scheme, we still need to fix an ordinal ϑ, a function K :ϑ× S −→ P(ϑ) satisfying 1.11.1.(2), and another function i with dom(i) = ϑ⊗ ϑ⊗ S such thatall the requirements of theorem 1.11.1 are satisfied. In our present particular case this may beachieved in a slightly different form that turns out to be simpler and more convenient for thepurposes of our forthcoming forcing argument.

If h is a function and a ⊂ dom(h) we write h[a] = ∪h(ξ) : ξ ∈ a (this piece of notation hasbeen used before). If x and y are two non-empty sets of ordinals with supx < sup y then we let

x ∗ y =x ∩ y if supx /∈ y,x \ y if supx ∈ y.

Note that this operation ∗ is not symmetric, on the contrary, if x ∗ y is defined then y ∗ x is not.

Definition 1.11.4.A pair of functions H : ϑ× ω −→ P(ϑ) and i : ϑ⊗ ϑ⊗ ω −→[ϑ]<ω are said

to be ϑ-suitable if the following three conditions hold for all α, β ∈ ϑ and n ∈ ω:(H1) α ∈ H(α, n) ⊂ H(α, 0) ⊂ α+ 1,(H2) i(α, β, n) ∈

[α]<ω,

(H3) if α < β then H(α, 0) ∗H(β, n) ⊂ H[i(α, β, n)].Concerning (H3) note that we have

maxH(α, 0) = α < maxH(β, n) = β,

hence H(α, 0) ∗H(β, n) is defined.

Given a ϑ-suitable pair (H, i) as above, let us define the functions

K : ϑ× S −→ P(ϑ) and i′ : ϑ⊗ ϑ⊗ S −→[ϑ]<ω

as follows:

K(α,C) = H(α, 0) ∩ α, (1.98)K(α, [n, 1]) = H(α, n) ∩ α, (1.99)

K(α, [n, 0]) = H(α, 0) \H(α, n), (1.100)

i′(α, β,C) = i(α, β, 0), (1.101)

i′(α, β, [n, ε]) = i(α, β, 0) ∪ i(α, β, n). (1.102)

It is straightforward to check then that K and i′ satisfy all the requirements of theorem 1.11.2.Because of this, with some abuse of notation, we shall denote the topology τK also by τH and thespace 〈ϑ× C, τK〉 by XH .

For our subbasic compact open sets we have

U(α) = U(α,C) = H(α, 0)× C, (1.103)

and to simplify notation we write

U(α, [n, ε]) = U(α, n, ε). (1.104)

Using this terminology, we may now formulate lemma 1.11.3 for this example in the followingmanner.

Lemma 1.11.5. If (H, i) is an ϑ-suitable pair then for every 〈α, x〉 ∈ ϑ × C the compact opensets

B(α, x, n, F ) =⋂U(α, j, x(j)) : 1 ≤ j ≤ n, \ U [F ]

with n ∈ N and F ∈ [α]<ω form a neighbourhood base of the point 〈α, x〉 in the space XH .

What we are set out to do now is to force an ω2-suitable pair (H, i) such that the space XH isas required. As mentioned, for this we need a special kind of ∆-function and this will be discussedin the next subsection.

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1.11.2. ∆-functions.

Definition 1.11.6. Let f :[ω2

]2 −→ [ω2

]≤ω be a function with f(α, β) ⊂ α ∩ β for α, β ∈[ω2

]2. Actually, in what follows, we shall simply write f(α, β) instead of f(α, β) .We say that two finite subsets x and y of ω2 are very good for f provided that for τ, τ1, τ2 ∈

x ∩ y, α ∈ x \ y, β ∈ y \ x and γ ∈ (x \ y) ∪ (y \ x) we always have∆1) τ < α, β =⇒ τ ∈ f(α, β),∆2) τ < α =⇒ f(τ, β) ⊂ f(α, β),∆3) τ < β =⇒ f(τ, α) ⊂ f(β, α),∆4) γ, τ1 < τ2 =⇒ f(γ, τ1) ⊂ f(γ, τ2).∆5) τ1 < γ < τ2 =⇒ τ1 ∈ f(γ, τ2).The sets x and y are said to be good for f iff ∆1)–∆3) hold.

We say that f :[ω2

]2 −→ [ω2

]≤ω with f(α, β) ⊂ α ∩ β is a very good ∆-function, if everyuncountable family of finite subsets of ω2 contains two sets x and y which are very good for f .

We will prove in Lemma 1.11.8 that it is consistent with CH that there is a very good ∆-function.

In the proof of the countable compactness of our space we shall need the following simpleconsequence of [11, Lemma 1.2] that yields an additional property of ∆-functions provided thatCH holds.

Lemma 1.11.7.Assume that CH holds, f is a ∆-function, and B ∈[ω2

]ω. Then for any finitecollection Ti : i < m ⊂

[ω2

]ω2 we may select a strictly increasing sequence 〈γi : i < m〉 withγi ∈ Ti such that B ⊂ f(γi, γj) whenever i < j < m .

Now, we have come to the main result of this section.

Lemma 1.11.8. It is consistent with CH that there is a very good ∆-function.

Proof of Lemma 1.11.8. There are several natural ways of constructing such a very good ∆-function f . One can do it by forcing, following and modifying a bit the construction given in[25]. One can use Velleman’s simplified morasses (see [104]) and put

f(α, β) = X ∩ βwhere X is an element of minimal rank of the morass that contains α and β.

In this section we chose to follow Todorčevic’s approach that uses his canonical coloringρ : [ω2]2 → ω1 obtained from a ω1-sequence (see [103, 7.3.2 and 7.4.8]). From this coloring ρ hedefines f by

f(α, β) = ξ < α : ρ(ξ, β) ≤ ρ(α, β)and proves that this f is a ∆-function in our terminology of 1.11.6 (see [103, 7.4.9 and 7.4.10]).(We should warn the reader, however, that he calls this a D-function instead of a ∆-function in[103].)

He also establishes the following canonical inequalities for ρ (see [103, 7.3.7 and 7.3.8]):

(i) |ξ < α : ρ(ξ, α) ≤ ν| < ω1

(ii) ρ(α, γ) ≤ maxρ(α, β), ρ(β, γ)

(iii) ρ(α, β) ≤ maxρ(α, γ), ρ(β, γ)for α < β < γ < ω2 and ν < ω1. We will now use these inequalities to prove that this f is even avery good ∆-function.

Let A be an uncountable family of finite subsets of ω2. Note that it is enough to find anuncountable A′ ⊆ A such that ∆4) and ∆5) of 1.11.6 hold for every two elements of A′, sincethen we may apply to A′ the fact that f is a ∆-function to obtain two elements of A that arevery good for f .

We may assume w.l.o.g. that A forms a ∆-system with root ∆ ⊆ ω2. Note that then the set

D = ξ ∈ ω2 : ∃τ1, τ2, τ3 ∈ ∆, ξ < τ1, ρ(ξ, τ1) ≤ ρ(τ2, τ3)

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is countable by (i). Define A′ ⊆ A to be the set of all elements a ∈ A which satisfy (a−∆)∩D = ∅.The countability of D implies that A′ is uncountable, moreover we have

(1) ρ(γ, τ1) > ρ(τ2, τ3)

for all τ1, τ2, τ3 ∈ ∆ and γ ∈ a−∆ with a ∈ A′ and γ < τ1.Now we prove that both ∆4) and ∆5) of 1.11.6 hold for every two sets x, y ∈ A′ which will

complete the proof of the lemma. Let τ1, τ2 ∈ ∆ = x ∩ y and γ ∈ (x \ y) ∪ (y \ x).Note that if τ1, γ < τ2, then

(2) ρ(γ, τ1) ≤ ρ(γ, τ2).

This follows from (iii) and (1).Now we prove ∆4). Consider two cases. First when τ1 < γ < τ2. Assume ξ ∈ f(τ1, γ), that

is ξ < τ1 and

(3) ρ(ξ, γ) ≤ ρ(τ1, γ),

By (ii) we have ρ(ξ, τ2) ≤ max(ρ(ξ, γ), ρ(γ, τ2)) which by (3) is less or equal to max(ρ(τ1, γ), ρ(γ, τ2)) =ρ(γ, τ2) by (2). But this means that ξ ∈ f(γ, τ2) and so gives the inclusion of ∆4).

The second case is when γ < τ1 < τ2. Assume ξ ∈ f(γ, τ1), that is ξ < γ and

(4) ρ(ξ, τ1) ≤ ρ(γ, τ1) .

By (ii) we have that ρ(ξ, τ2) ≤ max(ρ(ξ, τ1), ρ(τ1, τ2)) which by (4) is less or equal tomax(ρ(γ, τ1), ρ(τ1, τ2)). But we have

max(ρ(γ, τ1), ρ(τ1, τ2)) ≤ ρ(γ, τ2)

by (1) and (2), hence ρ(ξ, τ2) ≤ ρ(γ, τ2) and so ξ ∈ f(γ, τ2) that again gives the inclusion of ∆4).Finally, we prove ∆5). Assume τ1 < γ < τ2, then by (1) we have ρ(τ1, τ2) ≤ ρ(γ, τ2) and so

the definition of f gives that τ1 ∈ f(γ, τ2), as required in ∆5).

1.11.3. The forcing notion. Now we describe a natural notion of forcing with finite ap-proximations that produces an ω2-suitable pair (H, i). The forcing depends on a parameter fthat will be chosen to be a very good ∆-function, like the one constructed in 1.11.8.

Definition 1.11.9. For each function f :[ω2

]2 −→ [ω2

]≤ω satisfying f(α, β) ⊂ α ∩ β for anyα, β ∈

[ω2

]2 we define the poset (Pf , ≤) as follows. The elements of Pf are all quadruplesp = 〈a, h, n, i〉 satisfying the following five conditions (P1) – (P5):(P1) a ∈ [ω2]<ω, n ∈ ω, h : a× n→ P(a), i : a⊗ a⊗ n→ P(a),(P2) maxh(ξ, j) = ξ for all 〈ξ, j〉 ∈ a× n,(P3) h(ξ, j) ⊂ h(ξ, 0) for all 〈ξ, j〉 ∈ a× n,(P4) i(ξ, η, j) ⊆ f(ξ, η) whenever 〈ξ, η, j〉 ∈ a⊗ a⊗ n,(P5) if 〈ξ, η, j〉 ∈ a⊗ a⊗ n then h(ξ, 0) ∗ h(η, j) ⊂ h[i(ξ, η, j)],where, with some abuse of our earlier notation, we write

h[b] = ∪h(α, 0) : α ∈ b (1.105)

for b ⊂ a. We say that p ≤ q if and only if ap ⊇ aq, np ≥ nq, hp(ξ, j) ∩ aq = hq(ξ, j) for all〈ξ, j〉 ∈ aq × nq, moreover ip ⊃ iq.

Assume that the sets

Dα,n = p ∈ Pf : α ∈ ap and n < np

are dense in Pf for all pairs 〈α, n〉 ∈ ω2×ω. Then if G is a Pf -generic filter over V we may define, in V [G], the function H with domH = ω2 × ω and the function i with dom(i) = ω2 ⊗ ω2 ⊗ ω asfollows:

H(α, n) = ∪hp(α, n) : p ∈ G, 〈α, n〉 ∈ dom(hp), (1.106)i = ∪ip : p ∈ G. (1.107)

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Theorem 1.11.10.Assume that CH holds and f is a very good ∆-function. Then Pf is CCC and(H, i) is an ω2-suitable pair in V [G]. Moreover, the locally compact, 0-dimensional, and Hausdorffspace XH = 〈ω2 × C, τH〉 defined as in 1.11.4 satisfies, in V [G], the following properties:

(i) U(δ) = H(δ, 0)× C is compact open for each δ ∈ ω2,(ii) XH is first countable,

(iii) ∀A ∈ [ω2 × C]ω1 ∃α ∈ ω2 |A ∩ U(α)| = ω1,(iv) ∀Y ∈ [ω2×C]ω either the closure Y is compact or there is α < ω2 such that (ω2\α)×C ⊂ Y .Consequently, XH is a locally compact, 0-dimensional, normal, first countable, initially ω1-compact but non-compact space in V [G].

The rest of this section is devoted to the proof of Theorem 1.11.10.

1.11.4. The forcing is CCC. The CCC property of Pf is crucial for us because it impliesthat ω2 is preserved in the generic extension V [G]. Indeed, properties (H1)–(H3) of definition1.11.4 (for ϑ = ωV2 ) are easily deduced from of conditions (P1)–(P5) in 1.11.9 using straight-forward density arguments. So if ω2 is preserved then we immediately conclude that (H, i) is anω2-suitable pair in V [G].

Definition 1.11.11.Two conditions p0 = 〈a0, h0, n, i0〉 and p1 = 〈a1, h1, n, i1〉 from Pf are saidto be good twins provided that(1) p0 and p1 are isomorphic, i.e. |a0| = |a1| and the natural order-preserving bijection e between

a0 and a1 is an isomorphism between p0 and p1:(i) h1(e(ξ), j) = e[h0(ξ, j)] for ξ ∈ a0 and j < n,(ii) i1(e(ξ), e(η), j)) = e[i0(ξ, η, j)] for 〈ξ, η, j〉 ∈ a0 ⊗ a0 ⊗ n,(iii) e(ξ) = ξ whenever ξ ∈ a0 ∩ a1 and j < n ;

(2) i1(ξ, η, j) = i0(ξ, η, j) for each ξ, η ∈[a0 ∩ a1

]2 ;(3) a0 and a1 are good for f .The good twins p0 and p1 are called very good twins if a0 and a1 are very good for f .

Definition 1.11.12. If p = 〈a, h, n, i〉 and p′ = 〈a′, h′, n, i′〉 are good twins we define the amalga-mation p∗ = 〈a∗, h∗, n, i∗〉 of p and p′ as follows:

Let a∗ = a ∪ a′. For η ∈ h[a ∩ a′] ∪ h′[a ∩ a′] define

δη = minδ ∈ a ∩ a′ : η ∈ h(δ, 0) ∪ h′(δ, 0).

Now, for any ξ ∈ a∗ and m < n let

h∗(ξ,m) =

h(ξ,m) ∪ h′(ξ,m) if ξ ∈ a ∩ a′,h(ξ,m) ∪ η ∈ a′ \ a : δη is defined and δη ∈ h(ξ,m) if ξ ∈ a \ a′,h′(ξ,m) ∪ η ∈ a \ a′ : δη is defined and δη ∈ h′(ξ,m) if ξ ∈ a′ \ a.

(1.108)

Finally for 〈ξ, η,m〉 ∈ a∗ ⊗ a∗ ⊗ n let

i∗(ξ, η,m) =

i(ξ, η,m) if ξ, η ∈ a,i′(ξ, η,m) if ξ, η ∈ a′,f(ξ, η) ∩ a∗ otherwise.

(1.109)

(Observe that i∗ is well-defined because p and p′ are good twins). We will write p∗ = p + p′ forthe amalgamation of p and p′.

Lemma 1.11.13. If p and p′ are good twins then their amalgamation, p∗ = p+ p′, is a commonextension of p and p′ in Pf .

Proof. Lemma 1.11.13 corresponds to Lemma 1.10.14.The proof is quite technical, but it is similar to the proof of Lemma 1.10.14. So we omit it.

See in [11]

Proof of theorem 1.11.10: Pf is CCC. In every uncountable collection of conditions fromPf there are two which are good twins for f and, by Lemma 1.11.13, they are compatible.

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As was pointed out at the beginning of this section, we may now conclude that (H, i) is anω2-suitable pair in V [G]. This establishes the first part of Theorem 1.11.10 up to and including(i).

1.11.5. First countability.

Proof of theorem 1.11.10: XH is first countable. Since XH is locally compact and Haus-dorff it suffices to show that every point of XH has countable pseudo-character or, in otherwords, every singleton is a Gδ.

To see this, fix 〈α, x〉 ∈ ω2 × C. We claim that there is a countable set Γ ⊂ α such that⋂n∈N

U(α, n, x(n)) ⊂ U [Γ] ∪ 〈α, x〉. (1.110)

Since every U(γ) is clopen, this implies that

〈α, x〉 =⋂n∈N

U(α, n, x(n)) ∩⋂XH \ U(γ) : γ ∈ Γ

is indeed a Gδ.Our following lemma clearly implies (1.110). To formulate it, we first fix some notation. In

V [G] , for α ∈ ω2, 1 ≤ m < ω and Γ ⊂ ω2 we write

H1(α,m) = H(α,m) \ α, (1.111)

H0(α,m) = H(α, 0) \H(α,m), (1.112)H[Γ] = ∪H(γ, 0) : γ ∈ Γ. (1.113)

Recall that with this notation we have

U(α, n, ε) = (Hε(α, n)× C) ∪ (α × [n, ε]) .

Lemma 1.11.14. In V [G], for each 〈α, x〉 ∈ ω2 × C there is a countable set Γ ⊂ α such that⋂n∈N

Hx(n)(α, n) ⊂ H[Γ] . (1.114)

Proof. Suppose, arguing indirectly, that the lemma is false. Then, in V [G], for each countableset A ⊂ α there is γA ∈ α such that

γA ∈⋂n∈N

Hx(n)(α, n) \H[A]. (1.115)

>From now on, we work in the ground model V . For every ζ < ω1 let Aζ ⊆ α be a countablesubset such that ζ ′ ≤ ζ < ω1 implies Aζ′ ⊆ Aζ and

⋃ζ<ω1

Aζ = α.Let pζ = 〈aζ , hζ , nζ , iζ〉 ∈ Pf be a condition such that α ∈ aζ and for some γζ ∈ α ∩ aζ we

havepζ γζ ∈

⋂n∈N

Hx(n)(α, n) \H[Aζ ]. (1.116)

Using standard ∆-system and counting arguments and the properties of the very good ∆-functionf , we may find ζ1 < ζ2 < ω1 such that

α ∩ aζ1 ⊂ Aζ2 , (1.117)

moreover pζ1 , pζ2 are very good twins for f .Let p = pζ1 + pζ2 with p = 〈a, h, n, i〉 be their amalgamation as in 1.11.12. We now further

extend p to a condition of the form r = 〈a, hr, n+ 1, ir〉 with the following stipulations:(r1) hr ⊃ h,(r2) hr(ξ, n) = ξ for ξ ∈ a \ α,(r3) hr(α, n) = α ∪

(h(α, 0) ∩ h[α ∩ aζ1 ]

),

(r4) ir ⊃ i,(r5) ir(η, ξ, n) = ∅ for η < ξ ∈ a \ α,(r6) ir(η, α, n) = a ∩ f(η, α) for η < α.

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It is not clear at all that r is a condition, but if it is we have reached a contradiction. Indeed,if r ∈ Pf then r ≤ pζ2 , so r γζ2 /∈ H[Aζ2 ], hence γζ2 /∈ h[α ∩ aζ1 ] by (1.117). But then by (r3)we have

γζ2 /∈ hr(α, n). (1.118)On the other hand, since γζ1 ∈ α ∩ aζ1 ⊂ h[α ∩ aζ1 ] we have

γζ1 ∈ hr(α, n) (1.119)

by (r3). But this is a contradiction because, by (1.116), the first of these relations impliesr x(α) = 0 while the second implies r x(α) = 1.

So it remains to show that r ∈ Pf . Items (P1) - (P4) of Definition 1.11.9 are clear. Also, (P5)holds if j < n because p ∈ Pf . Thus we only have to check (P5) for triples of the form 〈η, ξ, n〉.

If η < ξ 6= α we have η 6∈ h(ξ, n) = ξ, and so hr(η, 0) ∗ hr(ξ, n) = hr(η, 0) ∩ hr(ξ, n) ⊆η ∩ ξ = ∅, hence (P5) of Definition 1.11.9 holds trivially. So assume now that η < α. In viewof the definition of r, our task is to show the following two assertions:

(I) if η ∈ hr(α, n) then h(η, 0) \ hr(α, n) ⊂ h[a ∩ f(η, α)],(II) if η 6∈ hr(α, n) then h(η, 0) ∩ hr(α, n) ⊂ h[a ∩ f(η, α)].

The fact that p = pζ1 + pζ2 and properties ∆4) and ∆5) of our very good ∆-function f will playan essential role in the proofs of (I) and (II).

Proof of (I). First note that by the definition of r we have

h(η, 0) \ hr(α, n) = h(η, 0) \ (h(α, 0) ∩ h[α ∩ aζ1 ]) =

(h(η, 0) \ h(α, 0)) ∪ (h(η, 0) \ h[α ∩ aζ1 ]). (1.120)

Since h(η, 0) \ h(α, 0) ⊂ h[i(η, α, 0)] ⊂ h[a ∩ f(η, α)] is obvious, it is enough to show that(I’) if η ∈ hr(α, n), then h(η, 0) \ h[α ∩ aζ1 ] ⊂ h[a ∩ f(η, α)].If η ∈ aζ1 then h(η, 0) \ h[α ∩ aζ1 ] = ∅ and we are done. So assume

now that η 6∈ aζ1 , that is η ∈ aζ2 \ aζ1 . Now η ∈ h[α∩ aζ1 ] means that there is a ξ ∈ α∩ aζ1 withη ∈ h(ξ, 0). By the definition 1.11.12 (1.108) of the amalgamation then there is δ ∈ aζ1 ∩aζ2 suchthat η < δ ≤ ξ and η ∈ hζ2(δ, 0). Since pζ2 ∈ Pf this implies

hζ2(η, 0) \ hζ2(δ, 0) ⊆ hζ2 [iζ2(η, δ, 0)]. (1.121)

A similar argument, referring back to definition 1.11.12 (1.108), yields us that h(η, 0)\hζ2(η, 0) ⊂h[α ∩ aζ1 ], and as hζ2(δ, 0) ⊂ h(δ, 0) ⊂ h[α ∩ aζ1 ] we may conclude that

h(η, 0) \ h[α ∩ aζ1 ] ⊂ hζ2 [iζ2(η, δ, 0)] ⊂ h[iζ2(η, δ, 0)]. (1.122)

Since η ∈ aζ2 \ aζ1 and δ, α ∈ aζ1 ∩ aζ2 , we have f(η, δ) ⊂ f(η, α) by ∆4). Consequently,

iζ2(η, δ, 0) ⊂ aζ2 ∩ f(η, δ) ⊂ a ∩ f(η, α), (1.123)

completing the proof of (I’) and hence of (I).

Proof of (II). If η 6∈ hr(α, n) then either η 6∈ h(α, 0) or η 6∈ h[α ∩ aζ1 ]. If η 6∈ h(α, 0) then p ∈ Pfimplies

h(η, 0) ∩ hr(α, n) ⊂ h(η, 0) ∩ h(α, 0) =

h(η, 0) ∗ h(α, 0) ⊂ h[i(η, α, 0)] ⊂ h[a ∩ f(η, α]). (1.124)

So assume that η /∈ h[α∩aζ1 ] , clearly then η 6∈ aζ1 as well. Consider any β ∈ h(η, 0)∩hr(α, n),we have to show that β ∈ h[a ∩ f(η, α)].

Case 1. β ∈ aζ1 . By using definition 1.11.12 (1.108) again, then β ∈ h(η, 0) implies that thereis a δ ∈ η ∩ aζ1 ∩ aζ2 with β ∈ hζ2(δ, 0). But then δ ∈ f(η, α) by property ∆5) of very good∆-functions, hence we are done.

Case 2. β 6∈ aζ1 . In this case β ∈ h[α ∩ aζ1 ] implies that there is a δ ∈ α ∩ aζ1 ∩ aζ2 such thatβ ∈ hζ2(δ, 0), hence β ∈ hζ2(η, 0)∩ hζ2(δ, 0). Moreover, η /∈ h[α∩ aζ1 ] implies η 6∈ hζ2(δ, 0). Thusif η < δ then pζ2 ∈ Pf and hζ2(η, 0) ∩ hζ2(δ, 0) = hζ2(η, 0) ∗ hζ2(δ, 0) imply that β ∈ hζ2(γ, 0) for

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some γ ∈ i(η, δ, 0) ⊂ f(η, δ). But we have f(η, δ) ⊂ f(η, α) by ∆4), so γ ∈ a∩ f(η, α) and we aredone.

Finally, if δ < η then δ ∈ f(η, α) because f satisfies ∆5), moreover we have β ∈ hζ2(δ, 0) ⊂h(δ, 0) and the proof of (II) is completed.

This then completes the proof of Lemma 1.11.14 and thus of the first countability of the spaceXH .

1.11.6. ω1-compactness. In this subsection we establish part (iii) of theorem 1.11.10. Thisimplies that every uncountable subset of XH has uncountable intersection with a compact set,hence every set of size ω1 has a complete accumulation point.

Lemma 1.11.15. If p = 〈a, h, n, i〉 ∈ Pf and β ∈ ω2 with β > max a then there is a conditionq ≤ p such that a ⊂ hq(β, 0).

Proof. We define the condition q = 〈a ∪ β, hq, n, iq〉 with the following stipulations: hq ⊃ h,iq ⊃ i, hq(β, j) = a ∪ β for j < n, iq(α, β, j) = ∅ for α ∈ a and j < n. It is straight-forward tocheck that q ∈ Pf is as required.

Lemma 1.11.16. In V [G], for each set A ∈ [ω2×C]ω1 there is β ∈ ω2 such that |A∩U(β)| = ω1.

Proof. Let A be a Pf -name for A and assume that p ∈ G with

p A = zξ : ξ < ω1 ∈[ω2 × C

]ω1.

We may assume that p also forces that zξ : ξ < ω1 is a one-one enumeration of A. For eachξ < ω1 we may pick pξ ≤ p and αξ ∈ ω2 with αξ ∈ apξ such that pξ zξ = 〈αξ, xξ〉. Letsupαξ : ξ < ω1 < β < ω2. By lemma 1.11.15 for each ξ < ω1 there is a condition qξ ≤ pξ suchthat αξ ∈ hqξ(β, 0), hence qξ zξ ∈ U(β). But Pf satisfies CCC, so there is q ∈ G such thatq |ξ ∈ ω1 : qξ ∈ G| = ω1. Clearly, then q |A ∩ U(β)| = ω1.

1.11.7. Countable compactness. In this subsection we show that part (iv) of theorem1.11.10 holds: in V [G], the closure of any infinite subset of XH is either compact or contains a"tail" of XH , that is (ω2 \ α)×C for some α < ω2. Of course, this implies that XH is countablycompact and thus, together with the results of the previous section, establishes the initial ω1-compactness of XH . Moreover, it also implies that XH is normal, for of any two disjoint closedsets in XH (at least) one has to be compact.

We start by proving an extension result for conditions in Pf . We shall use the followingnotation that is analogous to the one that was introduced before lemma 1.11.14.

h1(α,m) = h(α,m), (1.125)

h0(α,m) = h(α, 0) \ h(α,m). (1.126)

Lemma 1.11.17.Assume that p = 〈a, h, n, i〉 ∈ Pf , α ∈ a, and ε : n −→ 2 is a function withε(0) = 1. Then for every η ∈ α \ a there is a condition of the form q = 〈a ∪ η, hq, n, iq〉 ∈ Pfsuch that q ≤ p and

η ∈⋂m<n

hε(m)q (α,m) \ hq[a ∩ α]. (1.127)

Proof. We define hq and iq with the following stipulations:

hq(η,m) = η for m < n,hq(α,m) = h(α,m) ∪ η if m < n and ε(m) = 1,hq(α,m) = h(α,m) if m < n and ε(m) = 0,hq(ν,m) = h(ν,m) ∪ η if ν ∈ a \ α, m < n, and α ∈ h(ν,m),hq(ν,m) = h(ν,m) if ν ∈ a \ α, m < n, and α /∈ h(ν,m),iq ⊃ i, iq(η, ν,m) = ∅ if ν ∈ a \ η , and iq(ν, η,m) = ∅ if ν ∈ a ∩ η .

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To show q ∈ Pf we need to check only (P5). But this follows from the fact that if η ∈hq(ν, 0) ∗ hq(µ,m) then, as can be checked by examining a number of cases, we have ν, µ ∈ a andα ∈ h(ν, 0) ∗ h(µ,m) as well. By p ∈ Pf then there is a ξ ∈ i(ν, µ,m) with α ∈ h(ξ, 0) whichimplies η ∈ hq(ξ, 0) because ε(0) = 1, so we are done. Thus q ∈ Pf , q ≤ p, and q clearly satisfiesall our requirements.

Lemmas 1.11.15 and 1.11.17 can be used to show that

Dα,n = p ∈ Pf : α ∈ ap and n < np

is dense in Pf for all pairs 〈α, n〉 ∈ ω2×ω, showing that dom(H) = ω2×ω and dom(i) = ω2⊗ω2⊗ω.Our next lemma is a partial result on the way to what we promised to show in this section.

Lemma 1.11.18.Assume that, in V [G], we have D ∈ V ∩[ω2

]ω and Y = 〈δ, xδ〉 : δ ∈ D ⊂ω2 × C. Then

(ω2 \ sup(D))× C ⊂ Y .

Proof. By lemma 1.11.5 it suffices to prove that

V [G] |=( ⋂

1≤m<n

U(α,m, ε(m)) \ U [b])∩ Y 6= ∅ (1.128)

whenever α ∈ ω2 \ supD , n ∈ N, ε : n −→ 2 with ε(0) = 1, and b ∈[α]<ω. So fix these and

pick a condition p = 〈a, h, k, i〉 ∈ Pf such that α ∈ a, b ⊂ a , and n < k. (We know that the setE of these conditions is dense in Pf .) Let us then choose δ ∈ D \ a. By lemma 1.11.17 there is acondition q ≤ p such that

δ ∈⋂

1≤m<n

hε(m)q (α,m) \ hq[b]. (1.129)

Thenq 〈δ, xδ〉 ∈

⋂1≤m<n

U(α,m, ε(m)) \ U [b], (1.130)

henceq

( ⋂1≤m<n

U(α,m, ε(m)) \ U [b])∩ Y 6= ∅ . (1.131)

Since p ∈ E was arbitrary, the set of q’s satisfying the last forcing relation is also dense in Pf , sowe are done.

We need a couple more, rather technical, results before we can turn to the proof of part (iv)of theorem 1.11.10. First we give a definition.

Definition 1.11.19. (1) Assume that p = 〈a, h, n, i〉 ∈ Pf and a < b ∈ [ω2]<ω are such thata ⊂ f(γ, γ′) for any γ, γ′ ∈ [b]2. Then we define the b-extension of p to be the condition q ofthe form q = 〈a ∪ b, hq, n, iq〉 with h ⊂ hq, i ⊂ iq, and the following stipulations:(R1) hq(γ, `) = a ∪ γ for γ ∈ b and ` < n,(R2) iq(γ′, γ, `) = a for γ′, γ ∈ b with γ′ < γ and ` < n,(R3) iq(ξ, γ, `) = ∅ for ξ ∈ a, γ ∈ b, and ` < n.

(2) If q ∈ Pf and b ⊂ aq then s ≤ q is said to be a b-fair extension of q iff hs(γ, j) = hs(γ, 0)holds for any γ ∈ b and nq ≤ j < ns.

Our following result shows that the b-extension severely restricts any further extensions.

Lemma 1.11.20.Assume that p = 〈a, h, n, i〉 ∈ Pf , a < b, and q is the b-extension of p. If s ≤ qis any extension of q then

hs[a] = hs(γ′, 0) ∩ hs(γ, `) (1.132)

whenever 〈γ′, γ, `〉 ∈ b⊗ b⊗ n. If, in addition, s is a b-fair extension of q then (1.132) holds forall 〈γ′, γ, `〉 ∈ b⊗ b⊗ ns.

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Proof. We have γ′ /∈ hs(γ) by (R1) and s ≤ q, hence if ` < n then (P5) and (R2) imply

hs(γ′, 0) ∩ hs(γ, `) = hs(γ′, 0) ∗ hs(γ, `) ⊂ hs[is(γ′, γ, `)] = hs[a]. (1.133)

Similarly, for all ξ ∈ a, γ′′ ∈ b, and `′′ < n we have

hs(ξ, 0) \ hs(γ′′, `′′) = hs(ξ, 0) ∗ hs(γ′′, `′′) ⊂ hs[is(ξ, γ′′, `′′)] = hs[∅] = ∅, (1.134)

which implies hs[a] ⊂ hs(γ′′, `′′). But then hs[a] ⊂ hs(γ′, 0)∩hs(γ, `) which together with (1.133)yields (1.132).

Now, if s is a b-fair extension of q and 〈γ′, γ, `〉 ∈ b ⊗ b ⊗ ns with n ≤ ` < ns then we have(1.132) because hs(γ, 0) = hs(γ, `) and hs[a] = hs(γ′, 0) ∩ hs(γ, 0).

In our next result we are going to make use of the following simple observation.

Fact 1.11.21. If p = 〈a, h, n, i〉 ∈ Pf and X ⊂ a is an initial segment of a then p X =〈X, h X × n, n, i X ⊗X ⊗ n〉 ∈ Pf as well.

Lemma 1.11.22. Let p, q, s ∈ Pf be conditions and Q ⊂ S < E < F be sets of ordinals such that

ap = Q ∪ E , aq = Q ∪ E ∪ F , as = S ∪ E ∪ F ,q is the F -extension of p , and s is an F -fair extension of q. Assume, moreover, that |E| = kwith E = γi : i < k the increasing enumeration of E and |F | = 2k, F = γi,0, γi,1 : i < k withγi,0 < γi,1 satisfying

∀i < k ∀ξ ∈ S [ f(ξ, γi) = f(ξ, γi,0) = f(ξ, γi,1) ]. (1.135)

Let us now define r = 〈ar, hr, nr, ir〉 as follows:(A) ar = S ∪ E, nr = ns,(B) for ξ ∈ ar and j < nr let

hr(ξ, j) =hs(ξ, j) ∪ (S \ hs[ap]) if ξ = γi and γ0 ∈ hs(γi, j),hs(ξ, j) otherwise,

(C) for 〈ξ, η, j〉 ∈ ar ⊗ ar ⊗ nr

ir(ξ, η, j) =is(ξ, η, j) if ξ, η ∈ ap or ξ, η ∈ S,f(ξ, η) ∩ as otherwise.

Then r ∈ Pf , r ≤ p, r ≤ s S ∈ Pf , and S \ hs[ap] ⊂ hr(γ0, 0).

Proof. Lemma 1.11.22 corresponds to Lemma 1.10.23.The proof is quite technical, but it is similar to the proof of Lemma 1.10.14. So we omit it.

See in [11]

Proof of theorem 1.11.10: Property (iv). Our aim is to prove that the following statementholds in V [G]:(iv) If the closure Y of a set Y ∈ [XH ]ω is not compact then there is α < ω2 such that (ω2 \

α)× C ⊂ Y .We shall make use of the following easy lemma.

Lemma 1.11.23.A set Z ⊂ XH has compact closure if and only if

Γ = γ : ∃x 〈γ, x〉 ∈ Z ⊂ H[F ]

for some finite set F ⊂ ω2.

Proof of the lemma. If Z is compact then there is a finite set F ⊂ ω2 such that Z ⊂ U [F ].Clearly, then Γ ⊂ H[F ].

Conversely, if Γ ⊂ H[F ] for a finite F ⊂ ω2 then Z ⊂ U [F ], hence Z ⊂ U [F ] as well. But asU [F ] is compact, so is Z.

Given two sets X,E ⊂ ω2 with X < E we shall write

clf (X,E) = (the f -closure of X ∪ E) ∩ sup(X). (1.136)

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Fact 1.11.24. If ξ ∈ clf (X,E) and η ∈ clf (X,E) ∪ E then f(ξ, η) ⊂ clf (X,E).

Let us now fix a regular cardinal ϑ that is large enough so that Hϑ, the structure of setswhose transitive closure has cardinality < ϑ, contains everything relevant.

Lemma 1.11.25.Assume that

V [G] |= Γ ∈[ω2

]ω is not covered by finitely many H(ξ, 0) (1.137)

and Γ is a Pf -name for Γ. If M is a σ-closed elementary submodel of Hθ (in V ) such thatf, Γ ∈M , |M | = ω1, and δ = M ∩ ω2 ∈ ω2 then

V [G] |= Γ ∩H(δ, 0) \H[D] 6= ∅ for each finite D ⊂ δ. (1.138)

Proof of the lemma 1.11.25. Fix D ∈ [δ]<ω and a condition p ∈ Pf with D ∪ δ ⊂ ap suchthat

p “Γ ∈[ω2

]ω is not covered by finitely many H(ξ, 0)”. (1.139)We shall be done if we can find a condition r ≤ p and an ordinal α ∈ ar such that

r “α ∈ Γ” and α ∈ hr(δ, 0) \ hr[D]. (1.140)

Let Q = ap ∩ δ, E = ap \ δ, and γi : i < k be the increasing enumeration of E. In particular,then we have γ0 = δ.

To achieve our aim, we first choose a countable elementary submodel N of Hθ such thatM, Γ, p ∈ N and put

A = δ ∩N and B = clf (A ∪Q,E) .Note that we have A,B ∈ M because M is σ-closed. For each i < k the function f( . , γi) B isin M , hence so is the set

Ti = γ ∈ ω2 : ∀β ∈ B f(β, γ) = f(β, γi) ,and γi ∈ Ti \M implies |Ti| = ω2.

By Lemma 1.11.7 there is a set of 2k ordinals

F = γi, ε : i < k, ε < 2with γi, ε ∈ Ti and γi,0 < γi,1 for each i < k such that

B ∪ E ⊂⋂f(γi, ε , γi′, ε′) : 〈i, ε〉 , 〈i′, ε′〉 ∈ [k × 2]2. (1.141)

Since ap ⊂ B∪E < F , (1.141) implies that we can form the F -extension q = 〈ap ∪ F, hq, np, iq〉 ∈Pf of p , see definition 1.11.19.

As p “H[Q ∪ E] 6⊃ Γ”, there is a condition t ≤ q and an ordinal α such that

t “α ∈ Γ \H[Q ∪ E]′′. (1.142)

Clearly we can assume that α ∈ at, and then

t “α ∈ Γ” and α ∈ at \ ht[Q ∪ E]. (1.143)

Since Γ ∈ N ∩M and Pf is CCC, we have α ∈ M ∩ N ∩ ω2 = N ∩ δ. As Pf is CCC andα, Γ ∈ M ∩N we may choose a maximal antichain W ⊂ w ≤ p : w α ∈ Γ with W ∈ N ∩Mand henceW ⊂ N ∩M . By taking a further extension we can assume that t ≤ w for some w ∈W .

We claim that, putting S = B ∩ at, we have

it(ξ, η, j) ⊂ S ∪ E for each 〈ξ, η, j〉 ∈ S ∪ E ∪ F ⊗ S ∪ E ∪ F ⊗ np . (1.144)

Indeed, if ξ ∈ S ⊂ B then fact 1.11.24 and γi , ε ∈ Ti imply f(ξ, η) ⊂ B and so it(ξ, η, j) ⊂ S, andif ξ, η ∈ E ∪ F then

it(ξ, η, j) = iq(ξ, η, j) ⊂ ap = Q ∪ E ⊂ S ∪ Ebecause q is the F -extension of p.

Let us now make the following definitions:(s1) as = S ∪ E ∪ F ,(s2) hs(ξ, j) = ht(ξ, j) ∩ S = ht(ξ, j) ∩ as for ξ ∈ S and j < nt,(s3) is S ⊗ S ⊗ nt = it S ⊗ S ⊗ nt,

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(s4) for η ∈ E ∪ F and j < nt let

hs(η, j) =ht(η, j) ∩ as if j < np,ht(η, 0) ∩ as if np ≤ j < nt,

(1.145)

(s5) for η ∈ E ∪ F , ξ ∈ as ∩ η and j < nt let

is(ξ, η, j) =it(ξ, η, j) if j < np,it(ξ, η, 0) if np ≤ j < nt.

(1.146)

Then (1.144) and t ∈ Pf imply that s = 〈as, hs, nt, is〉 ∈ Pf , moreover s is an F -fair (evenE ∪ F -fair) extension of q.

Note that t ≤ w and aw ⊂ A ⊂ B implies aw ⊂ S, hence by the definition of the condition swe have s ≤ w and even s S ≤ w.

Things were set up in such a way that we can apply lemma 1.11.22 to the three conditionss ≤ q ≤ p and the sets Q ⊂ S < E < F to get a condition r ∈ Pf such that

• r ≤ p, r ≤ s S ≤ w,• α ∈ S \ hs[ap] ⊂ hs(γ0).

Since δ = γ0 andD ⊂ ap, we have α ∈ hr(δ)\hr[D]. Moreover, r ≤ s S ≤ w implies r “α ∈ Γ”.So r satisfies (1.140), which completes the proof of our lemma.

Assume now, to finish the proof of (iv), that

V [G] |= Y ∈[ω2 × C

]ω and Y is not compact. (1.147)

Then, by lemma 1.11.23, Γ = γ : ∃x ∈ C 〈γ, x〉 ∈ Y ∈ [ω2]ω can not be covered by finitelymany H(ξ, 0). Let Γ be a Pf -name for Γ.

Claim: If M is a σ-closed elementary submodel of Hθ with f, Γ ∈M , |M | = ω1, δ = M ∩ω2 ∈ ω2

then (δ × C) ∩ Y 6= ∅.Assume, on the contrary, that (δ×C)∩ Y = ∅. Then, as U(δ)∩ Y is compact, U(δ)∩ Y ⊂

U(δ) ∩ Y ⊂ U [D] for some finite set D ⊂ δ consequently we have Γ ∩ H(δ, 0) ⊂ H[D]. this,however, contradicts lemma 1.11.25 by which

Γ ∩H[δ,D] 6= ∅ for each finite D ⊂ δ. (1.148)

This contradiction proves our claim.

Since CH holds in V , the set S of ordinals δ ∈ ω2 that arise in the form δ = M ∩ ω2 for anelementary submodel M ≺ Hθ as in the above claim is unbounded (even stationary) in ω2. LetD be the set of the first ω elements of S. Then D ∈ V ∩ [ω2]ω and our claim implies that, inV [G], for each δ ∈ D there is xδ ∈ C with 〈δ, xδ〉 ∈ Y . But then, by lemma 1.11.18, for α = supDwe have

(ω2 \ α)× C ⊂ 〈δ, xδ〉 : δ ∈ D ⊂ Y . (1.149)This completes the proof of theorem 1.11.10.

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CHAPTER 2

Combinatorial principles

2.1. Combinatorial principles from adding Cohen reals

(This section is based on [12] and [1] )

The last 40 years have seen a furious activity in proving results that are independent ofthe usual axioms of set theory, that is ZFC. As the methods of these independence proofs (e.g.forcing or the fine structure theory of the constructible universe) are often rather sophisticated,while the results themselves are usually of interest to “ordinary” mathematicians (e.g. topologistsor analysts), it has been natural to try to isolate a relatively small number of principles, i.e.independent statements that a) are simple to formulate and b) are useful in the sense that theyhave many interesting consequences. Most of these statements, we think by necessity, are ofcombinatorial nature, hence they have been called combinatorial principles.

In this section we present several new combinatorial principles that are all statements aboutP(ω), the power set of the natural numbers. In fact, they all concern matrices of the form〈A(α, n) : 〈α, n〉 ∈ κ× ω〉, where A(α, n) ⊂ ω for each 〈α, n〉 ∈ κ × ω, and, in the interestingcases, κ is a regular cardinal with c = 2ω ≥ κ > ω1.

We show that these statements are valid in the generic extensions obtained by adding anynumber of Cohen reals to any ground model V , assuming that the parameter κ is a regular andω-inaccessible cardinal in V ( i.e. λ < κ implies λω < κ).

Then we present a large number of consequences of these principles, some of them com-binatorial but most of them topological, mainly concerning separable and/or countably tighttopological spaces. (This, of course, is not surprising because these are objects whose structuredepends basically on P(ω).)

2.1.1. The combinatorial principles. The principles we formulate here are all statementson κ × ω matrices of subsets of ω claiming – roughly speaking – that all these matrices containlarge “submatrices” satisfying certain homogeneity properties.

To simplify the formulation of our results we introduce the following pieces of notation. If Sis an arbitrary set and k is a natural number then let

(S)k = s ∈ Sk : | ran s| = k

and

(S)<ω =⋃k<ω

(S)k.

For D0, . . . , Dk−1 ⊂ S we let

(D0, . . . , Dk−1) = s ∈ (S)k : ∀i ∈ k (s(i) ∈ Di).Definition 2.1.1. If S is a set of ordinals denote by M(S) the family of all S × ω-matrices ofsubsets of ω, that is, A ∈ M(S) if and only if A = 〈A(α, i) : α ∈ S, i < ω〉, where A(α, i) ⊂ ωfor each α ∈ S and i < ω. If A = 〈A(α, i) : α ∈ S, i < ω〉 ∈ M(S) and R ⊂ S we define therestriction of A to R, A | R in the straightforward way: A | R = 〈A(α, i) : α ∈ R, i < ω〉. IfA = 〈A(α, i) : α ∈ S, i < ω〉 ∈ M(S), t ∈ ω<ω and s ∈ (S)|t| then we let

A(s, t) =⋂i<|t|

A(s(i), t(i)).

83

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84 2. COMBINATORIAL PRINCIPLES

Now we formulate our first and probably most important principle that we call Cs(κ). Wealso specify a weaker version of Cs(κ) denoted by C(κ) because in most of the applications (2.1.23,2.1.25, 2.1.30, 2.1.33, 2.1.37, 2.1.43 ) we don’t need the full power of Cs(κ).

Definition 2.1.2. For T ⊂ ω<ω a matrix A = 〈A(α, i) : α ∈ S, i ∈ ω〉 ∈ M(S) is called T -adic iffor each t ∈ T and s ∈ (S)|t| we have A(s, t) 6= ∅.Definition 2.1.3. For κ = cf(κ) > ω principle Cs(κ) (C(κ)) is the following statement:For every T ⊂ ω<ω and A ∈M(κ) we have (1) or (2) below:(1) there is a stationary (cofinal) set S ⊂ κ such that A | S is T -adic,(2) there are t ∈ T and stationary (cofinal) subsets D0, D1, . . . , D|t|−1 of κ such that for every

s ∈ (D0, . . . , D|t|−1) we haveA(s, t) = ∅.

Next we formulate a dual version of principles Cs(κ) and C(κ). Let us remark that we don’tknow whether Cs(κ) (C(κ)) implies Cs(κ) (C(κ)) or vice versa.

Definition 2.1.4. If κ = cf(κ) > ω, then principle Cs(κ) (C(κ)) is the following statement:For every T ⊂ ω<ω and A ∈M(κ) we have (1) or (2) below:(1) there is a stationary (cofinal) set S ⊂ κ such that for each t ∈ T and s ∈ (S)|t|

|A(s, t)| < ω,

(2) there are t ∈ T and stationary (cofinal) subsets D0, D1, . . . , D|t|−1 of κ such that for everys ∈ (D0, . . . , D|t|−1) we have

|A(s, t)| = ω.

Let us remark that in the “plain” dual of principle Cs(κ) we should have |A(s, t)| = ∅ in 2.1.41and |A(s, t)| 6= ∅ in 2.1.42, but this “principle” is easily provable in ZFC.

The principles D(κ) and Ds(κ) that we introduce next easily follow from C(κ) and Cs(κ),respectively, but as their formulation is much simpler, we thought it to be worth while to havethem as separate principles. We first give two auxiliary definitions.

Definition 2.1.5. If A = 〈A(α, i) : α < κ, i < ω〉 ∈ M(κ), then we set

A = Y ⊂ ω : |α < κ : ∃i < ω A(α, i) ⊂ Y | = κ

andAs = Y ⊂ ω : α < κ : ∃i < ω A(α, i) ⊂ Y is stationary in κ.

Now we can formulate Ds(κ) (D(κ)) as follows.

Definition 2.1.6. For κ = cf(κ) > ω principle Ds(κ) (D(κ)) is the following statement:If A ∈M(κ) and As (A) is centered then there is a stationary (cofinal) set S ⊂ κ such that A | Sis ω<ω-adic.

Theorem 2.1.7.Cs(κ) (C(κ)) implies Ds(κ) (D(κ)).

Proof. We give the proof only for Ds(κ) because the same argument works for D(κ).Let A ∈M(κ) so that As is centered and put T = ω<ω. By Cs(κ) either 2.1.3(1) or 2.1.3(2)

holds.If S ⊂ κ witnesses 2.1.3(1) for our T then A | S is clearly ω<ω-adic. So it is enough to show

that 2.1.3(2) can not hold.Assume, on the contrary, that there are t ∈ T = ω<ω and stationary subsets D0, D1, . . . ,

D|t|−1 of κ such that for each s ∈ (D0, . . . , D|t|−1) we have

A(s, t) = ∅. (+)

We can obviously assume that the sets Di are pairwise disjoint. Let Xi =⋃A(δ, t(i)) : δ ∈ Di

for every i < |t|. Then clearlyXi ∈ As for every i < |t|, while (+) implies⋂i<k

Xi = ∅, contradicting

that As is centered.

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2.1. COMBINATORIAL PRINCIPLES FROM ADDING COHEN REALS 85

Definition 2.1.8. If κ = cf(κ) > ω, then principle F s(κ) (F (κ)) is the following statement:For every T ⊂ ω<ω and A ∈M(κ) (1) or (2) below holds:(1) there is a stationary (cofinal) set S ⊂ κ such that

|A(s, t) : t ∈ T and s ∈ (S)|t|| ≤ ω.

(2) there are t ∈ T and stationary (cofinal) subsets D0, D1, . . . , D|t|−1 of κ such that if s0, s1 ∈(D0, . . . , D|t|−1) with s0(i) 6= s1(i) for each i < |t| then we have

A(s0, t) 6= A(s1, t).

Clearly, if t, D0, . . . ,D|t|−1 satisfy (2) then

|A(s, t) : s ∈ (D0, . . . , D|t|−1)| = κ.

There is a surprising connection between these principles and the dual versions Cs(κ) (C(κ))of Cs(κ) (C(κ)), respectively.

Theorem 2.1.9. F s(κ) (F (κ)) implies Cs(κ) (C(κ)).

Proof. Let A ∈ M(κ) and T ⊂ ω<ω and apply F s(κ) to A and T . Assume first that there is astationary set S ⊂ κ such that the family

I = A(s, t) : t ∈ T and s ∈ (S)|t|

is countable.Now for t ∈ T , i < |t| and I ∈ I ∩

[ω]ω set

D(I, t, i) = α ∈ S : A(α, t(i)) ⊃ I.

If for some t ∈ T and I ∈ I ∩[ω]ω the set D(I, t, i) is stationary for each i < |t| then this t and

the sets D(I, t, 0), . . . , D(I, t, |t| − 1) witness 2.1.42.So we can assume that for all t ∈ T and I ∈ I ∩

[ω]ω the set

b(I, t) = i < |t| : D(I, t, i) is non-stationary in κ

is not empty. Then the set

D =⋃D(I, t, i) : I ∈ I ∩

[ω]ω, t ∈ T, i ∈ b(I, t)

is not stationary and so S′ = S \ D is stationary. We claim that S′ witnesses 2.1.41. Assumeon the contrary that t ∈ T , s ∈ (S′)|t| and I = A(s, t) is infinite. Then I ∈ I ∩

[ω]ω and

s(i) ∈ D(I, t, i) for each i < |t|. Since s(i) /∈ D it follows that D(I, t, i) is stationary for eachi < |t|, that is, b(I, t) = ∅, which is a contradiction.

Assume now that there are t ∈ T and stationary subsets D0, D1, . . . , D|t|−1 of κ such thatif s0, s1 ∈ (D0, . . . , D|t|−1) with s0(i) 6= s1(i) for each i < |t| then

A(s0, t) 6= A(s1, t).

We show that in this case again 2.1.42 holds. Indeed, for each I ∈[ω]<ω pick sI ∈ (D0, . . . , D|t|−1)

such that A(sI , t) = I provided that there is such an s. Let R =⋃sI(i) : I ∈

[ω]<ω

, i < |t| andD′i = Di \R for i < |t|. Now if s ∈ (D′0, . . . , D

′|t|−1) then for any I ∈

[ω]<ω we have sI(i) 6= s(i)

for each i < |t|, hence I = A(sI , t) 6= A(s, t). As I was an arbitrary element of[ω]<ω we conclude

that |A(s, t)| = ω.

The following observation is almost trivial.

Proposition 2.1.10. If κ = cf(κ) > c then Cs(κ) and F s(κ) are valid.

As was mentioned, our principles are of interest only for κ > ω1. In fact, for κ = ω1, theyare all false!

Theorem 2.1.11 ([12]). C(ω1) (and so F (ω1) too) and D(ω1) are both false.

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2.1.2. Consistency of the principles in the Cohen model. A cardinal κ is ω-inaccessibleif λω < κ holds for each λ < κ. Given any infinite set I we denote by CI the poset Fn(I, 2, ω),i.e. the standard one adding |I|-many Cohen reals.

In this subsection we prove that if κ is a regular ω-inaccessible cardinal in some ground modelV and we add λ-many Cohen reals to V , where λ is an arbitrary cardinal, then in the extensionthe principles Cs(κ), Cs(κ) and F s(κ) are all satisfied. As we remarked in subsection 2.1.1 abovethe case κ > λ is trivial, while the case κ < λ can be reduced to the case κ = λ.

Since the proof of the latter is long and technical, we first sketch the main idea. So let usbe given a matrix A ∈ M(κ) and a set T ⊂ ω<ω in V [G], where G is Cκ-generic over V . In thefirst part of the proof we find a set I ∈

[κ]ω and a stationary set S ⊂ κ such that in V [G | I] the

sequences 〈A(α, i) : i < ω〉 for α ∈ S have also pairwise isomorphic names with disjoint supports(contained in κ \ I). This reduction, carried out in lemma 2.1.17, will be the place where weuse that κ is regular and ω-inaccessible in V. In the second part of the proof, using slightlydifferent arguments for Cs(κ) and for F s(κ), we show that if some A ∈ M(S) has names withthese properties then either S witnesses 2.1.3(1) (or 2.1.8(1), respectively) or some stationarysets Di ⊂ S witness 2.1.3(2) (or 2.1.8(2), respectively). In this second step we don’t use that κ isω-inaccessible or regular.

In our forcing arguments we follow the notation of Kunen [77]. Let us first recall definition[77, 5.11].

Definition 2.1.12.A CI -name B of a subset of some ordinal µ is called nice if for each ν < µthere is an antichain Bν ⊂ CI such that

B = 〈p, ν〉 : ν ∈ µ ∧ p ∈ Bν =⋃Bν × ν : ν ∈ µ.

We let supp(B) =⋃dom(p) : p ∈

⋃ν<µ

Bν.

It is well-known (see e.g. lemma [77, 5.12]) that every set of ordinals in V [G] has a nice namein V .

If ϕ is a bijection between two sets I and J then ϕ lifts to a natural isomorphism between CIand CJ , which will be also denoted by ϕ, as follows: for p ∈ CI let dom(ϕ(p)) = ϕ′′ dom(p) andϕ(p)(ϕ(ξ)) = p(ξ). Moreover ϕ also generates a bijection between the nice CI -names and the niceCJ -names (see [77, 7.12]): if B is a nice CI -name then let ϕ(B) =

⟨ϕ(p), ξ

⟩:⟨p, ξ⟩∈ B. If

I and J are sets of ordinals with the same order type then ϕI,J is the natural order-preservingbijection from I onto J .

Definition 2.1.13.Assume I, J ⊂ κ, moreover Ai and Bi are nice Cκ-names of subsets of ωfor i < ω, such that supp(Ai) ⊂ I and supp(Bi) ⊂ J . We say that the structures of names⟨I, Ai : i < ω

⟩and

⟨J, Bi : i < ω

⟩are twins if I and J have the same order type and for the

order preserving bijection ϕI,J we have(1) ϕI,J is the identity on I ∩ J ,(2) ϕI,J(Ai) = Bi for each i < ω.

Definition 2.1.14.Assume that I ⊂ κ, G is a Cκ-generic filter over V and H = G | I. If B is anice Cκ-name of a subset of some ordinal µ we define in V [H] the Cκ\I name πH(B) as follows:

πH(B) = 〈p | κ \ I, ν〉 : 〈p, ν〉 ∈ B ∧ p | I ∈ H.

Lemma 2.1.15. πH(B) is a nice Cκ\I-name in V [H] and supp(πH(B)) ⊂ supp(B) \ I, moreover

val(πH(B), G | (κ \ I)) = val(B,G).

Proof. Straightforward from the construction.

Definition 2.1.16.Assume that S ⊂ κ. A matrix B =⟨B(α, i) : α ∈ S, i < ω

⟩of nice Cκ-names

of subsets of ω is called a nice S-matrix if conditions (i) and (ii) below hold:(i) putting Jα =

⋃i<ω supp(B(α, i)) the sets Jα : α ∈ S are pairwise disjoint,

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(ii) the structures of names ⟨Jα, B(α, i) : i < ω

⟩: α ∈ S are pairwise twins.

We denote by N (S) the family of nice S-matrices.

Lemma 2.1.17. (Reduction lemma) Assume that κ is a regular, ω-inaccessible cardinal, Gis Cκ-generic over V and A ∈ M(κ) in V [G]. Then there are a countable set I ⊂ κ and astationary set S ⊂ κ in V such that, in V [G | I], there is B ∈ N (S) satisfying V [G] |= “A(α, i) =val(B(α, i), G | (κ \ I))” for each α ∈ S and i ∈ ω.

Proof. Assume that

1Cκ “A =⟨A(α, i) : α < κ, i < ω

⟩∈M(κ).”

We can assume that all the names A(i, α) are nice. Let Iα =⋃i<ω supp(A(α, i)).

We need a strong version of Erdős-Rado ∆-system theorem saying that there is a stationaryset T ⊂ κ such that Iα : α ∈ T forms a ∆-system with some kernel I, moreover sup I < min Iα\Ifor each α ∈ T . Although this statement is well-known, in [12] we presented a proof because wecould not find any reference to it.

Erdős-Rado Theorem . If κ is an ω-inaccessible regular cardinal and X = Xα : α < κ isa family of countable sets then there is a stationary set I ⊂ κ such that Xα : α ∈ I forms a∆-system.

Since 2ω < κ = cf(κ) and there are only 2ω different isomorphism types of structures of namesthere is a stationary set S ⊂ T such that the structures of names

⟨Iα, A(α, i) : i < ω

⟩: α ∈ S

are pairwise twins.From now on we work in V [G | I]. Let B(α, i) = πG | I(A(α, i)) for α ∈ S and i ∈ ω. Then

supp(B(α, i)) ⊂ Jα = Iα \ I and the structures of names⟨Jα, B(α, i) : i < ω

⟩are pairwise twins

by lemma 2.1.15 above. Thus B =⟨B(α, i) : α ∈ S, i < ω

⟩∈ N (S).

Definition 2.1.18.Assume that S ⊂ κ. A sequence B =⟨⟨Jα, Bα

⟩: α ∈ S

⟩is called a nice

S-sequence if conditions (i) and (ii) below hold:(i) Jα ∈

[κ]ω, Bα is a nice CJα -name, and Jα for α ∈ S are pairwise disjoint,

(ii) the structures of names⟨Jα, Bα

⟩for α ∈ S are pairwise twins.

We denote by S(S) the family of nice S-sequences.

Lemma 2.1.19. (Homogeneity lemma) Assume that S ⊂ κ and B =⟨⟨Jα, Bα

⟩: α ∈ S

⟩is a

nice S-sequence. If ϕ(x0, x1, . . . , xn−1, z) is a formula with free variables x0, x1, . . . , xn−1, z andZ is an element of the ground model, then (1) or (2) below holds:(1) 1Cκ “ ϕ(Bs(0)Bs(1), . . . , Bs(n−1), Z) for all s ∈ (S)n”,(2) for some r ∈ Cκ we have

r “ there are subsets D0, D1, . . . , Dn−1 of S such that(a) for each i < n and A ∈

[S]ω ∩ V we have Di ∩A 6= ∅,

(b) ¬ϕ(Bs(0)Bs(1), . . . , Bs(n−1), Z) for all s ∈ (D0, D1, . . . , Dk−1)”.

Proof. Assume that (1) fails, that is, there are p ∈ Cκ and s ∈ (S)k such that

p “¬ϕ(Bs(0), . . . , Bs(n−1), Z)”.

Let J =⋃i<k

Js(i) and p′ = p | J and r = p \ p′. Since the sets Jα are pairwise disjoint we can

assume that dom(r) ∩ Jα = ∅ for each α ∈ S.For 〈α, β〉 ∈ S2 we denote by ϕα,β the natural order preserving bijection between Jα and Jβ .

For β ∈ S and i < k let p(β, i) = ϕs(i),β(p | Js(i)). For i < k define the Cκ-name Di of a subset of

S as follows: Di = ⟨p(β, i), β

⟩: β ∈ S. Then

V [G] |= “Di = val(Di, G) = β ∈ S : p(β, i) ∈ G”,

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where G is Cκ-generic over V . Since the supports of p(β, i) for β ∈ S are pairwise disjoint astandard density argument gives that Di ∩A 6= ∅ whenever A ∈

[S]ω ∩ V , hence (a) holds.

To show (b) assume that r ∈ G and

V [G] |= “u ∈ (D0, . . . , Dk−1).”

Since u is finite we have u ∈ V . Let J∗ =⋃i<k

Ju(i) and ψ =⋃i<k

ϕu(i),s(i). Then ψ is a bijection

between J∗ and J and so it extends to isomorphisms between CJ∗ and CJ , and between the familiesof nice CJ∗ -names and of nice CJ -names. Let Ψ be the natural extension of ψ to a permutationof κ:

Ψ(ν) =

ψ(ν) if ν ∈ J∗,ψ−1(ν) if ν ∈ J ,ν if ν ∈ κ \ (J ∪ J∗).

Then Ψ extends to an automorphism of Cκ, and also to an automorphism of nice Cκ-names.Clearly if q ∈ CJ∗ and B is a nice CJ∗ -name then ψ(q) = Ψ(q) and ψ(B) = Ψ(B). Observe thatΨ(r) = r and Ψ(Z) = Z.

Let G∗ = Ψ′′G. Then G∗ is also a Cκ-generic filter over V and since Ψ(Bu(i)) = Bs(i) itfollows that

val(Bu(i), G) = val(Bs(i), G∗). (•)But p(u(i), i) ∈ G, so p | Js(i) = ψ(p(u(i), i)) ∈ G∗. Thus p = r ∪

⋃i<k

p | Js(i) ∈ G∗ as well. Since

p ¬ϕ(Bs(0), . . . , Bs(n−1), Z) and so V [G∗] |= “¬ϕ(Bs(0), . . . , Bs(n−1), Z)”, by (•) this implies

V [G] |= “¬ϕ(Bu(0), . . . , Bu(n−1), Z)”

which was to be proved.

Theorem 2.1.20. If κ is a regular, ω-inaccessible cardinal then for each cardinal λ we have

V Cλ |= “Cs(κ) and Cs(κ) hold.”

Proof. We deal only with Cs(κ) because the same argument works for Cs(κ). As we observedin section 2.1.1 we can assume that κ ≤ λ. First we investigate the case λ = κ.

Assume that

1Cκ “A =⟨A(α, i) : α < κ; i < ω

⟩∈M(κ) and T ⊂ ω<ω.”

Applying the reduction lemma 2.1.17 and that T is countable we can find a countable set I ⊂ κand a stationary set S ⊂ κ in V and a nice S-matrix B in V [G | I] such that

V [G] |= “ val(A(α, i), G) = val(B(α, i), G | (κ \ I))”

for α ∈ S and i ∈ ω, moreover T ∈ V [G | I].We show that for each q ∈ Cκ there is a condition r ≤ q in Cκ such that r “2.1.31 or 2.1.32

holds”. Let I ′ = I ∪ dom(q).For each t ∈ T let ϕt(x0, . . . , x|t|−1) be the following formula:

ϕ(〈B0,k : k < ω〉 , . . . ,⟨B|t|−1,k : k < ω

⟩)⇐⇒

⋂i<|t|

Bi,t(i) 6= ∅.

Applying the homogeneity lemma 2.1.19 to V [G | I ′] as our ground model and to every ϕt we getthat either q “2.1.31 holds” or q ∪ p “2.1.32 holds”. Let us remark that 2.1.19(2)(a) impliesthat as S is stationary, so is each Di.

Thus we have proved the theorem in the case κ = λ. If λ > κ and A ∈ (M(κ))V [G], whereG is Cλ-generic over V , then there is J ∈

[λ]κ such that A ∈ V [G | J ]. The stationary sets that

witness 2.1.3(1) or 2.1.3(2) in V [G | J ] remain stationary in V [G], and so we are done.

Theorem 2.1.21. If κ is a regular, ω-inaccessible cardinal then for each cardinal λ we have

V Cλ |= “F s(κ) holds.”

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Proof. As in 2.1.20 the important case is when λ = κ, because the case λ < κ is trivial and thecase κ < λ can be reduced to the case κ = λ.

So assume that1Cκ “A =

⟨A(α, i) : α < κ, i < ω

⟩∈M(κ).”

Applying lemma 2.1.17 we can find a countable set I ⊂ κ, a stationary set S ⊂ κ in V and inV [G | I] a nice S-matrix B such that V [G] |= “A(α, i) = val(B(α, i), G | (κ \ I))” for each α ∈ Sand i ∈ ω, moreover T ∈ V [G | I].

We need the following lemma that is probably well-known.

Lemma 2.1.22 ([12]). If H is a Cκ-generic filter over V and I, J are disjoint subsets of κ then

V [H] |= “P(ω) ∩ V [H | I] ∩ V [H | J ] = P(ω) ∩ V.”

To conclude the proof we show that if q ∈ Cκ then there is a condition r ≤ q in Cκ such thatr “2.1.81 or 2.1.82 holds”. Let I ′ = I ∪ dom(q).

For each t ∈ T let ϕt(x0, . . . , x|t|−1) be the following formula:

ϕ(〈B0,k : k < ω〉 , . . . ,⟨B|t|−1,k : k < ω

⟩)⇐⇒

⋂i<|t|

Bi,t(i) ∈ (P(ω))V .

Applying the homogeneity lemma 2.1.19 to V [G | I ′] as our ground model we get that (A) or (B)below holds:(A) 1Cκ “ B(s, t) ∈ (P(ω))V for each t ∈ T and s ∈ (S)|t|,”(B) for some t ∈ T and p ∈ Cκ we have

p “ there are subsets D0, D1, . . . , D|t|−1 of S such that(a) for each A ∈

[S]ω ∩ V we have A ∩ Di 6= ∅ for each i < |t|,

(b) If s ∈ (D0, D1, . . . , D|t|−1) we have

˙B(s, t) /∈ (P(ω))V .”

Let Jα =⋃i<ω supp(B(α, i)) for α ∈ S and denote by ϕα,β the natural order preserving

bijection between Jα and Jβ for 〈α, β〉 ∈ S2.Assume first that (A) holds. Fix t ∈ T and s ∈ (S)|t|. Write αi = s(i) for i < |t|. Since Cκ is

c.c.c , there is in V a countable set It ⊂ P(ω) such that 1Cκ “⋂i<k

B(αi, ni) ∈ It”.

Assume that⟨δ0, . . . , δ|t|−1

⟩∈ (S)k.

Let J∗ =⋃i<k

Jδi , J =⋃i<k

Jαi , and ψ =⋃i<k

ϕδi,αi . Then ψ is a bijection between J∗ and J and

so it lifts up to an isomorphism between CJ∗ and CJ and between the families of nice CJ∗ -namesand nice CJ -names.

Let G be Cκ-generic and put G0 = G | J∗. Since supp(B(δi, ni)) ⊂ J∗ it follows thatval(B(δi, ni), G) = val(B(δi, ni), G0). Let G1 = ψ′′G0. Then G1 is also a CJ -generic filter andsince ψ(B(δi, ni)) = B(αi, ni) it follows that

val(B(δi, ni), G0) = val(B(αi, ni), G1). (•)

Since 1Cκ “⋂i<k

B(αi, ni) ∈ It”, by (•) we have 1Cκ “⋂i<k

val(B(δi, ni), G) ∈ It” as well.

From this it is obvious that we have

1Cκ ˙B(t, s) : t ∈ T ∧ s ∈ (S)|t| ⊂ I =⋃It : t ∈ T.

where I is countable as T is.Assume now that (A) fails and so (B) holds.Let G be Cκ-generic with p ∈ G and 〈γ0, . . . , γk−1〉, 〈δ0, . . . , δk−1〉 ∈ (D0, . . . , Dk−1) such that

V [G] |= “γi, δi ∈[Di

]2 for i < k are pairs of distict ordinals”.

Let J∗ =⋃i<k

Jγi and J? =⋃i<k

Jδi . Then J∗ ∩ J? = ∅, hence by lemma 2.1.22 we have

P(ω) ∩ V [G | J∗] ∩ | V [G | J?] = P(ω) ∩ V and so V [G] |= “⋂i<k B(δi, ni) /∈ V ” implies that

V [G] |= “⋂i<k B(δi, ni) 6=

⋂i<k B(γi, ni)”

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The theorem is proved.

2.1.3. Applications. We start with presenting some combinatorial applications becausethey are quite simple and so they nicely illustrate the use of our principles.

Kunen [76] proved that if one adds Cohen reals to a model of CH, then in the genericextension there is no strictly ⊂∗-increasing chain of subsets of ω of length ω2. The first theoremwe prove easily yields Kunen’s above result.

Theorem 2.1.23. If C(κ) holds then for each A ⊂[ω]ω of size κ either

(a) ∃B ∈[A]κ ∀B 6= B′ ∈ B |B \B′| = ω

or(b) ∃X ∈

[ω]ω |A ∈ A : A ⊂ X| = |A ∈ A : X ⊂∗ A| = κ.

Proof. Fix a 1–1 enumeration Aξ : ξ < κ of A. Let A(ξ, 2n) = Aξ \ n and A(ξ, 2n + 1) =(ω \ Aξ) \ n. Put T = 〈2i, 2i+ 1〉 : i ∈ ω. If S ∈

[κ]κ witnesses 2.1.31, then B = Aξ : ξ ∈ S

satisfies (a). If on the other hand D,E ∈[κ]κ, D ∩E = ∅, with 〈2i, 2i+ 1〉 ∈ T show that 2.1.32

holds, then let X = ∪Aξ : ξ ∈ D. Then Aξ ⊂ X for each ξ ∈ D and X \ i ⊂ Aζ for eachζ ∈ E.

Theorem 2.1.24. If D(κ) or C(κ) holds, then κ is not embeddable into P (ω)/fin.

Remark . In the original version of the paper the statement of theorem 2.1.24 above was derivedonly from principle C(κ). This strengthening of our result was pointed out by the referee.

Proof. Assume that Aα : α < κ is a strictly ⊂∗-increasing chain in[ω]ω. For α < κ and n < ω

let

A(α, n) =ω \Aα if n = 0,Aα \ n otherwise.

We show that A = 〈A(α, n) : α < κ, n < ω〉 is a counterexample to D(κ) and C(κ).To see that A is centered, observe that if Y ∈ A, then there is αY < κ such that Aβ ⊂

AαY ⊂∗ Y for each αY < β < κ. Thus if Y0, . . . , Yn−1 ∈ A then taking α = maxαYi : i < n wehave Aα+1 ⊂ Aα ⊂∗

⋂i<n

Yi. On the other hand, if α < β < κ then Aα ⊂∗ Aβ , thus Aα ∩ (ω \Aβ)

is finite and so A(α, k) ∩A(β, 0) = ∅ for some large enough k. Thus there is no S of size κ (evenof size 2) such that A | S is ω<ω-adic. Thus D(κ) fails.

Next, let t = 〈0, 1〉 and T = t. Then for any S ⊆ κ of size κ, if s ∈ S2 is such thats(0) < s(1), then A(s, t) is infinite. Hence 2.1.4.1 does not hold. Likewise if D0, D1 ⊆ κ are ofsize κ, taking s ∈ (D0, D1)2 such that s(0) > s(1), A(s, t) is finite. This shows that 2.1.4.2 doesnot hold, i.e. C(κ) fails.

The next theorem can be considered as a kind of dual to 2.1.23.

Theorem 2.1.25 ([12]). If C(κ) holds then for each A ⊂[ω]ω of size κ and for each natural

number k either(a) there is a family B ∈

[A]κ such that for each B′ ∈

[B]k we have |

⋂B′| = ω

or(b) there are k subfamilies B0, . . . , Bk−1 of B of size κ such that

|⋂i<k

⋃Bi| < ω.

Next we prove a consequence of theorem 2.1.25, but first we give a definition.

Definition 2.1.26. Let κ be a regular cardinal and A ⊂[ω]ω be an almost disjoint family. A is

called a κ-Luzin gap if |A| = κ and there is no X ∈[ω]ω such that both |A ∈ A : |A \X| <

ω| = κ and |A ∈ A : |A ∩X| < ω| = κ. A Luzin-gap is an ω1-Luzin gap.

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An ω1-Luzin gap can be constructed in ZFC and simple forcings give models in which thereare 2ω-Luzin gaps while 2ω is as large as you wish. The next corollary of theorem 2.1.25 impliesthat one can not construct ω2-Luzin gaps from the assumption 2ω ≥ ω2 alone .

Corollary 2.1.27. If C(κ) holds then there is no κ-Luzin gap.

Proof. Assume that A ⊂[ω]ω is an almost disjoint family of size κ. Then we can not get a even

a two element subfamily B ⊂ A satisfying 2.1.25(a). So applying theorem 2.1.25 for this A andfor k = 2 there are subfamilies B ⊂ A and D ⊂ A of size κ such that (

⋃B) ∩ (

⋃D) is finite.

Hence X =⋃B witnesses that A is not a κ-Luzin gap.

Now we turn to applying our principles to topology. We start with an application of therelatively weak principle D(κ).

A. Dow [36] proved that if we add ω2 Cohen reals to a model of GCH then in the genericextension βω can be embedded into every separable, compact T2 space of size > c = ω2. Here weshow that c = ω2 = 2ω1 together with D(ω2) suffice to imply this statement.

First we need a lemma based on the observation that large separable spaces contain many“similar” points.

Given a topological space X and a point x ∈ X we denote by VX(x) the neighbourhoodfilter of x in X, that is, VX(x) = U ⊂ X : x ∈ intX(U). If D is a dense subset of X letVX(x) | D = U ∩D : D ∈ VX(x). We omit the subscript X if it may not cause any confusion.

In section 2.1.1 we defined the operation A for A ∈M(κ). By an abuse of notation we defineA for every family A of subsets of ω as follow:

A = X ⊂ ω : |A ∩ P(X)| = |A|.Lemma 2.1.28.Assume that X is a separable regular topological space of size > c<c, wherec = 2ω, D ∈

[X]ω, D = X. Then there are a point x ∈ X and a family A = Aα, Bα : α < c ⊂

P(D) such that(1) Aα ∩Bα = ∅ for each α < c,(2) A ⊂ V(x) | D.

Proof. Fix an enumeration Dξ : ξ < c of P(D) and let Dα = Dξ : ξ < α for α < c. Forx ∈ X and α < c let V(x, α) = (V(x) | D) ∩ Dα. A point x ∈ X is called special if there is anα < c such that V(x, α) 6= V(y, α) for each y ∈ X \ x. Clearly there are at most c<c specialpoints in X. Since |X| > c<c we can pick a point x ∈ X which is not special. Then for each α < cwe can find a point xα 6= x in X such that V(xα, α) = V(x, α). Since X is regular the points xand xα have neighbourhoods Uα and Wα, respectively, with Uα ∩Wα = ∅. Let Aα = Uα ∩D andBα = Wα ∩D.

Now assume that E ∈ A and pick ξ < c with E = Dξ. We can find α < c such that ξ < αand either Aα ⊂ E or Bα ⊂ E. Hence E ∈ V(x, α)∪V(xα, α) = V(x, α). Therefore E ∈ V(x) | Dwhich was to be proved.

Let us now recall the definition of a µ-dyadic system from [62].

Definition 2.1.29. If X is a topological space a family 〈A(α, 0), A(α, 1)〉 : α ∈ µ of pairs ofclosed subsets of X is a µ-dyadic system such that

(1) A(α, 0) ∩A(α, 1) = ∅ for each α < µ,(2) for each ε ∈ Fn(µ, 2, ω) we have

⋂α∈dom(ε)

A(α, ε(α)) 6= ∅.

Theorem 2.1.30. If D(c) holds, X is a separable compact T2 space of size > c<c then X containsa c-dyadic system, consequently X maps continuously onto [0, 1]c ( and so βω can be embeddedinto X ).

Proof. Fix a countable dense subset D of X. By lemma 2.1.28 there is a family A = Aα, Bα :α < c ⊂ P(D) such that Aα ∩ Bα = ∅ for α < c and A is centered. Let D(α, 0) = Aα,D(α, 1) = Aα and D(α, n) = D for α < κ and n ≥ 2 and consider the κ × ω-matrix D =〈D(α, i) : α < κ, i < ω〉. Since A = D we can apply D(c) to get a cofinal S ⊂ c such that

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the family⟨Aα, Bα : α < c

⟩is c-dyadic. Now we can apply theorem [62, 3.18] to get the other

consequences.

Since the cardinality of a locally compact, scattered separable space is at most 2ω by [88],the height of such a space is less then (2ω)

+. So under CH there is no such a space of height

ω2. M. Weese asked whether the existence of such a space of height ω2 follows from ¬CH. Thisquestion was answered in the negative by W. Just, who proved, [64, theorem 2.13 ], that if oneadds Cohen reals to a model of CH then in the generic extension there are no locally compactscattered thin spaces of height ω2.

The next theorem is a generalization of the above mentioned result of Just.

Theorem 2.1.31. If Cs(κ) holds then there is no locally compact, thin scattered space of heightκ.

Proof. Assume on the contrary that there is such a space X. We can assume that Iα(X) =α × ω for α < ht(X). For each α < ht(X) fix compact open neighbourhoods U(α, n) of 〈α, n〉for n ∈ ω such that U(α, n) ⊂ 〈α, n〉 ∪

⋃Iβ(X) : β < α and the sets U(α, n) for n < ω are

pairwise disjoint.Put A(α, 2n) = U(α, n) ∩ I0(X) and A(α, 2n+ 1) = I0(X) \

⋃U(α,m) : m ≤ n. Let

T = t ∈ ω<ω : t(0) is even and t(i) is odd for i > 0 .

Now apply Cs(κ) to the matrix 〈A(α, n) : α < κ, n < ω〉 ∈ M(κ) and T .Observe that A(β, 2n) ∩

⋂i<k

A(αi, 2ni + 1) = ∅ iff

U(β, n) ∩ I0(X) ⊂⋃i<k

U(αi, ni) ∩ I0(X) iff

U(β, n) ⊂⋃i<k

U(αi, ni).

Thus if t = 〈2n, 2n0 + 1, . . . , 2nk−1 + 1〉 ∈ T and 〈β, α0, . . . , αk−1〉 ∈ (κ)k+1 then A(β, 2n) ∩⋂i<k

A(αi, 2ni + 1) = ∅ implies β ≤ maxi<k

αi. This excludes 2.1.3(2). So 2.1.3(1) holds, that

is we have a stationary set S ⊂ κ such that if t = 〈2n, 2n0 + 1, . . . , 2nk−1 + 1〉 ∈ T and〈β, α0, . . . , αk−1〉 ∈ (S)k+1 then

A(β, n) ∩⋂i<k

A(αi, 2ni + 1) 6= ∅,

that isU(β, n) \

⋃i<k

⋃j≤ni

U(αi, j) 6= ∅.

But U(β, n) is compact and each U(α, n) is open so it follows that for every β ∈ S and n ∈ ω theset

D(β, n) = U(β, n) \⋃U((α,m) : α ∈ S \ β ∧m ∈ ω

is not empty. For every such β and n let 〈γ(β, n),m(β, n)〉 ∈ D(β, n).Since Iβ(X) is dense in X \

⋃Iα(X) : α < β for every β ∈ κ there is k(β) ∈ ω such that

〈β, k(β)〉 ∈ U(β∗, 0), where β∗ = minS\β+1. Thus 〈β, k(β)〉 /∈ D(β, k(β)) and so γ(β, k(β)) < βfor each β ∈ S. The set S is stationary so there are a stationary set S′ ⊂ S, and ordinals γ < κand k,m < ω such that k(β) = k, γ(β, k) = γ and m(β, k) = m whenever β ∈ S′. Thus〈γ,m〉 ∈ D(β, k) for each β ∈ S′, while D(β, k) ∩ D(β′, k) = ∅ for any β, β′ ∈

[S′]2 by the

construction. This is a contradiction, hence the theorem is proved.

Remark . It is easy to see that the proof of theorem 2.1.31 goes through if, instead of assumingthat all levels of the space are countable, we only require that(i) (i) all levels are of size < κ,(ii) (ii) there are stationary many countable levels.

In [64] W. Just also proved that if one adds at least ω2 Cohen reals to a model of CH thenin the generic extension there is no locally compact, scattered topological space X such that

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ht(X) = ω1 + 1, I0(X) is countable, | Iα(X)| ≤ ω1 for α < ω1 and | Iω1(X)| = ω2. The nexttheorem shows how to get a generalization of this result from our principles.

Theorem 2.1.32. If cf(λ) ≥ ω1 and F (λ+) holds then there is no locally compact, scatteredtopological space X such that ht(X) = λ + 1, I0(X) is countable, | Iα(X)| ≤ λ for all α < λ and| Iλ(X)| = λ+.

Proof. See in [12].

Following the terminology of [49] a Hausdorff space is called P2 if it does not contain twouncountable disjoint open sets. Hajnal and Juhász in [49] constructed a ZFC example of a firstcountable, P2 space of size ω1 as well as consistent examples of size 2ω with 2ω as large as youwish . On the other hand, using a result of Z. Szentmiklóssy they proved that it is consistentwith ZFC that 2ω is as large as you wish and there are no first countable P2 spaces of size ≥ ω3.However their method was unable to replace here ω3 with ω2. Our next result does just thisbecause, as is shown in [49], every P2 space is separable.

Theorem 2.1.33. If C(κ) holds then every first countable, separable T2 topological space X ofsize κ contains two disjoint open sets U and V of cardinality κ.

Proof. Let D be a countable dense subset of X. For each x ∈ X fix a neighborhood baseU(x, n) : n ∈ ω of x in X. Apply C(κ) to the matrix 〈U(x, n) ∩D : x ∈ X,n < ω〉 and T = ω2.Since X is T2, there is no S ∈

[X]κ satisfying 2.1.31. So there are S0, S1 ∈

[X]κ and n,m ∈ ω

such that U(x, n)∩U(y,m)∩D = ∅ whenever x ∈ S0 and y ∈ S1. But D is dense in X, thereforeU =

⋃U(x, n) : x ∈ S0 and V =

⋃U(y,m) : y ∈ S1 are disjoint open sets of size κ.

Definition 2.1.34. Let X be a topological space and D ⊂ X. We say that D is sequentiallydense in X iff for each x ∈ X there is a sequence Sx from D which converges to x. A space Y issaid to be sequentially separable if it contains a countable sequentially dense subset.

Definition 2.1.35.Given a topological space 〈X, τ〉 and a subspace Y ⊂ X a function f is calleda neighbourhood assignment on Y in X iff f : Y −→ τ and y ∈ f(y) for each y ∈ Y .

Our next result says that under C(κ) if a sequentially separable space X does not contain adiscrete subspace of size κ, (i.e. s(X) ≤ κ using the notation of [62]) then X does not contain leftor right separated subspaces of size κ either. This can be written as h(X) z(X) ≤ κ. Since in [11]a normal, Frechet-Urysohn, separable (hence sequentially separable) space X is forced such thatz(X) ≤ ω1 but h(X) = ω2, this result is not provable in ZFC. First, however, we need a lemma.

Lemma 2.1.36.Assume that C(κ) holds. Let X be a sequentially separable space with Y ⊂ X,|Y | = κ. If f is a neighbourhood assignment on Y in X, then either (a) or (b) below holds:(a) there is Y ′ ∈

[Y]κ such that f(y) ∩ Y ′ = y for each y ∈ Y ′ (hence Y ′ is discrete),

(b) there are Y0, Y1 ∈[Y]κ such that y ∈ f(x) whenever x ∈ Y0 and y ∈ Y1.

Proof. We can assume that D = ω is sequentially dense in X. For each y ∈ Y choose a sequenceSy ⊂ D converging to y. Let A(y, 2n) = D \ f(y), A(y, 2n + 1) = Sy \ n, T = 〈2n, 2m+ 1〉 :n,m ∈ ω and apply C(κ). Assume first that Y ′ ∈

[Y]κ witnesses 2.1.31 and let x 6= y ∈ Y ′.

Then for each n ∈ ω we have (Sy \ f(x)) \ n 6= ∅, i.e. Sy \ f(x) is infinite. But Sy converges to y,so y /∈ f(x), and so Y ′ satisfies (a). Assume now that 2.1.32 holds. Then there are Y0, Y1 ∈

[ω]ω

and m ∈ ω such that (D\f(x))∩(Sy \m) = ∅ for each x ∈ Y0 and y ∈ Y1. But then Sy \m ⊂ f(x)hence y ∈ f(x) which was to be proved.

Theorem 2.1.37. If C(κ) holds, X is a regular, sequentially separable space with s(X) ≤ κ thenh(X) z(X) ≤ κ.

Proof. Assume on the contrary that Y ∈[X]κ and the neighbourhood assignment f : Y −→ τX

witnesses that Y is left (right) separated. We can assume that Y = κ and Y is left (right)separated under the natural ordering of κ. Since X is regular we can find a neighbourhoodassignment g : Y −→ τ with g(y) ⊂ f(y) for each y ∈ Y . Apply lemma 2.1.36 to Y and g. Now2.1.36(a) can not hold because s(X) ≤ κ, hence there are Y0, Y1 ∈

[Y]κ satisfying 2.1.36(b). Since

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both Y0 and Y1 are cofinal in Y = κ under the natural ordering of the ordinals, applying left (orright) separatedness of Y we can pick x ∈ Y0 and y ∈ Y1 such that y /∈ f(x). By the choice of gthis implies y /∈ g(x) which contradicts 2.1.36(b).

The Sorgenfrey line L is weakly separated and is of size c with s(L) = ω1. This shows thattheorem 2.1.37 does not remain valid if you put weakly separated subspaces instead of right orleft separated ones.

As an easy consequence of 2.1.37 we get the following result in which (sequential) separabilityis no longer assumed. We also note that under CH the assumption of X being Frechet-Urysohnis not necessary in this result.

Theorem 2.1.38.Assume C(ω2). If X is regular, Frechet-Urysohn space and s(X) = ω thenh(X) ≤ ω1.

Proof. If C(ω2) and X is separable, then by theorem 2.1.37 even s(X) ≤ ω1 implies h(X) z(X) ≤ω1. Now, every uncountable space X which is both right and left separated contains an uncount-able discrete subspace, hence every right separated subspace of X is (hereditarily) separable. Soby the above if Y ⊂ X is right separated then |Y | ≤ ω1, i.e. h(X) ≤ ω1.

In [57] we investigated the following question: What makes a space have weight larger thanits character? To answer this question we introduced the notion of an irreducible base of a spaceand proved that any weakly separated space has such a base, moreover the weight of a spacepossessing an irreducible base can not be smaller than its cardinality. We asked [57, Problem1] whether every first countable space of uncountable weight contains an uncountable subspacewith an irreducible base? In theorem 2.1.42 and corollary 2.1.43 we will give a partial positiveanswer to this problem, using the principle C(κ). First we recall some definitions from [57].

Definition 2.1.39. Let X be a topological space. A base U of X is called irreducible if it has anirreducible decomposition U =

⋃Ux : x ∈ X, i.e., (i) and (ii) below hold:

(i) Ux is a neighbourhood base of x in X for each x ∈ X,(ii) for each x ∈ X the family U−x =

⋃y 6=xUy is not a base of X (hence U−x does not contain a

neighbourhood base of x in X).

Definition 2.1.40. Let X be a topological space with Y ⊂ X. Similarly as above, an outer baseU of Y in X is called irreducible if it has an irreducible decomposition U =

⋃Uy : y ∈ Y , i.e.,

(i) and (ii) below hold:(i) Uy is a neighbourhood base of y in X for each y ∈ Y ,(ii) for each y ∈ Y the family U−y =

⋃Uz : z ∈ Y \y does not contain a neighbourhood base

of y in X.

Note that in general, a subspace Y having an irreducible outer base in X does not necessarilypossess an irreducible base in itself. However, if Y is dense in an open set and the irreducibleouter base of Y consists of regular open sets then clearly this is the case. Moreover, by our nextresult, under certain conditions we can at least find another subspace of the same size as Y thatdoes have an irreducible base.

Lemma 2.1.41. If X is a regular, separable space and Y ⊂ X has an irreducible outer base inX consisting of regular open sets, then there is Z ⊂ X with |Z| = |Y | such that the subspace Zhas an irreducible base.

Proof. Let B =⋃By : y ∈ Y be an irreducible outer base of Y in X consisting of regular open

sets and D be a countable dense subset of X. We distinguish two cases:

Case 1. |(intY ) ∩ Y | = |Y |.

Let Z = (intY ) ∩ Y . Since Z is dense in the open set intY , by our above remark Z has anirreducible base.

Case 2. |(intY ) ∩ Y | < |Y |.

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In this case the set Y1 = Y \intY is nowhere dense, so D1 = D\Y1 is dense in X. Let Z = D1∪Y1,then |Z| = |Y1| = |Y |. Write D1 = dn : n < ω and for each dn ∈ D let Bdn be a neighbourhoodbase of dn in X consisting of regular open sets, that are disjoint to Y1 ∪ dm : m < n. Thenclearly

⋃Bz : z ∈ Z is an irreducible outer base of Z in X consisting of regular open sets and

Z is dense in X, so again we are done.

Theorem 2.1.42.Assume C(κ). If X is a separable, first countable, regular space with w(X) ≥ κ,then there is subspace Y ⊂ X of cardinality κ that has an irreducible base.

Proof. Let D ⊂ X be a countable, dense subset of X. For each x ∈ X fix a neighbourhood baseU(x, n) : n ∈ ω consisting of regular open sets and set V (x, n) = U(x, n)∩D. Since the U(x, n)are regular open and D is dense, we clearly have U(x, n) ⊂ U(y,m) iff V (x, n) ⊂ V (y,m).

Since w(X) ≥ κ, by transfinite recursion on β < κ we can choose points xα : α < κ ⊂ Xsuch that for any β < κ the family U(xα, n) : α < β, n < ω does not contain a neighbourhoodbase of xβ , in other words, there is a natural number kβ such that for all α < β < κ and n ∈ ωwe have

¬(xβ ∈ U(xα, n) ⊂ U(xβ , kβ)). (∗)We can assume that kβ = 0 for each β < κ. Let X ′ = xα : α < κ. For x ∈ X ′ and n < ω put

A(x, 2n) = [V (x, n)× 0] ∪ [(D \ V (x, n))× 1]

andA(x, 2m+ 1) = [(D \ V (x, 0))× 0] ∪ [V (x,m)× 1] .

Note that A(x, 2n) ∩ A(y, 2m + 1) = ∅ iff V (y,m) ⊂ V (x, n) ⊂ V (y, 0). Apply C(κ) to〈A(x, i) : x ∈ X ′, i < ω〉 and T = 〈2n, 2m+ 1〉 : n,m < ω. By (∗) (and kβ = 0) there areno D,E ∈

[X ′]κ and n,m ∈ ω such that

V (y,m) ⊂ V (x, n) ⊂ V (y, 0) (†)

whenever x ∈ D and y ∈ E, because (†) fails if x = xα, y = xβ and α < β. So there is Y ∈[X ′]κ

such that for all n,m ∈ ω and x 6= y ∈ Y the intersection of A(x, 2n) and A(y, 2m + 1) is notempty. This means that ¬(V (y,m) ⊂ V (x, n) ⊂ V (y, 0)), i.e. if we set By = U(y, n) : n < ωthen it follows that B =

⋃By : y ∈ Y ′ is a an irreducible outer base of Y in X consisting of

regular open sets. Now applying lemma 2.1.41 we can conclude the proof.

Unfortunately, as C(ω1) is false, the above result is not applicable in the perhaps mostinteresting case when w(X) = ω1. The annoying assumption of separability, however, can becircumvented as follows.

Corollary 2.1.43.Assume C(κ). If X is a first countable, regular space with w(X) ≥ κ, thenthere is an uncountable subspace Y ⊂ X that has an irreducible base.

Proof. If X is separable, then the previous theorem can be applied. If X is not separable, thenX contains an uncountable left separated subspace Y and again Y has an irreducible base.

2.1.4. A recent application concerning Banach algebras. The following statement areproved in [1]

Proposition 2.1.44. If C(κ) holds then the following statement denoted by ⊗κ holds:(⊗κ): ∀fα : α < κ ⊂ `∞/c0 ∀` ∈ ω ∀c1, . . . , c` ∈ R ∀x ∈ R ∃S1, . . . S` ∈

[κ]κ ∃ρ ∈ ≤, >

∀a ∈ (S1, . . . , S`)

∥∥∥∥∥∑i=1

ci · fa(i)

∥∥∥∥∥ ρx.Proposition 2.1.45. If ⊗κ holds then C([0, κ]) 6→ `∞/c0.

Proposition 2.1.46. If ⊗ω2 holds then there is a Banach space X with density ≤ ω2 such thatC([0, ω2]) 6→ X and X 6→ `∞/c0.

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2.2. LCS∗ spaces in Cohen real extension


W. Just proved, [64, theorem 2.13], that if one blows up the continuum by adding Cohenreals to a model of CH then in the resulting generic extension there is no LCS∗ space of heightω2 and width ω. In this section we strengthen the result of Just by proving, in particular, that inthe same Cohen real extension no LCS space may have ω2 many countable (non-empty) levels. Itseems to be an intriguing (and natural) problem if the non-existence of an LCS∗ space of widthω and height ω2 implies in ZFC the above conclusion, or more generally: when is a subsequenceof the cardinal sequence of an LCS∗ space again such a cardinal sequence? In connection withthis problem let us remark that, (as is shown in [30] or [63]), in the side-by-side random realextension of a model of CH the combinatorial principle Cs(ω2) introduced in [12, definition 2.3]holds, consequently in such an extension there is no LCS∗ space X of height ω2 and width ω.In fact, by [12, theorem 4.12], Cs(ω2) implies that α ∈ ω2 : |Iα(X)| = ω is non-stationary inω2. However, we do not know if our above mentioned result, namely theorem 2.2.1, is implied byCs(ω2) .

Let us formulate then the promised strengthening of Just’s result. We note that no assumption(such as CH) is made on our ground model.

Theorem 2.2.1. Let us set κ = (2ω)+ and add any number of Cohen reals to our ground model.Then in the resulting extension no LCS∗ space contains a κ-sequence Eα : α < κ of pairwisedisjoint countable subspaces such that Eα ⊃ Eβ holds for all α < β < κ. In particular, for anyLCS∗ space X we have

∣∣α : | Iα(X)| = ω∣∣ < κ.

In fact, we shall prove a more general statement, but to formulate that we need a definition. Afamily of pairs (of sets) D =

〈Dα

0 , Dα1 〉 : α ∈ I

is said to be dyadic over a set T iff Dα

0 ∩Dα1 = ∅

for each α ∈ I andD[ε] =

⋂Dα

ε(α) : α ∈ dom ε

intersects T for each ε ∈ Fn(I, 2). We simply say that D is dyadic iff it is dyadic for some T , i.e.D[ε] 6= ∅ for each ε ∈ Fn(I, 2). (As usual, Fn(I, 2) denotes the set of all finite partial functionsfrom I into 2.)

Now, it is obvious that in an LCS∗ space• the compact open sets form a base that is closed under finite unions,• there is no infinite dyadic system of pairs of compact sets.

Consequently, theorem 2.2.2 below immediately yields theorem 2.2.1 above.

Theorem 2.2.2. Set κ = (2ω)+ and add any number of Cohen reals to the ground model. Thenin the resulting generic extension the following statement holds: If X is any T2 space containingpairwise disjoint countable subspaces Eα : α < κ such that Eα ⊃ Eβ for α < β < κ and X = E0

(i. e. E0 is dense in X) , moreover, for each x ∈ X, we have fixed a neighbourhood base B(x) ofx in X that is closed under finite unions then there is an infinite set a ∈

[κ]ω, for each α ∈ a

there are disjoint finite subsets L0α and L1

α of Eα, and for each x ∈ L0α ∪ L1

α there is a basicneighbourhood V (x) ∈ B(x) such that the infinite family of pairs⟨ ⋃

x∈L0α

V (x),⋃x∈L1

α

V (x)⟩

: α ∈ a

is dyadic.

This topological statement in the Cohen extension in turn will follow from a purely combi-natorial one concerning certain matrices, namely theorem 2.2.7.

To formulate this theorem we again need some notation and definitions.For an ordinal α the interval [ωα, ωα+ ω) will be denoted by Iα.Given two sets A and B we write f : A

p−→ B to denote that f is a partial function from Ato B, i. e. a function from a subset of A into B. As usual, we let

Fn(A,B) = f : |f | < ω and f : Ap−→ B.

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2.2. LCS∗ SPACES IN COHEN REAL EXTENSION 97

If A ⊂ On then for any partial function f : Ap−→ B we set

γ(f) =

min dom f if dom f 6= ∅,supA if dom f = ∅.

We letΩ =

〈A,B〉 ∈

[ω]<ω × [ω]<ω : A ∩B = ∅

,

and for ` = 〈A,B〉 ∈ Ω we set π0(`) = A and π1(`) = B.If S and T are sets of ordinals, we denote by M(S, T ) the family of all S × ω-matrices

consisting of subsets of T , i. e. A ∈ M(S, T ) means that A = 〈Aα,i : α ∈ S, i ∈ ω〉, whereAα,i ⊂ T for each α ∈ S and i < ω.

For A ∈ M(S, T ), f : Sp−→ S, and s : S

p−→ Ω the pair (f, s) is said to be A-dyadic (overU) iff the family of pairs⟨

∪Af(α),n : n ∈ π0(s(α))

,∪Af(α),n : n ∈ π1(s(α))

⟩: α ∈ dom f ∩ dom s

.

is dyadic (over U). If the pair 〈idS , s〉 is A-dyadic (over U) then s is simply called A-dyadic (overU). It is this latter notion of A-dyadicity of a single partial function that is really important (thatfor pairs is only of technical significance). Hence we state below an alternative characterisationof it.

For A ∈M(S, T ), s : Sp−→ Ω, and ε ∈ Fn(dom s, 2) we write

A[s, ε] =⋂

α∈dom ε

⋃Aα,n : n ∈ πε(α)(s(α)).

Observation 2.2.3. If A ∈M(S, T ) then s : Sp−→ Ω is A-dyadic over U iff A[s, ε] ∩ U 6= ∅ for

each ε ∈ Fn(dom s, 2) and⋃Aα,n : n ∈ π0(s(α)) ∩

⋃Aα,n : n ∈ π1(s(α)) = ∅

for each α ∈ dom s.

The following easy observation will be applied later, in the proof of lemma 2.2.9:

Observation 2.2.4. If g : Sp−→ S and s : S

p−→ Ω satisfy dom s ⊂ ran g, and the pair (g, s g)is A-dyadic over U then s is A-dyadic over U , as well.

Definition 2.2.5. Fix a cardinal κ and let D ∈ M(κ, κ). For s : κp−→ Ω we say that s is

D-min-dyadic (m.d.) iff s is D-dyadic over Iγ(s).Moreover, we say that the matrix D is m.d.-extendible iff for each finite D-min-dyadic partial

function s : κp−→ Ω and for each γ < γ(s) there is an ` ∈ Ω such that s ∪ 〈γ, `〉 is also

D-min-dyadic, i. e. D-dyadic over Iγ .

Since I0 = ω, we clearly have the following.

Observation 2.2.6. If D ∈ M(κ, κ) is m.d-extendible and s : κp−→ Ω is a finite D-min-dyadic

partial function then s is D-dyadic over ω.

Finally, a matrix D ∈ M(κ, κ) will be called ω-determined iff Dα,n ∩Dα,m ∩ ω = ∅ impliesDα,n ∩Dα,m = ∅ whenever α < κ and n < m < ω.

With this we now have all the necessary ingredients to formulate and prove the promisedcombinatorial statement that will be valid in any Cohen real extension.

Theorem 2.2.7. Set κ = (2ω)+ and add any number of Cohen reals to the ground model. Then inthe resulting generic extension for every ω-determined and m.d.-extendible matrix D ∈ M(κ, κ)there is an infinite D-dyadic partial function h : κ

p−→ Ω.

Before proving theorem 2.2.7, however, we show how theorem 2.2.2 can be deduced from it.

Proof of theorem 2.2.2 using theorem 2.2.7. We can assume without any loss of generalitythat Eα = Iα for each α < κ and then will define an appropriate matrix D ∈M(κ, κ).

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To this end, for coding purposes, we first fix a bijection ρ :[ω]2 −→ ω and let η : ω −→ ω

and ν : ω −→ ω be the "co-ordinate" functions of its inverse, i. e. k = ρ(ν(k), η(k)) andν(k) < η(k) for each k < ω.

Since X is T2, for each n < ω we can simultaneously pick basic neighbourhoods Bαn (m) ∈B(ωα+m) of the points ω · α+m ∈ Eα = Iα for all m < n such that the sets Bαn (m): m < nare pairwise disjoint.

Now we define D = 〈Dα,k : 〈α, k〉 ∈ κ× ω〉 ∈ M(κ, κ) as follows:

Dα,k = Bαη(k)(ν(k)) ∩ κ.

This matrix D is clearly ω-determined because E0 = I0 = ω is dense in X. It is a bit lesseasy to establish the following

Claim .D is also m.d.-extendible.

Proof of the claim. Let s : κp−→ Ω be a finite D-min-dyadic partial function and let γ < γ(s).

Since the sets D[s, ε] : ε ∈ dom s2 are all open in the subspace κ and they all intersect Iγ(s),moreover every element of Iγ(s) is an accumulation point of Iγ , it follows that D[s, ε] ∩ Iγ mustbe infinite for each ε ∈ dom s2. Thus we can easily pick two disjoint finite subsets A0 and A1 ofIγ such that every D[s, ε] intersects both A0 and A1. Let n < ω be chosen in such a way thatA0 ∪A1 ⊂ ωγ +m : m < n, and set Ki = ρm,n : m < n ∧ ωγ +m ∈ Ai for i < 2. Since %is one-to-one we have K0 ∩K1 = ∅, hence ` = 〈K0,K1〉 ∈ Ω, moreover( ⋃

m∈K0

Dγ,m

)∩( ⋃m∈K1

Dγ,m

)= ∅ (?)

because the elements of the family Bγn(m) : m < n are pairwise disjoint.Now put t = s ∪ 〈γ, `〉. Then for each ε ∈ dom t2 we clearly have

Aε(γ) ∩ D[t, ε] 6= ∅, (??)

hence (?) and (??) together yield that the extension t of s is D-dyadic over Iγ = Iγ(t).

Thus we may apply theorem 2.2.7 to the matrix D to obtain an infinite D-dyadic partialfunction h : κ

p−→ Ω. Set a = domh and for each α ∈ a and i < 2 put Liα = ωα + ν(k) : k ∈πi(h(α)). For x ∈ Liα put

V (x) = ∪Bαη(k)(ν(k)) : x = ωα+ ν(k) and k ∈ πi(h(α)).

Then V (x) ∈ B(x) because B(x) is closed under finite unions. Since for i < 2(∪V (x) : x ∈ Liα

)∩ κ = ∪Dα,k : k ∈ πi(h(α))

and∪Dα,k : k ∈ π0(h(α)) ∩ ∪Dα,k : k ∈ π1(h(α)) = ∅,

we have (∪V (x) : x ∈ Liα

)∩(∪V (x) : x ∈ Liα

)= ∅

because the latter intersection is an open set which does not intersect the dense set I0 ⊂ κ. Hencethe infinite family ⟨ ⋃

x∈L0α

V (x),⋃x∈L1

α

V (x)⟩

: α ∈ a

is indeed dyadic. 2.2.2

Proof of theorem 2.2.7. The proof will be based on the following two lemmas, 2.2.9 and 2.2.10.For these we need some more notation and a new and rather technical notion of extendibility forset matrices.

Given a set A we set

F(A) = f ∈ Fn(A,A) : f is injective and dom(f) ∩ ran(f) = ∅.

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2.2. LCS∗ SPACES IN COHEN REAL EXTENSION 99

Each function f ∈ F(A) can be extended in natural way to a bijection f∗ : A −→ A as follows:

f∗(a) =

f(a) if a ∈ dom f ,f−1(a) if a ∈ ran f ,a otherwise.

Definition 2.2.8. If S and T are sets of ordinals then the matrix A ∈ M(S, T ) is called nicelyextendible iff for each f ∈ F(S) there are a family N(f) ⊂ Fn(S,Ω) and a functionKf : N(f) −→[S]≤ω such that

(1) the pair (f, s) is A-dyadic whenever f ∈ F(S) and s ∈ N(f),(2) ∅ ∈ N(f) for each f ∈ F(S),(3) for f, g ∈ F(S) and s ∈ N(f) if f∗ | Kf (s) = g∗ | Kf (s) then s ∈ N(g).(4) for any f ∈ F(S), s ∈ N(f) and α ∈ S ∩ γ(s) there is ` ∈ Ω such that s ∪ 〈α, `〉 ∈ N(f).

Clearly, this last condition (4) is what explains our terminology.

Lemma 2.2.9. If κ > ω1 is regular and A ∈ M(κ, ω) is a nicely extendible matrix then there isan infinite partial function h : κ

p−→ ω that is A-dyadic .

Proof. By induction on n ∈ ω we will define functions h0 ⊂ h1 ⊂ . . . hn ⊂ . . . from Fn(κ,Ω)such that |hn| = n and for each ν ∈ κ(∗)nν there is g ∈ F(κ) such that γ(g) > ν, ran g = domhn and hn g ∈ N(g).

First observe that h0 = ∅ satisfies our requirements because, according to (2), condition (∗)0ν

holds trivially for each ν < κ.Next assume that the construction has been done and the induction hypothesis has been

established for n. For each ν < κ choose a function gν ∈ F(κ) witnessing (∗)nν+ω1and then write

Kν = Kgν (hn gν) and pick ζν ∈ (ν, ν + ω1) \Kν . Clearly the set

L = ξ ∈ κ : |ν < κ : ξ /∈ Kν| < κis countable and so we can pick ξn ∈ κ \ (L ∪ domhn); then the set

J = ν < κ : ξn /∈ Kνis of size κ.

Now set g′ν = gν ∪ 〈ζν , ξn〉 for every ν ∈ J . For every such ν then ζν , ξn /∈ Kν impliesgν∗ | Kν = g′ν

∗ | Kν , hence hn gν ∈ N(g′ν) by (3). Since ζν < ν + ω1 < γ(gν) = γ(hn gν), wecan now apply (4) to get `ν ∈ Ω such that (hn gν) ∪ 〈ζν , `ν〉 ∈ N(g′ν).

We can then fix `n ∈ Ω such that Jn = ν ∈ J : `ν = `n is of size κ and let hn+1 =hn ∪ 〈ξn, `n〉.

If ν ∈ Jn then hn+1 g′ν = (hn gν) ∪ 〈ζν , `n〉 ∈ N(g′ν) and γ(g′ν) > ν, so g′ν witnesses(∗)n+1

ν . But Jn is unbounded in κ, hence the inductive step is completed.By (∗)n0 , for each n < ω there is gn such that domhn = ran gn and hngn ∈ N(gn). Hence, by

(1), (gn, hn gn) is A-dyadic, and so hn is A-dyadic according to observation 2.2.4. Consequentlyh =

⋃hn : n < ω is as required: it is A-dyadic and infinite. 2.2.9

Given any infinite set I we denote by CI the poset Fn(I, 2), i.e. the standard notion of forcingthat adds |I| many Cohen reals.

Lemma 2.2.10. Let κ = (2ω)+. Then for each λ we have

V Cλ |= If D ∈M(κ, κ) is both ω-determined and m.d.-extendible

then there is I ∈[κ]κ such that

D∗ = 〈Dα,n ∩ ω : 〈α, n〉 ∈ I × ω〉 is nicely extendible.

Proof. Assume that1Cλ‖—D ∈ M(κ, κ) is m.d.-extendible.

Let θ be a large enough regular cardinal and consider the structure Hθ =⟨Hθ,∈, /, κ, λ, D

⟩,

where Hθ =x : |TC(x)| < θ

and / is a fixed well-ordering of Hθ.

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Working in V , for each α < κ choose a countable elementary submodel Nα ofHθ with α ∈ Nα.Then there is I ∈

[κ]κ such that the models Nα : α ∈ I are not only pairwise isomorphic but,

denoting by σα,β the unique isomorphism between Nα and Nβ , we have(i) the family Nα ∩ θ : α ∈ I forms a ∆-system with kernel Λ,(ii) σα,β(ξ) = ξ for each ξ ∈ Λ,(iii) σα,β(α) = β.

For each α < κ and n < ω let Dα,n be the /-minimal Cλ-name of the 〈α, n〉th entry of D.Since / is in Hθ and σα,β(α) = β we have

Claim 2.2.10.1. σα,β(Dα,n) = Dβ,n for each α, β ∈ I and n ∈ ω.Let G be any Cλ-generic filter over V . We shall show that

V [G] |= “D∗ = 〈Dα,n ∩ ω : 〈α, n〉 ∈ I × ω〉 is nicely extendible.”

For each f ∈ F(I) define the bijection ρf : λ −→ λ as follows:

ρf (ξ) =σα,f∗(α)(ξ) if ξ ∈ Nα ∩ λ for some α ∈ I,ξ otherwise.

In a natural way ρf extends to an automorphism of Cλ, which will be denoted by ρf as well.Clearly, we have

Claim 2.2.10.2. If f ∈ F(I), f(α) = β, p ∈ Cλ ∩Nα then σα,β(p) = ρf (p).

For f ∈ F(I) let Gf = ρ−1f (p) : p ∈ G and then set

N(f) = s ∈ Fn(I,Ω) : s is D[Gf ]-min-dyadic =

s ∈ Fn(I,Ω) : ∃q ∈ Gf q‖—“s is D-min-dyadic”.

To define Kf , for each s ∈ N(f) pick a condition ps ∈ G such that

ρ−1f (ps)‖—s is D-min-dyadic

and letKf (s) = α ∈ I : (Nα \ Λ) ∩ dom ps 6= ∅.

Note that Kf (s) as defined above is finite, although 2.2.8.(3) only requires Kf (s) to becountable.

To check property 2.2.8.(3) assume that f, g ∈ F(I) and s ∈ N(f) with g∗ | Kf (s) =f∗ | Kf (s). Then ρ−1

g (ps) = ρ−1f (ps) and so

ρ−1g (ps)‖—s is D-min-dyadic,

hence s is also D[Gg]-min-dyadic , i.e. s ∈ N(g).Before checking 2.2.8.(1) we need one more observation.

Claim 2.2.10.3. Df(α),n[G] ∩ ω = Dα,n[Gf ] ∩ ω whenever f ∈ F(I), α ∈ dom f , and n < ω.

Proof of claim 2.2.10.3. Let k ∈ ω. Then k ∈ Df(α),n[G] iff ∃p ∈ G p‖— “k ∈ Df(α),n”iff ∃p ∈ G ∩ Nf(α) p‖— “k ∈ Df(α),n” iff ∃q ∈ Gf ∩ Nα p = σα,f(α)(q)‖— “k ∈ Df(α),n” iff∃q ∈ Gf ∩Nα q‖— “k ∈ Dα,n” iff ∃q ∈ Gf q‖— “k ∈ Dα,n” iff k ∈ Dα,n[Gf ]. 2.2.10

Now let f ∈ F(I) and s ∈ N(f). By the definition of N(f), s is D[Gf ]-min-dyadic and so byobservation 2.2.6 s is D[Gf ]-dyadic over ω. But it follows from 2.2.10.3, that s is D[Gf ]-dyadicover ω if and only if the pair (f, s) is D[G]-dyadic over ω.

2.2.8.(2) is clear because ∅ is trivially A-min-dyadic for any A ∈ M(κ, ω). Finally 2.2.8.(4)follows from the definition of N(f) because D[Gf ] is m.d.-extendible. 2.2.10

Now, to complete the proof of theorem 2.2.7, first apply lemma 2.2.10 to get I ∈[κ]κ such

thatD∗ = 〈Dα,n ∩ ω : 〈α, n〉 ∈ I × ω〉

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2.3. WEAK FREEZE-NATION PROPERTY OF POSETS 101

is nicely extendible. Then applying lemma 2.2.9 to D∗ we obtain an infinite D∗-dyadic functionh : κ

p−→ Ω. Since the matrix D is ω-determined the function h is D-dyadic, as well. 2.2.7

2.3. Weak Freeze-Nation property of posets


Definition 2.3.1. For a regular κ, a quasi ordering P is said to have the κ Freese-Nation property(the κ-FN for short) if there is a mapping (κ-FN mapping) f : P → [P ]<κ such that

for any p, q ∈ P if p ≤ q then there is r ∈ f(p) ∩ f(q) such that p ≤ r ≤ q.A mapping f as above is called a κ-FN mapping on P . ℵ1-FN is called as weak Freese-Nationproperty (wFN-property, in short). A weak Freese-Nation mapping is a ℵ1-FN mapping.

Freese and Nation [44] used the ℵ0-FN in a characterization of projective lattices and asked ifthis property alone already characterizes the projectiveness. L. Heindorf gave a negative answerto the question showing that the Boolean algebras with the ℵ0-FN are exactly those which areopenly generated. It is known that the class of openly generated Boolean algebras containsprojective Boolean algebras as a proper subclass (see [51] — openly generated Boolean algebrasare called ‘rc-filtered’ there). Heindorf and Shapiro [51] then studied the ℵ1-FN which they calledthe weak Freese-Nation property and proved some elementary properties of the Boolean algebraswith this property. Partial orderings with the κ-FN for arbitrary regular κ were further studiedin Fuchino, Koppelberg and Shelah [46]. Koppelberg [69] gives some nice applications of theℵ1-FN.

In the following we shall quote some elementary facts from [46] which we need later. Firstof all, it can be readily seen that every small partial ordering has the κ-FN:

Lemma* 2.3.2. ([46]) Every partial ordering P of cardinality ≤ κ has the κ-FN.

For a partial ordering P and a sub-ordering Q ⊆ P , we say that Q is a κ-sub-ordering of Pand denote it with Q ≤κ P if, for every p ∈ P , the set q ∈ Q : q ≤ p has a cofinal subset ofcardinality < κ and the set q ∈ Q : q ≥ p has a coinitial subset of cardinality < κ.

Lemma* 2.3.3. ([46]) Suppose that δ is a limit ordinal and (Pα)α≤δ a continuously increasingchain of partial orderings such that Pα ≤κ Pδ for all α < δ. If Pα has the κ-FN for every α < δ,then Pδ also has the κ-FN.

For application of Lemma 2.3.3, it is enough to have Pα ≤κ Pδ and the κ-FN of Pα for everyα < δ such that either α is a successor or of cofinality ≥ κ: Pα ≤κ Pδ for α < δ of cofinality < κfollows from this since such Pα can be represented as the union of < κ many κ-sub-orderings ofPδ. Hence by inductive application of Lemma 2.3.3, we can show that Pα satisfies the κ-FN forevery α ≤ δ. Similarly, if δ is a cardinal > κ, then it is enough to have Pα <κ Pδ and the κ-FNof Pα for every limit α < δ of cofinality ≥ κ.Proposition* 2.3.4 ([46]). For a regular κ and a poset P , the following statements are equivalent:(1) P has the κ-FN;(2) For some, or equivalently, any sufficiently large χ, if M ≺ Hχ = (Hχ,∈) is such that P ∈M ,

κ ⊆M and |M | = κ then P ∩M ≤κ P holds;(3) C ∈ [P ]κ : C ≤κ P contains a club set.

Though 2.3.4,(2) is quite useful to show that a partial ordering has the κ-FN, sometimes itis quite difficult to check Proposition 2.3.4(2) as in the case of the ℵ1-FN of P (ω) or [κ]<ω: inthese cases it is independent if Proposition 2.3.4(2) holds. Applications like Corollary 2.3.15 inmind, we could think of another possible variant of Proposition 2.3.4, (2) in terms of the followingweakening of the notion of internal approachability from [42]:

Definition 2.3.5. For a regular κ and a sufficiently large χ, we shall call an elementary submodelM of Hχ Vκ-like if, either κ = ℵ0 andM is countable, or there is an increasing sequence (Mα)α<κof elementary submodels of M of cardinality less than κ such that Mα ∈Mα+1 for all α < κ andM =

⋃α<κMα.

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In [46], a characterization of the κ-FN using Vκ-like elementary submodels in place of elemen-tary submodels in Proposition 2.3.4(2) was discussed. Unfortunately, in the proof an inaccuracywas found by the present author (see “Added in Proof” in [46]). In this section, we introduce aweakening of the very weak square principle from [43] — the principle ∗∗∗κ,µ . In subsection 2.3.1we show the equivalence of ∗∗∗κ,µ with the existence of a matrix (Mα,β)α<µ+,β<cf(µ) — which wecalled (weak) (κ, µ)-dominating matrix — of elementary submodels of H(χ) for sufficiently largeχ with certain properties. This fact is used in subsection 2.3.2 to show that ∗∗∗κ,λ together witha very weak version of the Singular Cardinals Hypothesis yields the characterization of partialorderings with the κ-FN in terms of Vκ-like elementary submodels (Theorem 2.3.14). ZFC oreven ZFC + GCH is not enough for this characterization: in section 2.3.3, we show that, underChang’s conjecture for ℵω, there is a counter-example to the characterization. Together withTheorem 2.3.14, this counter-example also shows that ∗∗∗ℵ1,ℵω is not a theorem in ZFC + GCH.

One of the most natural questions concerning the κ-FN would be if (P(ω),⊆) has the ℵ1-FN. It is easy to see that (P(ω),⊆) has the ℵ1-FN iff (P(ω)/fin,⊆∗) does (see [46]). See alsoKoppelberg [69] for some consequences of the ℵ1-FN of P(ω)/fin. By Lemma 2.3.2, P(ω) hasthe ℵ1-FN under CH. In section 2.3.4, we show that P(ω) and a lot of other ccc complete Booleanalgebras still have the ℵ1-FN in a model obtained by adding arbitrary number of Cohen reals toa model of, say, V = L. On the other hand it can happen very easily that P(ω) does not havethe ℵ1-FN. In [46], it was shown that this is the case when b > ℵ1 or, more generally, if there isa ⊆∗-sequence of elements of P(ω) of order-type ≥ ω2.

When κ is fixed, for C ⊆ κ, we denote with (C)′ the set of limit points of C other than κ.

2.3.1. Very weak square and dominating matrix. For a cardinal µ, the weak squareprinciple for µ (notation: ∗µ) is the statement: there is a sequence (Cα)α∈Lim(µ+) such that forevery α ∈ Lim(µ+)w1) Cα ⊆ P(α) and |Cα| ≤ µ;w2) every C ∈ Cα is club in α and if cf(α) < µ then otp(C) < µ;w3) for every C ∈ Cα and δ ∈ (C)′, we have C ∩ δ ∈ Cδ.

Clearly we have µ → ∗µ. Jensen [53] proved that ∗µ is equivalent to the existence of a specialAronszajn tree on µ+. Ben-David and Magidor [27] showed that the weak square principle fora singular µ is actually weaker than the square principle: they constructed a model of ∗ℵω and¬ℵω starting from a model with a supercompact cardinal.

Foreman and Magidor considered in [43] the following principle which is, e.g. under GCH, aweakening of ∗ principle: for a cardinal µ, the very weak square principle for µ holds if there isa sequence (Cα)α<µ+ and a club D ⊆ µ+ such that for every α ∈ Dv1) Cα ⊆ α, Cα is unbounded in α;v2) for all bounded x ∈ [Cα]<ω1 , there is β < α such that x = Cβ .

In this paper, we shall use the following yet weaker variant of the very weak square principle.For a regular cardinal κ and µ > κ, let ∗∗∗κ,µ be the following assertion: there exists a sequence(Cα)α<µ+ and a club set D ⊆ µ+ such that for α ∈ D with cf(α) ≥ κy1) Cα ⊆ α, Cα is unbounded in α;y2) [α]<κ ∩ Cα′ : α′ < α dominates [Cα]<κ (with respect to ⊆).

Since y2) remains valid when Cα’s for α ∈ D are shrunk, we may replace y1) byy1’) Cα ⊆ α, Cα is unbounded in α and otp(Cα) = cf(α).

A corresponding remark holds also for the sequence of the very weak square principle.

Lemma 2.3.6 ([7]). a) The very weak square principle for µ implies ∗∗∗ω1,µ.b) For a singular cardinal µ and a regular κ such that cf([λ]<κ,⊆) ≤ µ for every λ < µ, ∗µimplies ∗∗∗κ,µ.

∗∗∗κ,µ has some influence on cardinal arithmetic of cardinals below µ:

Lemma 2.3.7 ([7]). Suppose that κ is regular and µ is such that cf(µ) < κ. If ∗∗∗κ,µ holds, thenwe have cf([λ]<κ,⊆) < µ for every λ < µ.

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Suppose now that κ is a regular cardinal and µ > κ is such that cf(µ) < κ. Let µ∗ = cf(µ). Fora sufficiently large χ and x ∈ H(χ), let us call a sequence (Mα,β)α<µ+,β<µ∗ a (κ, µ)-dominatingmatrix over x — or just dominating matrix over x if it is clear from the context which κ and µare meant — if the following conditions hold:(j1) Mα,β ≺ H(χ), x ∈Mα,β , κ+ 1 ⊆Mα,β and |Mα,β | < µ for all α < µ+ and β < µ∗

(j2) (Mα,β)β<µ∗ is an increasing sequence for each α < µ+ ;(j3) if α < µ+ is such that cf(α) ≥ κ, then there is β∗ < µ∗ such that, for every β∗ ≤ β < µ∗,

[Mα,β ]<κ ∩Mα,β is cofinal in ([Mα,β ]<κ,⊆) ;j1) Mα,β ≺ H(χ), x ∈Mα,β , κ+ 1 ⊆Mα,β and |Mα,β | < µ for all α < µ+ and β < µ∗ ;j2)j3) if α < µ+ is such that cf(α) ≥ κ, then there is β∗ < µ∗ such that, for every β∗ ≤ β < µ∗,

[Mα,β ]<κ ∩Mα,β is cofinal in ([Mα,β ]<κ,⊆) ;For α < µ+, let Mα =

⋃β<µ∗Mα,β . By j1) and j2), we have Mα ≺ H(χ).

j4) (Mα)α<µ+ is continuously increasing and µ+ ⊆⋃α<µ+ Mα.

Foreman and Magidor [43] called a sequence of subsets of µ+ having some properties similar tothose of the sequence (µ+ ∩Mα,β)α<µ+,β<µ∗ for (Mα,β)α<µ+,β<µ∗ as above Jensen matrix. Ourdefinition of dominating matrices and some ideas in the proofs in this section are inspired by theirpaper. Note that, in the case of 2<κ = κ, j3) can be replaced by the following seemingly strongerproperty:j3’) if α < µ+ is such that cf(α) ≥ κ, then there is β∗ < µ∗ such that, for every β∗ ≤ β < µ∗,

[Mα,β ]<κ ⊆Mα,β .This is simple because of the following observation:

Lemma 2.3.8 ([7]). Suppose that 2<κ = κ and M is an elementary submodel of H(χ) for somesufficiently large χ and κ ⊆M . If [M ]<κ∩M is cofinal in [M ]<κ, then we have [M ]<κ ⊆M .

Note also that, if M ≺ H(χ) is Vκ-like, then [M ]<κ ∩M is cofinal in [M ]<κ. Hence, under2<κ = κ, M ≺ H(χ) is Vκ-like if and only if |M | = κ and [M ]<κ ⊆M .

In the following theorem, we show that ∗∗∗κ,µ together with a very weak version of the SingularCardinals Hypothesis below µ implies the existence of a dominating matrix:

Theorem 2.3.9. Suppose that κ is a regular cardinal and µ > κ is such that cf(µ) < κ. If wehave cf([λ]<κ,⊆) = λ for cofinally many λ < µ and ∗∗∗κ,µ holds, then, for any sufficiently large χand x ∈ H(χ), there is a (κ, µ)-dominating matrix over x.

Proof. Let µ∗ = cf(µ) and (µβ)β<µ∗ be an increasing sequence of cardinals below µ such thatµ0 > µ∗, supµβ : β < µ∗ = µ and cf([µβ ]<κ,⊆) = µβ for every β < µ∗. Let (Cα)α∈µ+

and D ⊆ µ+ be as in the definition of ∗∗∗κ,µ . Without loss of generality, we may assume that|Cα| ≤ cf(α) for all α < µ+. We may also assume that α > µ for every α ∈ D.

In the following, we fix a well ordering E on H(χ) and, when we talk about H(χ) as astructure, we mean H(χ) = (H(χ),∈,E). X ⊆ H(χ) as a substructure of H(χ) is thus thestructure (X,∈ ∩X2,E∩X2) — for notational simplicity we shall denote such a structure simplyby (X,∈,E).

Let N ∈ H(χ) be an elementary substructure of H(χ) such that N contains every thingneeded below — in particular, we let µ+ ⊆ N and x, (Cα)α<µ+ , D, (µα)α<µ∗ ∈ N . Let (Nξ)ξ<κbe an increasing sequence of elementary submodels of H(χ) such that(0) N0 = N ;(1) Nξ ∈ H(χ) for every ξ < κ and(2) (Nη)η≤ξ ∈ Nξ+1 for every ξ < κ.

Now, for each ξ < κ, let

Nξ = (Nξ,∈,E, Rξ, κ, µ, µ∗, D, (Cα)α<µ+ , (µα)α<µ∗ , x, η)η<ξwhere Rξ is the relation (η,Nη) : η < ξ . For X ⊆ µ+, let us denote with skξ(X) the Skolem-hull of X with respect to the built-in Skolem functions of Nξ (induced from E). For ξ < ξ′ < κ,Nξ is an element of Nξ′ by (2) and the Skolem functions of Nξ are also elements of Nξ′ . In

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particular, we have skξ(X) ⊆ skξ′(X). It follows that sk(X) =⋃ξ<κ skξ(X) is an elementary

submodel of H(χ). Note also that, if X ⊆ µ+ is an element of skξ′(Y ) then, since skξ(X) isdefinable in skξ′(Y ), we have skξ(X) ∈ skξ′(Y ).

For the proof of the theorem, it is clearly enough to construct Mα,β with j1) — j4) for everyα in the club set D and for every β < µ∗. Let

Mα,β = sk(µβ ∪ Cα)

for α ∈ D and β < µ∗. We show that (Mα,β)α∈D,β<µ∗ is as desired. It is clear that j1) and j2)hold. We need the following claim to show the other properties:

Claim 2.3.9.1.Mα = sk(α) for every α ∈ D.

Proof of the Claim. “⊆” is clear since µβ ∪ Cα ⊆ α for every α ∈ D and β < µ∗. For “⊇”, itis enough to show that α ⊆ Mα. Let γ < α. By y1), there is γ1 ∈ Cα such that γ < γ1. Letf ∈ Mα,0 be a surjection from µ to γ1, and let δ < µ be such that f(δ) = γ. Then we haveγ = f(δ) ∈Mα,β∗ ⊆Mα for β∗ < µ∗ such that δ < µβ∗ .

For j3), suppose that α ∈ D and cf(α) ≥ κ. Let β∗ be such that |Cα| < µβ∗ and [α]<κ∩Cα′ :α′ < α, α′ ∈Mα,β∗ dominates [Cα]<κ. The last property is possible by y2), Claim 1 and µ∗ < κ.We show that this β∗ is as needed in j3). Let β < µ∗ be such that β∗ ≤ β and suppose thatx ∈ [Mα,β ]<κ. Then there are u ∈ [µβ ]<κ and v ∈ [Cα]<κ such that x ⊆ sk(u∪v). Since µβ ∈Mα,β

and cf([µβ ]<κ,⊆) = µβ , there is X ∈ Mα,β such that X ⊆ [µβ ]<κ, |X| = µβ and X is cofinal in([µβ ]<κ,⊆). Since µβ ⊆Mα,β , it follows that X ⊆Mα,β . Hence there is u′ ∈Mα,β ∩ [µβ ]<κ suchthat u ⊆ u′. On the other hand, by definition of β∗, cf(α) ≥ κ, there is α′ ∈ α ∩Mα,β such thatCα′ ∈ [α]<κ and v ⊆ Cα′ . Thus, we have x ⊆ sk(u ∪ v) ⊆ sk(u′ ∪ Cα′). By regularity of κ, thereis ξ < κ such that x ⊆ skξ(u′ ∪ Cα′). But skξ(u′ ∪ Cα′) ∈Mα,β and |skξ(u′ ∪ Cα′)| < κ.

j4) follows immediately from Claim 2.3.9.1.

Note that, in the proof above, the sequence (Mα,β)α<µ+,β<µ∗ satisfies also:j5) for α < α′ < µ+ and β < µ∗, there is β′ < µ∗ such that Mα,β ⊆Mα′,β′ .

[ Suppose that α, α′ ∈ D are such that α < α′ and β < µ∗. By Claim 2.3.9.1, there is β′ < µ∗

such that β < β′, α ∈ Mα′,β′ and otp(Cα) ≤ µβ′ . Then we have Cα ∈ Mα′,β′ and Cα ⊆ Mα′,β′ .Also µβ ∈Mα′,β′ and µβ ⊆ µβ′ ⊆Mα′,β′ . Hence it follows that Mα,β = sk(µβ ∪ Cα) ⊆Mα′,β′ .]

Conversely, the existence of a (κ, µ)-dominating matrix (over some/any x) implies ∗∗∗κ,µ :

Theorem 2.3.10. Suppose that κ is a regular cardinal and µ > κ is such that cf(µ) < κ. If thereexists a (κ, µ)-dominating matrix, then ∗∗∗κ,µ holds.

Proof. Let µ∗ = cf(µ) and (Mα,β)α<µ+,β<µ∗ be a (κ, µ)-dominating matrix. LetMα =⋃β<µ∗Mα,β

for each α < µ+. For

X =⋃ [Mα,β ]<κ ∩Mα,β : α < µ+, β < µ∗ ,

let Cα+1 : α < µ+ be an enumeration of X. Let

D = α < µ+ : Mα ∩ µ+ = α,Cα′+1 : α′ < α ⊇ [Mα†,β ]<κ ∩Mα†,β

for every α† < α, β < µ∗ .

By j4), D is a club subset of µ+. For α ∈ D with cf(α) ≥ κ, let Cα = Mα,βα ∩ α where βα < µ+

be such that Mα,βα ∩ α is cofinal in α (this is possible as Mα ∩ µ+ = α and µ∗ < κ) and that[Mα,βα ]<κ ∩ Mα,βα is cofinal in [Mα,βα ]<κ (possible by j3) ). For α ∈ Lim(µ+) \ α ∈ D :cf(α) ≥ κ , let Cα be anything, say Cα = ∅. We claim that (Cα)α<µ+ and D as above satisfythe conditions in the definition of ∗∗∗κ,µ : y1) is clear by definition of Cα’s. To show y2), let α ∈ Dbe such that cf(α) ≥ κ and x ∈ [Cα]<κ. Then, by the choice of βα, there is y ∈ [α]<κ ∩Mα,βα

such that x ⊆ y. By j4), there are α† < α and β† < µ∗ such that y ∈Mα†,β† . By definition of D,it follows that y = Cα′+1 for some α′ < α. This shows that [α]<κ ∩ Cα′ : α′ < α dominates[Cα]<κ.

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Thus, by Theorem 2.3.9 and Theorem 2.3.10, if cf([λ]<κ,⊆) = λ for cofinally many λ < µ, ∗∗∗κ,µis equivalent to the existence of a (κ, µ)-dominating matrix. The assumption “cf([λ]<κ,⊆) = λfor cofinally many λ < µ” cannot be removed from this equivalence theorem since ∗∗∗κ,µ impliesthis. However, using the following weakening of the notion of dominating matrix, we obtain acharacterization of ∗∗∗κ,µ in ZFC: for a regular cardinal κ and µ > κ such that µ∗ = cf(µ) < κ, letus call a matrix (Mα,β)α<µ+,β<µ∗ of elementary submodels of H(χ) for a sufficiently large χ, aweak (κ, µ)-dominating matrix over x, if it satisfies j1), j2), j4) for Mα =

⋃β<µ∗Mα,β , α < µ+,

andj3−) if α < µ+ is such that cf(α) ≥ κ, then there is β∗ < µ∗ such that, for every β∗ ≤ β < µ∗,

[Mα,β ]<κ ∩Mα is cofinal in ([Mα,β ]<κ,⊆) .Since µ∗ < κ, the condition above is equivalent to:j3∗) if α < µ+ is such that cf(α) ≥ κ, then there is β∗ < µ∗ such that, for every β∗ ≤ β < µ∗,

there is β′ < µ∗ such that [Mα,β ]<κ ∩Mα,β′ is cofinal in ([Mα,β ]<κ,⊆) .

Theorem 2.3.11. Suppose that κ is a regular cardinal and µ > κ is such that µ∗ = cf(µ) < κ.Then ∗∗∗κ,µ holds if and only if there is a weak (κ, µ)-dominating matrix over some/any x.

Proof. For the forward direction the proof is almost the same as that of Theorem 2.3.9. Welet (µβ)β<µ∗ here merely an increasing sequence of regular cardinals with the limit µ. Then(Mα,β)α∈D,β<µ∗ is constructed just as in the proof of Theorem 2.3.9. Lemma 2.3.7 is then usedto see that j3−) is satisfied by this matrix. For the converse, just the same proof as that ofTheorem 2.3.10 will do.

Existence of a (weak) dominating-matrix is not a theorem in ZFC: we show in section 2.3.3 that theChang’s conjecture for ℵω together with 2ℵω = ℵω+1 implies that there is no (ℵn,ℵω)-dominatingmatrix for any n ≥ 1.

2.3.2. A characterization of the κ-Freese-Nation property. The following game overa partial ordering P was considered in [46, 45].

Definition 2.3.12. Let Gκ(P ) be the following game played by Players I and II: in a play inGκ(P ), Players I and II choose subsets Xα and Yα of P of cardinality less than κ alternately forα < κ such that

X0 ⊆ Y0 ⊆ X1 ⊆ Y1 ⊆ · · · ⊆ Xα ⊆ Yα ⊆ · · · ⊆ Xβ ⊆ Yβ ⊆ · · ·

for α ≤ β < κ. Thus a play in Gκ(P ) looks like

Player I : X0, X1, . . . , Xα, . . .

Player II : Y0, Y1, . . . , Yα, . . .

where α < κ. Player II wins the play if⋃α<κXα =

⋃α<κ Yα is a κ-sub-ordering of P . Let us

call a strategy τ for Player II simple if, in τ , each Yα is decided from the information of the setXα ⊆ P alone (i.e. also independent of α).

Another notion we need here is the following generalization of Vκ-likeness.

Definition 2.3.13. Let κ be regular and χ be sufficiently large. For D ⊆ M ≺ H(χ) : |M | <κ , we say that M ∈ [H(χ)]κ is D-approachable if there is an increasing sequence (Dα)α<κ ofelements of D such that

a) Dα ∪ Dα ⊆ Dα+1 for every α < κ; andb) M =

⋃α<κDα.

Clearly M ≺ H(χ) is Vκ-like if and only if M is D-approachable for D = M ≺ H(χ) :|M | < κ .

A slightly weaker version of the following theorem was announced in [46]:

Theorem 2.3.14. Let κ be a regular uncountable cardinal and κ ≤ λ. Suppose thati) ([µ]<κ,⊆) has a cofinal subset of cardinality µ for every µ such that κ < µ < λ and cf(µ) ≥ κ ;and

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ii) ∗∗∗κ,µ holds for every µ such that κ ≤ µ < λ and cf(µ) < κ.Then, for a partial ordering P of cardinality ≤ λ, the following are equivalent:

1) P has the κ-FN ;2) Player II has a simple winning strategy in Gκ(P ) ;3) for some, or equivalently any sufficiently large χ, and any Vκ-like M ≺ H(χ) with P , κ ∈M ,we have P ∩M ≤κ P ;4) for some, or equivalently any sufficiently large χ, there is D ⊆ [H(χ)]<κ such that D iscofinal in [H(χ)]<κ and for any D-approachable M ⊆ H(χ), we have P ∩M ≤κ P .

Note that ¬0# implies the conditions i) and ii). Also note that, for every λ < κ+ω, the con-dition i) holds in ZFC. Hence the characterization above holds for partial orderings of cardinality≤ κ+ω without any additional assumptions.

Proof. A proof of 1)⇒ 2)⇒ 3) is given in [46]. For 3)⇒ 4), suppose that P satisfies 3). Then Ptogether with D = M ≺ H(χ) : |M | < κ, P, κ ∈ M satisfies 4). The proof of 4)⇒ 1) is doneby induction on ν = |P | ≤ λ. If ν ≤ κ, then P has the κ-FN by Lemma 2.3.2. For ν > κ, letP and D be as in 4) and assume that 4)⇒ 1) holds for every partial ordering of cardinality < ν.We need the following claims:

Claim 2.3.14.1. Let χ∗ be sufficiently large above χ. Suppose that M is an elementary submodelof H(χ∗) such that P , H(χ), D ∈M , κ+ 1 ⊆M and [M ]<κ ∩M is cofinal in [M ]<κ with respectto ⊆. Then we have P ∩M ≤κ P .

Proof of the Claim. Suppose not. then there is b ∈ P such that either

a) (P ∩M) | b has no cofinal subset of cardinality < κ ; orb) (P ∩M) ↑ b has no coinitial subset of cardinality < κ.

To be definite, let us assume that we have the case a) — for the case b), just the same argumentwill do. We can construct an increasing sequence (Nα)α<κ of elements of D such that

c) Nα ∈M and |Nα| < κ for α < κ (since κ+ 1 ⊆M , it follows that Nα ⊆M) ;d) Nα ∈ Nα+1 for every α < κ ;e) (P ∩Nα) | b is not cofinal in (P ∩Nα+1) | b for every α < κ.

Then N =⋃α<κNα is D-approachable elementary submodel of H(χ) by c) and d). Hence, by

4), we have P ∩M ≤κ P . But, by e), (P ∩N) | b has no cofinal subset of cardinality < κ. Thisis a contradiction.

To see that the construction of Nγ is possible at a limit γ < κ, assume that Nα, α < γhave been constructed in accordance with c),d) and e). Let N ′ =

⋃α<γ Nα. By c), we have

N ′ ⊆ M and |N ′| < κ. Since [M ]<κ ∩M is cofinal in [M ]<κ, there is some N ′′ ∈ M such thatN ′ ⊆ N ′′ ⊆ H(χ) and |N ′′| < κ. Hence by elementarity ofM and by D ∈M there is N ′′′ ∈M∩Dsuch that N ′′ ⊆ N ′′′. Clearly, we may let Nγ = N ′′′. For the construction at a successor step,assume that Nα have been chosen in accordance with c),d) and e). By assumption, there isc ∈ (P ∩M) | b such that there is no c′ ∈ (P ∩Nα) | b with c ≤ c′. By elementarity of M there isN∗ ∈M ∩ D such that Nα ∪ Nα, c ⊆ N∗ and |N∗| < κ. Then Nα+1 = N∗ is as desired.

Claim 2.3.14.2. If Q ≤κ P , then for every D-approachable M ≺ H(χ) with Q ∈ M we haveQ ∩M ≤κ Q. In particular, such Q also satisfies the condition 4).

Proof of the Claim. Let M ≺ H(χ) be D-approachable with Q ∈M . By assumption, we haveP ∩M ≤κ P . By elementarity of M and since Q ∈M , we have Q∩M ≤κ P ∩M . It follows thatQ ∩M ≤κ P and hence Q ∩M ≤κ Q. Now, let D0 = M ∈ D : Q ∈M . Then it is clear thatQ satisfies the condition 4) with D0 in place of D.

Now we are ready to prove the induction steps.Case I : ν is a limit cardinal or ν = µ+ with cf(µ) ≥ κ. Let ν∗ = cf ν. Then, by i), we can findan increasing sequence (Mα)α<ν∗ of elementary submodels of H(χ∗) such that, for every α < ν∗,

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|Mα| < ν and Mα satisfies the conditions in Claim 2.3.14.1; and P ⊆⋃α<ν∗Mα. By Claim

2.3.14.1, P ∩Mα ≤κ P for every α < ν∗. For α < ν∗ let

Pα =

P ∩Mα, if α is a successor,P ∩ (

⋃β<αMβ), otherwise.

Then (Pα)α<ν∗ is a continuously increasing sequence of sub-orderings of P such that |Pα| < ν forevery α < ν∗ and P =

⋃α<ν∗ Pα. We have also Pα ≤κ P for every α < ν∗: for a successor α < ν∗

this is clear. If a limit α < ν∗ has cofinality < κ then Pα can be represented as the union of anincreasing sequence of < κ many κ-sub-ordering of P and hence Pα ≤κ P . If α < ν∗ is a limitwith cofinality ≥ κ, then Pα = P ∩M where M =

⋃β<αMβ . Now it is clear that M satisfies

the conditions in Claim 2.3.14.1. Hence we again obtain that Pα = P ∩M ≤κ P . Now, by Claim2.3.14.2, each of Pα, α < ν∗ satisfies the condition 4) and hence, by induction hypothesis, has theκ-FN. Thus, by Lemma 2.3.3, P also has the κ-FN.

Case II : ν = µ+ with cf(µ) < κ. Let µ∗ = cf(µ). Without loss of generality we may as-sume that the underlying set of P is ν. By Theorem 2.3.9, there is a (κ, µ)-dominating matrix(Mα,β)α<ν,β<µ∗ over (P,H(χ)). For α < ν and β < µ∗, let Pα,β = P∩Mα,β and Pα =

⋃β<µ∗ Pα,β .

By j4), the sequence (Pα)α<ν is continuously increasing and⋃α<ν Pα = P . |Pα| ≤ µ for every

α < ν by j1).

Claim 2.3.14.3. Pα ≤κ P for every α < ν.

Proof of the Claim. For α < ν such that cf(α) ≥ κ, we have Pα,β ≤κ P for every sufficientlylarge β < µ∗ by j3) and Claim 2.3.14.1. Since µ∗ < κ, it follows that Pα ≤κ P . If cf(α) < κ,then, by the argument above, we have Pα′ ≤κ P for every α′ < α with cf(α′) ≥ κ. Since Pα canbe represented as the union of < κ many of such Pα′ ’s, it follows again that Pα ≤κ P .

Now, by Claim 2.3.14.2, each of Pα, α < ν satisfies the condition of 4). Hence, by inductionhypothesis, they have the κ-FN. By Lemma 2.3.3, it follows that P also has the κ-FN.

In [46], it is shown that 〈P(ω1),⊆〉 does not have the weak Freese-Nation property. If acomplete Boolean algebra B does not have the c.c.c., then 〈P(ω1),⊆〉 can be completely embeddedinto B. Hence, in this case, B can not have the weak Freese-Nation property (see Proposition2.4.1 a) or [46, Lemma 2.7]).

Corollary 2.3.15. Suppose that κ and λ satisfy i), ii) in Theorem 2.3.14 and 2<κ = κ. Then:a) Every κ-cc complete Boolean algebra of cardinality ≤ λ has the κ-FN.b) For any µ such that µ<κ ≤ λ, the partial ordering ([µ]<κ,⊆) has the κ-FN.

Proof. Let χ be sufficiently large. For a), let B be a κ-cc complete Boolean algebra. We showthat B satisfies 3) in Theorem 2.3.14. Let M ≺ H(χ) be Vκ-like with B, κ ∈ M . By 2<κ = κ,Lemma 2.3.8 and by the remark after the lemma, we have [M ]<κ ⊆ M . Hence B ∩ M is acomplete subalgebra of B. It follows that B ∩ M ≤κ B. For b), let M ≺ H(χ) be Vκ-likewith λ, κ ∈ M . Then as above we have [M ]<κ ⊆ M . Hence, letting X = λ ∩ M , we have[λ]<κ ∩M = [X]<κ ≤κ [λ]<κ.

2.3.3. Chang’s Conjecture for ℵω. Recall that (κ, λ)→→ (µ, ν) is the following assertion:For any structure A = (A,U, . . .) of countable signature with |A| = κ, U ⊆ A and |U | = λ,there is an elementary substructure A′ = (A′, U ′, . . .) of A such that |A′| = µ and |U ′| = ν.

In [80], a model of ZFC + GCH + Chang’s Conjecture for ℵω, i.e. (ℵω+1,ℵω) →→ (ℵ1,ℵ0), isconstructed starting from a model with a cardinal having a property slightly stronger than huge.The following theorem together with Corollary 2.3.15 shows that the ℵ1-FN of the partial ordering([ℵω]ℵ0 ,⊆) is independent from ZFC (or even from ZFC + GCH).

Theorem 2.3.16. Suppose that (ℵω)ℵ0 = ℵω+1 and (ℵω+1,ℵω) →→ (ℵ1,ℵ0). Then ([ℵω]ℵ0 ,⊆)does not have the ℵ1-FN.

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Proof. Assume to the contrary that there is an ℵ1-FN mapping F : [ℵω]ℵ0 → [[ℵω]ℵ0 ]ℵ0 . Let usfix an enumeration (bα)α<ℵω+1 of [ℵω]ℵ0 and consider the structure:

A = (ℵω+1,ℵω,≤, E, f, g, h),

where

1) ≤ is the canonical ordering on ℵω+1;2) E = (α, β) : α ∈ ℵω, β ∈ ℵω+1, α ∈ bβ ;3) f : ℵω+1 × ℵω+1 → ℵω+1 is such that, for each α ∈ ℵω+1,

F (bα) = bf(α,n) : n ∈ ω ;4) g : ℵω+1 × ℵω+1 → ℵω is such that, for each α ∈ ℵω+1, g(α, ·) | α is an injective mappingfrom α to ℵω; and5) h : ℵω+1 × ℵω+1 × ℵω+1 → ω + 1 is such that for each α, β ∈ ℵω+1, h(α, β, ·) | (bα ∩ bβ) isinjective.

Note that, by 5) and since ω is definable in A, we can express “bα∩bβ is finite” as a formula ϕ(α, β)in the language of A. Now, applying the Chang’s conjecture for A with A = ℵω+1 and U = ℵω,we obtain elementary substructure A′ = (A′, U ′,≤′, E′, f ′, g′, h′) of A such that |A′| = ℵ1 and|U ′| = ℵ0.

Claim 2.3.16.1. otp(A′) = ω1.

Proof. By 4) and elementarity ofA′, every initial segment of A′ can be mapped into U ′ injectivelyand hence countable. Since |A′| = ℵ1, it follows that otp(A′) = ω1.

Claim 2.3.16.2. For any α < ℵω+1, there is γ < ℵω+1 such that bβ ∩ bγ is finite for every β < α.

Proof. Since |α| ≤ ℵω, we can find a partition (In)n∈ω of α such that |In| < ℵω for everyn < ω. For n < ω, let ηn = min(ℵω \

⋃ bξ : ξ ∈

⋃m≤n Im ). Let z = ηn : n ∈ ω and

γ < ℵω+1 be such that bγ = z. For any β < α, if β ∈ Im0 for some m0 < ω, then we havebβ ∩ bγ ⊆ ηn : n < m0 . Thus this γ is as desired.

Claim 2.3.16.3. For any countable I ⊆ A′, there is γ ∈ A′ such that bβ ∩ bγ is finite for everyβ ∈ I.

Proof. By Claim 2.3.16.1, there is α ∈ A′ such that I ⊆ α. By elementarity of A′, the formulawith the parameter α expressing the assertion of Claim 2.3.16.2 for this α holds in A′. Hencethere is some γ ∈ A′ such that bβ ∩ bγ is finite for every β ∈ A′ ∩ α.

LetI = ξ ∈ A′ : bξ ∈ F (U ′) .

Then I is countable. Hence, by Claim 2.3.16.3, there is γ ∈ A′ such that bβ ∩ bγ is finite forevery β ∈ I. As bγ ⊆ U ′ (this holds in virtue of h(γ, γ, ·)), there is b ∈ F (bγ) ∩ F (U ′) such thatbγ ⊆ b ⊆ U ′. Let b = bξ0 . Then ξ0 ∈ I and |bγ ∩ bξ0 | = |bγ | = ℵ0. This is a contradiction to thechoice of γ.

At this point it is unclear if the assumption of Theorem 2.3.16 yields a counter example toCorollary 2.3.15, a) for κ = ℵ1. Or, more generally:

Question 2.3.17 ([7, Problem 2]). Is there a model of ZFC + GCH where some ccc completeBoolean algebra does not have the ℵ1-FN?

Of course, we need the consistency strength of some large cardinal to obtain such a model byCorollary 2.3.15.

We answer this question in the positive in Theorem 2.4.7.The following corollary slightly improves [43, Theorem 4.1].

Corollary 2.3.18. Suppose that (ℵω)ℵ0 = ℵω+1 and (ℵω+1,ℵω)→→ (ℵ1,ℵ0). Then the equivalenceof the assertions in Theorem 2.3.14 fails. Hence we have ¬∗∗∗ℵ1,ℵω under these assumptions.

Proof. By Theorem 2.3.16 and Corollary 2.3.15, b).

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Similarly we can prove ¬∗∗∗ℵn,ℵω for every n ∈ ω under the assumption of 2ℵω = ℵω+1 and(ℵω+1,ℵω)→→ (ℵ1,ℵ0).

2.3.4. Cohen models. Let V be our ground model and let G be a V -generic filter overP = Fn(τ, 2) for some τ . Suppose that B is a ccc complete Boolean algebra in V [G]. Withoutloss of generality we may assume that the underlying set of B is a set X in V . B is said to betame if there is a P -name ≤ of partial ordering of B and a mapping t : X → [τ ]ℵ0 in V suchthat, for every p ∈ P and x, y ∈ X, if p ‖–P “ x ≤ y ”, then p | (t(x) ∪ t(y)) ‖–P “ x ≤ y ”. A lotof ‘natural’ ccc complete Boolean algebras in V [G] are contained in the class of tame Booleanalgebras:

Lemma 2.3.19. Let G be as above. Suppose that V [G] |= “ B is a ccc complete Boolean algebra ”and either:i) there is a Boolean algebra B′ in V such that B′ is dense subalgebra of B in V [G]; orii) B is the completion of a Suslin forcing in V [G].Then B is tame.

For Suslin forcing, see e.g. [24].

Theorem 2.3.20. Let P , G be as above and λ an infinite cardinal. Assume that, in V ,i) µℵ0 = µ for every regular uncountable µ such that µ < λ; andii) ∗∗∗ℵ1,µ

holds for every µ such that ℵ0 < µ < λ and cf(µ) = ω.

Then, for any tame ccc complete Boolean algebra B in V [G] of cardinality ≤ λ we have V [G] |=“ B has the ℵ1-FN ”.

Proof. Let X, ≤ and t be as in the definition of tameness for B above. We may assume ‖–P “ Xis the underlying set of B ” and ‖–P “ B is a ccc complete Boolean algebra ” where B is a P -namefor B. Let χ be sufficiently large. The following is the key lemma to the proof:

Claim 2.3.20.1. Suppose that M ≺ H(χ) is such that τ , X, ≤, t ∈ M and [M ]ℵ0 ⊆ M . LetI = τ ∩M , P ′ = Fn(I, 2), G′ = G ∩ P ′, X ′ = X ∩M and ≤′= ≤ ∩M . Then

a) ≤′ is a P ′-name, ‖–P “ B′ = (X ′, ≤′) is a ccc complete Boolean algebra ”, B′ is tame (inV [G′]) and the infinite sum ΣB in V [G] is an extension of the infinite sum ΣB

′in V [G′].

b) ‖–P “ (X ′, ≤′) ≤σ (X, ≤) ”.

Proof of the Claim. a): It is easy to see that ‖–P ′ “ (X ′, ≤′) is a Boolean algebra ” and ‖–P “ (X ′, ≤′)is a subalgebra of (X, ≤) ”. Since ‖–P “ (X, ≤) has the ccc ”, it follows that ‖–P ′ “ (X ′, ≤′) hasthe ccc ”. By elementarity of M , it is also easy to see that t | X ′ witnesses the tameness of B′ inV [G′]. To see that ‖–P ′ “ (X ′, ≤′) is complete ” it is enough to see that every countable subset ofB′ has its supremum in V [G′]. Let C be a P ′-name of countable subset of X ′. Without loss ofgenerality, we may assume that C is countable and consists of sets of the form (p, x) where p ∈ P ′and x ∈ X ′. Since [M ]ℵ0 ⊆ M , C ∈ M . Clearly, we have M |= “ C is a P -name of countablesubset of X ”. Hence, M |= “ ∃p ∈ P ∃y ∈ X(p ‖–P “ ΣC = y ”) ”. Let p ∈ P and y ∈ X be suchelements of M . Then p ∈ P ′ and y ∈ X ′. By elementarity of M , we have p ‖–P “ ΣBC = y ”. Onthe other hand, from M |= “ p ‖–P “ ΣC = y ” ” it follows that p ‖–P ′ “ ΣB

′C = y ”.

b): By assumption, for x ∈ X and y ∈ X ′, we have y ≤ x in V [G], if and only if thereis p ∈ G, dom(p) ⊆ t(x) ∪ t(y) such that p ‖–P “ y ≤ x ”. For q ∈ G ∩ Fn(t(y), 2), the setUq = y ∈ X ′ : ∃p ∈ G′ (p ∪ q ‖–P “ y ≤ x ”) is in V [G′]. Hence, by a), ΣBUq is an elementin B′. Since X ′ is closed under t, it follows that ΣBUq : q ∈ G ∩ Fn(t(y), 2) is cofinal inB′ | y.

Now, let ν = |X|. Without loss of generality, we may assume that X = ν. We show by inductionon ν that ‖–P “ B has the ℵ1-FN ”. For ν ≤ ℵ1 the assertion follows from Lemma 2.3.2 (appliedin V P ). In the induction steps, we mimic the proof of Theorem 2.3.14. Let χ be sufficiently large.

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Case I : ν is a limit cardinal or ν = µ+ for some µ with cf(µ) ≥ ω1. By i), we can construct acontinuously increasing sequence (Mα)α<ν of elementary submodels of H(χ) such that

0) τ , ν, ≤, t ∈M0;1) |Mα| < ν for every α < ν;2) [Mα+1]ℵ0 ⊆ Mα+1 for every α < ν (note that it follows that the inclusion also holds for

every limit < ν of cofinality ≥ ω1); and3) ν ⊆

⋃α<νMα.

For each α < ν, let Xα = X ∩Mα and ≤α=≤ ∩Mα and let Bα be the P -name corresponding to(Xα, ≤

α). By Claim 2.3.20.1, we have ‖–P “ Bα is ccc complete Boolean algebra and Bα ≤ℵ1 B ”,

for all α < ν such that either α is a successor or of cofinality ≥ ω1. By induction hypothesis,we have ‖–P “ Bα has the ℵ1-FN ” for such α’s. Hence by Lemma 2.3.3 and the remark after thelemma (applied in V P ) it follows that ‖–P “ B has the ℵ1-FN ”.

Case II : ν = µ+ for a µ with cf(µ) = ω. By ii), there is an (ℵ1, µ)-dominating matrix(Mα,n)α<ν,n<ω over (τ, ν, ≤, t). For α < ν, let Mα =

⋃n<ωMα,n. For α < ν and n < ω,

let Xα,n = X ∩Mα,n, ≤α,n

=≤ ∩Mα,n and Bα,n be the P -name corresponding to (Xα,n, ≤α,n

).Likewise, letXα = X∩Mα, ≤

α=≤ ∩Mα and Bα be the P -name corresponding to (Xα, ≤

α). Then

we have Xα =⋃n<ωXα,n, ≤

α=⋃n<ω ≤

α,n and ‖–P “ Bα =⋃n<ω Bα,n ”. By Lemma 2.3.8 and

i), j3’) holds for the dominating matrix. Hence, by Claim 2.3.20.1, we have ‖–P “ Bα,n ≤ℵ1 Bαand Bα,n is a ccc complete Boolean algebra ” for every α < ν with cf(α) > ω and every sufficientlylarge n < ω. By induction hypothesis, it follows that ‖–P “ Bα,n has the ℵ1-FN ” for such α andn. By Lemma 2.3.3 (applied in V P ) it follows that ‖–P “ Bα has the ℵ1-FN ” for every α < ν withcf(α) > ω. Hence again by Lemma 2.3.3 and the remark after that (applied in V P ) we obtainthat ‖–P “ B has the ℵ1-FN ”.

Corollary 2.3.21. Suppose that V = L holds and let G be V -generic over P = Fn(τ, 2) for someτ . Then (among others) the following ccc complete Boolean algebras have the ℵ1-FN: Cκ (∼= thecompletion of Fn(κ, 2)) for any κ; P(ω) (hence also P(ω)/fin); Borel(IR)/Null-sets.

Theorem 2.3.20 raises the following question:

Question 2.3.22 ([7, Problem 3]).Assume that V [G] is a model obtained by adding Cohen realsto a model of ZFC + CH. Is it true that P(ω) has the ℵ1-FN in V [G] ?

By Theorem 2.3.20, the answer to this question is positive if the number of added Cohen reals isless than ℵω.

In Theorem 2.4.8, we answer this question in the negative.

Question 2.3.23.Does Theorem 2.3.20 hold also without the assumption of tameness?

Or, more generally:

Question 2.3.24 ([7, Problem 5]).Are the following equivalent?(i) P(ω) has the ℵ1-FN;(ii) every ccc complete Boolean algebra has the ℵ1-FN.

The answer is no, as we will see in Corollary 2.4.4.

2.4. Weak Freeze-Nation property of Boolean algebras


To answer Questions 2.3.17, 2.3.22, 2.3.23. and 2.3.24 we prove the following in this section.(a) In a model obtained by adding ℵ2 Cohen reals, there is always a c.c.c. complete Boolean

algebra without the weak Freese-Nation property. (See Theorem 2.4.3)(b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. com-

plete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (SeeTheorem 2.4.7)

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2.4. WEAK FREEZE-NATION PROPERTY OF BOOLEAN ALGEBRAS 111

(c) Modulo consistency of (ℵω+1,ℵω)→→ (ℵ1,ℵ0), it is consistent with GCH that C(ℵω) does nothave the weak Freese-Nation property and hence the assertion in (c) does not hold, and alsothat adding ℵω Cohen reals destroys the weak Freese-Nation property of 〈P(ω),⊆〉. (SeeTheorem 2.4.8)After giving a negative answer to Question 2.3.23, the following problem still remains: For

which Boolean algebra B, the weak Freese-Nation property of B is equivalent with the weak Freese-Nation property of 〈P(ω),⊆〉 ? The following was proved in [3] :(d) If a weak form of µ and cof([µ]ℵ0 ,⊆) = µ+ hold for each µ > cf(µ) = ω, then the weak

Freese-Nation property of 〈P(ω),⊆〉 is equivalent to the weak Freese-Nation property of anyof C(κ) or R(κ) for uncountable κ.The following criteria of the weak Freese-Nation property are used in the later sections. A

partial ordering Q is said to be a retract of a partial ordering P if there are order preservingmappings i : Q→ P and j : P → Q such that j i = idQ.

Q is said to be a σ-subordering of P (notation: Q ≤σ P ) if, for every p ∈ P , Q | p = q ∈Q : q ≤ p has a countable cofinal subset and Q ↑ p = q ∈ Q : q ≥ p has a countable coinitialsubset. Note that if C is a complete subalgebra of a complete Boolean algebra B (notation:C ≤c B) or a countable union of complete subalgebras of B, then it follows that C ≤σ B.

Proposition 2.4.1. (a) (Lemma 2.7 in [46]) If Q is a retract of P and P has the weak Freese-Nation property then Q has the weak Freese-Nation property.(b) Suppose that Q is a complete Boolean algebra and there is a strictly order-preserving em-bedding f of Q into P (i.e. f preserves ordering and incomparability). If P has the weak Freese-Nation property then Q also has the weak Freese-Nation property.(c) (Lemma 2.3 (a) in [46]) If Q ≤σ P and P has the weak Freese-Nation property, then Q alsohas the weak Freese-Nation property.(d) (Lemma 2.6 in [46]) If Pα, α < δ is an increasing sequence of partial orderings with the weakFreese-Nation property such that Pα ≤σ Pα+1 for every α < δ and Pγ =

⋃α<γ Pα for all γ < δ

with cf(γ) > ω, then P =⋃α<δ Pα also has the weak Freese-Nation property.

Proposition 2.4.1 (b) follows easily from Proposition 2.4.1 (a): the mapping g : P → Q definedby g(p) =

∑q ∈ Q : f(q) ≤ p for p ∈ P witnesses that Q is a retract of P .

2.4.1. The partial ordering PS . In this subsection we introduce a construction of partialorderings PS and Boolean algebras BS which will be used in the following Section 2.4.2 and 2.4.3.For S ⊆ κ and an indexed family S = 〈Sα : α ∈ S〉 of subsets of κ, let

PS = xi : i ∈ κ ∪ yα : α ∈ Swhere xi’s and yα’s are pairwise distinct, and let ≤S be the partial ordering on PS defined by

p ≤S q ⇔ p = q orp = xi and q = yα for some i ∈ κ and α ∈ S with i ∈ Sα .

Let BS be the Boolean algebra generated freely from PS except the relation ≤S . Note that theidentity map on PS canonically induces a strictly order-preserving embedding of PS into BS .

Proposition 2.4.2. Suppose that cf(κ) ≥ ω2, S ⊆ κ is stationary such that S ⊆ α < κ :cf(α) ≥ ω1 and S = Sα : α ∈ S is such that Sα is a cofinal subset of α for each α ∈ S. IfPS is embedded into a partial ordering P by a strictly order-preserving mapping then P does nothave the weak Freese-Nation property. In particular, BS and its completion do not have the weakFreese-Nation property.

Proof. Without loss of generality, we may assume that PS is a subordering of P . Assume to thecontrary that there is a weak Freese-Nation mapping f : P → [P ]≤ℵ0 . Let

C = ξ < κ : ∀η < ξ ∀p ∈ f(xη)[∃α ∈ S [xη ≤ p ≤ yα ] → ∃α′ ∈ S ∩ ξ [xη ≤ p ≤ yα′ ]

].

Then C is a club subset of κ. Let α ∈ C ∩ S and let

A = p ∈ f(yα) : ∃η ∈ Sα [ p ∈ F (xη) ∧ xη ≤ p ≤ yα ] .

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Since α ∈ C, for each p ∈ A there is αp < α such that p ≤ yαp . Let α∗ = supαp : p ∈ A.Since A is countable we have α∗ < α. Let β ∈ Sα \α∗. Since xβ ≤ yα, there is a p ∈ A such thatxβ ≤ p ≤ yα. Hence xβ ≤ yαp . But this is impossible since αp < α∗ ≤ β.

2.4.2. Cohen models. Consider the following principle:(∗∗) There is a sequence 〈Sα : α ∈ Lim(ω2)〉 such that each Sα is a cofinal subset of α

and for any pairwise disjoint 〈xβ : β < ω1〉 with xβ ∈ [ω2]<ℵ0 for β < ω1, there areβ0 < β1 < ω1 such that xβ0 ∩ Sα = ∅ for all α ∈ xβ1 ∩ Lim(ω2) and that xβ1 ∩ Sα = ∅for all α ∈ xβ0 ∩ Lim(ω2) .

Proposition . Let P = Fn(ω2, 2). Then ‖–P “ (∗∗) ”.

Proof. Without loss of generality we may assume P = Fn(⋃α∈Lim(ω2) α × α, 2). For α ∈

Lim(ω2), let Sα be a P -name such that ‖–P “ Sα = β ∈ α : g(β, α) = 1 ” where g is thecanonical name for the generic function. By genericity, ‖–P “ Sα is cofinal in α ” for every α ∈Lim(ω2). Let S be a P -name such that ‖–P “ S = 〈Sα : α ∈ Lim(ω1)〉 ”.

To show that S is forced to satisfy the property in (∗∗), let 〈xβ : β < ω1〉 be a P -name ofa sequence of pairwise disjoint finite subsets of ω2. For each β < ω1, let pβ and xβ ∈ [ω2]<ℵ0

be such that pβ ‖–P “ xβ = xβ ”. By thinning out the index set ω1, we may assume without lossof generality that dom(pβ), β < ω1 form a ∆-system with the root d and pβ | d, β < ω1 areall equal to the same p ∈ P . Since pβ , β < ω1 are then pairwise compatible, xβ , β < ω1 arepairwise disjoint. Further, we may assume that sβ , β < ω1 form a ∆-system with the root swhere sβ = γ : 〈γ, α〉 ∈ dom(pβ) for some α ∈ Lim(ω2). Note vthat xβ , β < ω1 are pairwisedisjoint since pβ ’s are pairwise compatible.

Let β0 < β1 < ω1 be such that xβ0 ∩ s = ∅, xβ1 ∩ s = ∅, xβ0 ∩ sβ1 = ∅ and xβ1 ∩ sβ0 = ∅.To take such β0 and β1, first fix a β0 such that xβ0 ∩ s = ∅. This is possible since s is finiteand xβ ’s are pairwise disjoint. By the same argument we can finde infinitely many β’s such thatxβ ∩ (s ∪ sβ0) = ∅. Now for such β’s, since sβ \ s are pairwise disjoint, there are infinitely may βamong them with xβ0 ∩ sβ = ∅. Let β1 be one of such β’s.

Letp∗ = pβ0 ∪ pβ1 ∪ 〈〈β, α〉, 0〉 : β ∈ xβ0 , α ∈ xβ1 ∩ Lim(ω2)

∪〈〈β, α〉, 0〉 : β ∈ xβ1 , α ∈ xβ0 ∩ Lim(ω2)Then p∗ ∈ P . Clearly p∗ ‖–P “ xβ0 ∩ Sα = ∅ ” for all α ∈ xβ1 ∩Lim(ω2) and p∗ ‖–P “ xβ1 ∩ Sα = ∅ ”for all α ∈ xβ0 ∩ Lim(ω2).

Proposition . Suppose that 〈Sα : α ∈ Lim(ω2)〉 is as in (∗∗). Let S = α < ω2 : cf(α) = ω1and S = 〈Sα : α ∈ S〉. Then BS satisfies the c.c.c.

Proof. Otherwise we can find Iα ∈ [ω2]<ℵ0 , Jα ∈ [S]<ℵ0 for α < ω1 and t(α, i), u(α, ξ) ∈+1,−1 for each i ∈ Iα, ξ ∈ Jα and α < ω1 such that

zα =∏i∈Iα t(α, i)xi ·

∏ξ∈Jα u(α, ξ) yξ, α < ω1

form a pairwise disjoint family of elements of BS+.By ∆-system argument, we may assume that Iα ∪Jα, α < ω1 are pairwise disjoint. Applying

(∗∗) to 〈Iα ∪ Jα : α < ω1〉, we find β0 < β1 < ω1 such that Iβ0 ∩ Sξ = ∅ for all ξ ∈ Jβ1 andthat Iβ1 ∩ Sξ = ∅ for all ξ ∈ Jβ0 . By definition of BS , it follows that zβ0 · zβ1 6= 0 . This is acontradiction.

Theorem 2.4.3. In a Cohen model (i.e. any model obtained by adding ≥ ℵ2 Cohen reals) thereis a c.c.c. complete Boolean algebra B of density ℵ2 without the weak Freese-Nation property.

Proof. By Proposition , (∗∗) holds in a Cohen model. Hence BS as in Proposition satisfiesthe c.c.c. By Proposition 2.4.2, the completion of BS does not have the weak Freese-Nationproperty.

Corollary 2.4.4.The weak Freese-Nation property of 〈P(ω),⊆〉 does not imply the weak Freese-Nation property of all c.c.c. complete Boolean algebras.

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2.4. WEAK FREEZE-NATION PROPERTY OF BOOLEAN ALGEBRAS 113

Proof. If we start from a model of CH and add ℵ2 Cohen reals, then 〈P(ω),⊆〉 has the weakFreese-Nation property in the resulting model (see e.g. [7]). On the other hand, by Theorem2.4.3, there is a c.c.c. complete Boolean algebra without the weak Freese-Nation property in sucha model.

Under CH, every c.c.c. complete Boolean algebra of size ℵ2 has the weak Freese-Nationproperty ([46]). Hence it follows from the result above that

Proposition 2.4.5.CH implies ¬(∗∗).

2.4.3. The Weak Freese-Nation property of c.c.c. complete Boolean algebras un-der GCH. In [7] it is proved that, assuming CH and a weak form of square principle at singularcardinals of cofinality ω, every c.c.c. complete Boolean algebra has the weak Freese-Nation prop-erty. In this section we show that even GCH does not suffice for this result.

Hajnal, Juhász and Shelah [50] showed that, starting from a model with a supercompactcardinal, a model of GCH and the following principle can be constructed:

(∗∗∗) There are a stationary S ⊆ α < ωω+1 : cf(α) = ω1 and a family S = 〈Sα : α ∈ S〉such that each Sα is a cofinal subset of α of ordertype ω1 and that, for all distinct α,β ∈ S, Sα ∩ Sβ is finite.

Proposition 2.4.6. Suppose that S = 〈Sα : α ∈ S〉 is as in (∗∗∗). Then BS satisfies the c.c.c.

Proof. Otherwise we can find Iα ∈ [ωω+1]<ℵ0 , Jα ∈ [S]<ℵ0 , α < ω1 and t(α, i), u(α, ξ) ∈+1,−1 for each α < ω1, i ∈ Iα and ξ ∈ Jα such that

zα =∏i∈Iα t(α, i)xi ·

∏ξ∈Jα u(α, ξ) yξ, α < ω1

form a pairwise disjoint family of elements of BS+.By ∆-system argument, we may assume that Iα ∪ Jα, α < ω1 are pairwise disjoint and each

Iα has the same size, say n.For α < β < ω1, since zα · zβ = 0, either (I) there is η ∈ Jα such that Iβ ∩ Sη 6= ∅ or else (II)

there is ξ ∈ Jβ such that Iα ∩ Sξ 6= ∅. If (I) holds then let us say that 〈α, β〉 is of type (I).Now, one of the following two cases should hold. We show that both of them lead to a

contradiction.Case I. There is an infinite subset S of ω such that for every β ∈ ω1 \ ω, k ∈ S : 〈k, β〉 is

of type (I) is cofinite in S. In this case, by thinning out the index set ω1, we may assume that,for any k ∈ ω and β ∈ ω1 \ ω, 〈k, β〉 is of type (I). For all β ∈ ω1 \ ω, since | Iβ | = n, there are0 ≤ i0(β) < i1(β) < n + 1 such that I∗β = Iβ ∩ Sα0(β) ∩ Sα1(β) 6= ∅ for some αk(β) ∈ Jik(β) fork = 0, 1 by Pigeonhole Principle. Hence we can find an infinite X ⊆ ω1 \ ω, 0 ≤ i0 < i1 < n+ 1and α0, α1 ∈ ω1, α0 6= α1 such that α0(β) = α0 and α1(β) = α1 for all β ∈ X. But then⋃β∈X I

∗β ⊆ Si0 ∩ Si1 . Since the set on the left side is infinite as a disjoint union of pairwise

disjoint non-empty sets, this is a contradiction to (∗∗∗).Case II. For any infinite subset S ⊆ ω, there is β ∈ ω1 \ ω such that for infinitely many

k ∈ S, 〈k, β〉 is not of type (I). In this case, by thinning out the first ω elements of the index setω1, we may assume that for each k ∈ ω, there is ξ(k) ∈ Jω such that Ik ∩ Sξ(k) 6= ∅. Note thatJω is finite. So by thinning out further the first ω elements of the index set ω1, we may assumethat there is ξ0 ∈ Jω such that Ik ∩ Sξ0 6= ∅ for every k < ω. Similarly we may also assumethat there are ξi ∈ Jω+i for 1 ≤ i ≤ n such that Ik ∩ Sξi 6= ∅ for every k < ω. Then for eachk < ω, we can find i0(k) < i1(k) ≤ n such that I∗k = Ik ∩ Sξi0(k)

∩ Sξi1(k)6= ∅ by Pigeonhole

Principle. Since there are only n(n− 1)/2 possibilities of i0(k) < i1(k) ≤ n, there are i0 < i1 ≤ nand an infinite set X ⊆ ω such that for every k ∈ X, i0(k) = i0 and i1(k) = i1. It follows thatSξi0 ∩ Sξi1 ⊇

⋃k∈X I

∗k . As Sξi0 ∩ Sξi1 is finite, while

⋃k∈X I

∗k is infinite as a union of infinitely

many pairwise disjoint non-empty sets, this is a contradiction.

Theorem 2.4.7. It is consistent with GCH (modulo the consistency strength of a supercompactcardinal) that there is a c.c.c. complete Boolean algebra without the weak Freese-Nation property.

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Proof. Let S be a family as in (∗∗∗). By Proposition 2.4.6 and Proposition 2.4.2 the completionof BS is a c.c.c. complete Boolean algebra without the weak Freese-Nation property.

2.4.4. An application of a Chang’s Conjecture.

Theorem 2.4.8. Suppose that V0 is a transitive model of ZFC such that

V0 |= GCH + (ℵω+1,ℵω)→→ (ℵ1,ℵ0) .

Let P be a c.c.c. partial ordering in V0 of cardinality ℵ1 adding a dominating real. Let η ∈ ωω bea dominating real over V0 generically added by P and let V1 = V0[η]. Note that V1 |= GCH. InV1 let Q = Fn(ℵω, ω) and let Q be a corresponding P -name. Then we have:

V1 |= ‖–Q “ 〈P(ω),⊆〉 does not have the weak Freese-Nation property ”.

Proof. In the proofs of this and the next theorem, we shall denote by a dotted symbol a nameof an element in a generic extension. By the same symbol without the dot, we denote thecorresponding element in a fixed generic extension. Without further mention, we shall identifyP ∗ Q names with the corresponding Q-name in V1 and vice versa.

Now toward a contradiction, assume that there is a Q-name F in V1 such that

(⊗) V1 |= ‖–Q “ F is a weak Freese-Nation mapping on 〈ωω,≤∗〉 ”.

Let ϕ be a P ∗ Q-name of the function ℵω → ω generically added by Q over V1. Let V2 = V1[ϕ].By GCH, we can find in V0 a scale 〈fα : α < ℵω+1〉 in 〈

∏n∈ω ℵn,≤∗〉. Without loss of generality,

we may assume that for every α < ℵω+1 and n ∈ ω, fα(n) ∈ ℵn \ ℵn−1 (where we set ℵ−1 = ∅).For each α ∈ ℵω+1, let gα : ω → ω be defined by

gα = ϕ fα .Let χ be sufficiently large and let N ≺ 〈H(χ),∈〉 be such that N contains everything we need

in the course of the proof: in particular 〈fα : α < ℵω+1〉 ∈ N , | ℵω∩N | = ℵ0 and otp(ℵω+1∩N) =ω1. The last two conditions are possible by the assumption of V0 |= (ℵω+1,ℵω)→→ (ℵ1,ℵ0).

In V0, let ξn,k : k ∈ ω be an enumeration of (ℵn \ ℵn−1) ∩N for each n ∈ ω. Here again,we use the convention that ℵ−1 = ∅. Let h∗ be a P ∗ Q-name of an element of ωω such that

‖–P∗Q “ h∗(n) = maxϕ(ξn,k) : k ≤ η(n) for all n ∈ ω ” .

Claim 2.4.8.1. For every α ∈ ℵω+1 ∩N , ‖–P∗Q “ gα ≤∗ h∗ ”.

Proof of the Claim. Since α ∈ N we have fα ∈ N . Let eα : ω → ω be the function in V0 suchthat fα(n) = ξn,eα(n) for all n ∈ ω. Since η is dominating, there is n∗ ∈ ω such that

V1 |= eα | (ω \ n∗) ≤ (η | ω \ n∗).

By definition of h∗, it follows that

V2 |= gα(n) = ϕ fα(n) = ϕ(ξn,eα(n)) ≤ h∗(n)

for all n ≥ n∗.

Let N0 = N , N1 = N0[η] and N2 = N1[ϕ]. Then we have N2 ≺ H(χ)[η][ϕ].Let hl ∈ N0, l ∈ ω be P ∗ Q-names such that

‖–P∗Q “ hl : l ∈ ω = F (h∗) ∩ N2 ”.

For l ∈ ω, Let Sl ∈ [ℵω]ℵ0 ∩N0 be such that, regarding hl as a P -name of Q-name,

V1 |= ‖–P “ hl is a Fn(Sl, ω)-name ”.

This is possible since P has the c.c.c., ‖–P “ Q has the c.c.c. ” and by the elementarity of N0. Foreach l ∈ ω, let sl ∈

∏n∈ω ℵn ∩ N0 be defined (in N0) by sl(k) = supSl ∩ ℵk for k ∈ ω. Since

〈fα : α < ℵω+1〉 was taken to be a scale on 〈∏n∈ω ℵn,≤∗〉, for each l ∈ ω there is αl ∈ ℵω+1∩N0

such that sl ≤∗ fαl . Since otp(ℵω+1 ∩ N) = ω1, we can find an α∗ ∈ ℵω+1 ∩ N0 such thatsupαl : l ∈ ω ≤ α∗.

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2.5. WEAK FREEZE-NATION PROPERTY OF P (ω) 115

Now, by choice of hl, l ∈ ω, the following claim together with Claim 2.4.8.1 contradicts to(⊗) and hence proves the theorem:

Claim 2.4.8.2. V1 |= ‖–Q “ gα∗ 6≤∗hl for all l ∈ ω ”.

Proof of the Claim. Assume to the contrary that, in V1, we have

q ‖–Q “ gα∗ | (ω \ k) ≤ hl | (ω \ k) ”

for some q ∈ Q and k, l ∈ ω. We may assume that

sl | (ω \ k) < fα∗ | (ω \ k)

and sup(dom(p)) < fα∗(m) for all m ∈ ω \ k as well. Working further in V1, let m∗ = k + 1 andlet q′ ≤ q be such that

q′ ‖–Q “ hl(m∗) = j∗ ”for some j∗ ∈ ω. Then q′′ = q ∪ (q′ | Sl) also forces the same statement. Since fα∗(m∗) ∈ℵm∗ \ (sl(m∗) ∪ dom(p)), we have

fα∗(m∗) 6∈ dom(q′′).

Henceq∗ = q′′ ∪ 〈fα∗(m∗), j∗ + 1〉

is an element of Q and q∗ ≤ q′′ ≤ q. Butq∗ ‖–Q “ gα∗(m∗) = ϕ fα∗(m∗) = j∗ + 1 > j∗ = hl(m∗). ”

This is a contradiction.

The theorem is proved.

2.5. Weak Freeze-Nation property of P (ω)

(This section is based on [4] and [2])In this section we study the consequences of the weak Freese-Nation property of (P(ω),⊆).

It is shown that the weak Freese-Nation property of P(ω) captures many of the features of Cohenmodels and hence may be considered as one of the principles axiomatizing some portion of thecombinatorics available in Cohen models. It is shown that the weak Freese-Nation property of(P(ω),⊆) is one of the strongest among such principles: e.g. we prove under this assumption thatmost of the known cardinal invariants including all of those which appear in Cichón’s diagramtake the same value as in a corresponding Cohen model.

Some consequences of the weak Freese-Nation property of P(ω) on algebraic behavior ofP(ω)/fin were studied in [69].

2.5.1. Luzin gap. For a regular κ, an almost disjoint family A ⊆ [ω]ℵ0 is said to be aκ-Luzin gap if | A | = κ and for any x ∈ [ω]ℵ0 either | a ∈ A : | a \ x | < ℵ0 | < κ or| a ∈ A : | a ∩ x | < ℵ0 | < κ.

Theorem 2.5.1.Assume that P(ω) has the ℵ1-FN. Then there is no ℵ2-Luzin gap.

Proof. Let f : P(ω) → [P(ω)]ℵ0 be an ℵ1-FN mapping. We may assume that f(a) = f(b) =f(ω \ b) for all a, b ∈ P(ω) with a =∗ b. Thus x ⊆∗ y implies that there is z ∈ f(x) ∩ f(y) suchthat x ⊆∗ z ⊆∗ y and |x ∩ y | < ℵ0 implies that there is z ∈ f(x) ∩ f(y) such that x ⊆∗ z and| z ∩ y | < ℵ0.

Suppose that A = aα : α < ω2 is an almost disjoint family of subsets of ω. We show thatA is not an ℵ2-Luzin gap. Let χ be sufficiently large regular cardinal and consider the modelH = (H(χ),∈,E) where E is any well-ordering on H. Let N be an elementary submodel of Hsuch that A, f ∈ N , N∩ω2 ∈ ω2 and cf(δ) = ω1 for δ = N∩ω2. For α ∈ N we have | aα∩aδ | < ℵ0

and hence aα ⊆∗ (ω \ aδ). Thus there is bα ∈ f(aα)∩ f(aδ) such that aα ⊆∗ bα ⊆∗ (ω \ aδ). Sincef(aδ) is countable and cf(δ) = ω1 there is b ∈ f(aδ) such that I = α < δ : bα = b is cofinal inδ. We show that b witnesses that A is not an ℵ2-Luzin gap, i.e., J = α < ω2 : aα ⊆∗ b andK = α < ω2 : | b ∩ aα | < ℵ0 both have cardinality ℵ2.

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Claim 2.5.1.1. | J | = ℵ2.

Proof of the Claim. First note that b ∈ N since b ∈ f(aα) for any α ∈ I ⊆ N . Hence we haveJ ∈ N and I ⊆ J . Since I is cofinal in N ∩ ω2, we have N |= J is cofinal in ω2. By elementarityit follows that H |= J is cofinal in ω2. Hence J is really cofinal in ω2.

Claim 2.5.1.2. |K | = ℵ2.

Proof of the Claim. Since b ∈ N it follows that K ∈ N . For β ∈ N ∩ ω2 = δ, we haveH |= δ ∈ K ∧ β < δ. Hence H |= K 6⊆ β and N |= K 6⊆ β by elementarity. It follows thatN |= Kis unbounded in ω2. Hence, again by elementarity, H |= Kis unbounded in ω2. Thus K isreally unbounded in ω2.

This proves the theorem.

Corollary 2.5.2. b = ℵ1 or even the statement “ P(ω) does not contain any strictly increasing⊆∗-chain of length ω2” does not imply that P(ω) has the ℵ1-FN.

Proof. Suppose that our ground model V satisfies the CH. Koppelberg and Shelah [70] provedthat the forcing with Fn(ω2, 2) can be represented as a two step iteration A∗ B where ‖–A “ P(ω)contains an ℵ2-Luzin gap ”. Thus, by Theorem 2.5.1, we have ‖–A “ P(ω) does not have theℵ1-WF ”. On the other hand, we have ‖–A∗B “ P(ω) does have the ℵ1-WF ” by Theorem 2.3.20.Hence ‖–A∗B “ there is no strictly increasing ⊆∗-chain in P(ω) of length ω2 ”. It follows that‖–A “ there is no strictly increasing ⊆∗-chain in P(ω) of length ω2 ”.

2.5.2. Cardinal invariants. In this subsection we show that the weak Freese-Nation prop-erty of P(ω) implies that all the cardinal invariants of reals appearing in [28] or [65] behave justas in a Cohen model.

We shall begin with a brief review of definitions and basic facts of some of these cardinalinvariants.

cov(meager) and non(meager) denote the covering number and “non” of the ideal of meagersets respectively:

cov(meager) = min| F | : ∀X ∈ F(X ⊆ R ∧X is meager) ∧⋃F = R,

non(meager) = min|X | : X ⊆ R ∧X is non-meager.The following characterization of cov(meager) will be used:

Lemma 2.5.3. (for the proof see e.g. [28])

cov(meager) = min| F | : F ⊆ ωω ∧ ∀f ∈ ωω∃g ∈ F∀n ∈ ω(f(n) 6= g(n)).In a Cohen model, cov(meager) = 2ℵ0 and non(meager) = ℵ1, and the values of cardinal

invariants in Cichón’s diagram are decided from these equations:

cov(null) ←− non(meager) ←− cof (meager) ←− cof (null)∣∣ y y ∣∣∣∣ b ←− d∣∣y y y y

add(null) ←− add(meager) ←− cov(meager) ←− non(null)

⇐ ℵ1

.................. 2ℵ0 ⇒The equations cov(meager) = 2ℵ0 and non(meager) = ℵ1 also imply that s = e = ℵ1 andr = u = i = 2ℵ0 .

The following variant of the bounding number is studied in [39] and [65].

b∗ = minκ : ∀F ⊆ ωω (F is unbounded→ ∃G ⊆ F (| G | ≤ κ ∧ G is unbounded)).

Also a similar variation of non(meager) is studied in [65] and [106]:

shr(meager) = minκ : ∀X ⊆ R (F is non-meager→ ∃Y ⊆ X (|Y | ≤ κ ∧ Y is non-meager)).

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Clearly b ≤ b∗ ≤ d and non(neager) ≤ shr(meager) ≤ cof(meager). The equation b∗ ≤shr(meager) is proved in [106].

Cichón’s diagram in Cohen models with these new cardinal invariants looks like this (see [65]and [106]):

cov(null)←− non(meager) ←− shr(meager)←− cof (meager) ←− cof (null)∣∣ y y y ∣∣∣∣ b ←− b∗ ←− d∣∣y y y y

add(null)←− add(meager) ←− cov(meager) ←− non(null)

⇐ ℵ1

.................. 2ℵ0 ⇒

G ⊆ [ω]ℵ0 is said to be groupwise dense if it is downward closed with respect to ⊆∗ and forevery strictly increasing f ∈ ωω, there is an infinite X ⊆ ω such that

⋃n∈X [f(n), f(n+ 1)) ∈ G.

The groupwise density number g is defined by

g = min| G | : ∀G ∈ G(G ⊆ [ω]ℵ0 ∧G is groupwise dense) ∧⋂G = ∅

(see [29]). G ⊆ [ω]ℵ0 is groupwise dense if, and only if, G is downward closed with respect to ⊆∗and G is non-meager, under the identification of [ω]ℵ0 with the subspace f ∈ ω2 : | n ∈ ω :f(n) = 1 | = ℵ0 of ω2 in the canonical way (see [28]). In [29], it is shown that g = ℵ1 in aCohen model.

(1) of the following proposition asserted originally the weaker equation:non(meager) = ℵ1. We thank M. Kada and S. Kamo for pointing out that our proof showsactually the equation in the present form.

Proposition 2.5.4.Assume that P(ω) has the weak Freese-Nation property. Then

(1) shr(meager) = ℵ1. Hence non(meager) = b∗ = ℵ1 and s = e = ℵ1;(2) a = ℵ1;(3) g = ℵ1.

Proof. (1): Suppose that S ⊆ R is non-meager. We show that there is a T ⊆ S of cardinality≤ ℵ1 such that T is still non-meager in R.

Let χ be large enough and M ≺ H(χ) be such that |M | = ℵ1, S ∈ M and that [M ]ℵ0 ∩Mis cofinal in [M ]ℵ0 with respect to ⊆. We show that S ∩ M is non-meager. This suffices as|S ∩M | ≤ ℵ1.

Let Q = (q, r) ∈ Q : q < r and let f ∈M be a weak Freese-Nation mapping on (P(Q),⊆).For each x ∈ P(Q), let o(x) denote the union of open intervals corresponding to each element ofx. Thus o(x) = t ∈ R : ∃(q, r) ∈ x (q < t < r).

Let D ∈ [P(Q)]ℵ0 be such that o(d) is a dense subset of R for all d ∈ D. We show below that⋂o(d) : d ∈ D ∩ S ∩M 6= ∅. From this it follows readily that S ∩M is non-meager.For each d ∈ D, let C(d) = f(d) ∩M ∩ x ∈ P(Q) : d ⊆ x . Then we have C(d) ⊆ D

and o(d) =⋂o(x) : x ∈ C(d). Let C(d) ∈ [D]ℵ0 ∩M be such that C(d) ⊆ C(d). Let o(d) =⋂

o(x) : x ∈ C(d). Then o(d) ∈M , o(d) is co-meager and o(d) ⊆ o(d). Let F ∈ [M ]ℵ0 ∩M besuch that o(d) : d ∈ D ⊆ F and that each X ∈ F is a co-meager set ⊆ R. Then, since

⋂F ∈M

is co-meager, there is an r ∈ S ∩M ∩⋂F . But r ∈

⋂F ⊆

⋂o(d) : d ∈ D ⊆

⋂o(d) : d ∈ D.

Hence⋂o(d) : d ∈ D ∩ S ∩M 6= ∅.

(2): Let χ, M ≺ H(χ) be as in the proof of (1) and let f ∈ M be a weak Freese-Nationmapping on (P(ω),⊆∗). Let F : P(ω)→ [P(ω)]≤ℵ0 be defined by F (x) = f(x) ∪ f(ω \ x). ThenF is again an element of M . Let 〈Sα : α < ω1〉 be an enumeration of [[ω]ℵ0 ]ℵ0 ∩M and let〈aα : α < ω1〉 be a sequence of pairwise almost disjoint elements of [ω]ℵ0 ∩M such that:

(∗) for any x ∈ Sα, if |x \⋃β∈u aβ | = ℵ0 for all u ∈ [α]<ℵ0 , then |x ∩ aα | = ℵ0.

We show that aα : α < ω1 is maximal almost disjoint. Otherwise, there is a b ∈ [ω]ℵ0 almostdisjoint from all aα, α < ω1. Let α∗ < ω1 be such that F (b) ∩ M ⊆ Sα∗ . Since b and aα∗

are almost disjoint, there is x ∈ F (b) ∩ F (aα∗) such that (i) b ⊆∗ x and (ii) |x ∩ aα∗ | < ℵ0.Since x ∈ F (b) ∩M ⊆ Sα∗ , (i) implies that x satisfies the if-clause in (∗) for α∗. But then (ii)contradicts the choice of aα∗ .

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(3): Let M be as before and let f ∈M be a weak Freese-Nation mapping on (P(ω),⊆∗). ForC ∈ [[ω]ℵ0 ]ℵ0 ∩M let

GC = x ∈ [ω]ℵ0 : ∃y ∈ [ω]ℵ0 ∩M(∀c ∈ C (c 6⊆ y) ∧ x ⊆ y).GC is groupwise dense: Clearly GC is downward-closed with respect to ⊆∗. To show that GC isnon-meager, let G′C = y ∈ [ω]ℵ0 ∩M : ∀c ∈ C c 6⊆∗ y. Then G′C = [ω]ℵ0 ∩M \

⋃c∈C,n∈ωy ∈

[ω]ℵ0 : c\n ⊆ y. Since [ω]ℵ0∩M is non-meager by the proof of (1) and each y ∈ [ω]ℵ0 : c\n ⊆ yis nowhere dense, it follows that G′C is non-meager. By G′C ⊆ GC , GC is non-meager as well.

Let G = GC : C ∈ [[ω]ℵ0 ]ℵ0 ∩M. We show that⋂G = ∅. Since | G | = ℵ1, this completes

the proof.For arbitrary x ∈ [ω]ℵ0 , let C ∈ [[ω]ℵ0 ]ℵ0 ∩M be such that f(x) ∩ [ω]ℵ0 ⊆ C. Then for any

y ∈ [ω]ℵ0 ∩M with x ⊆∗ y, there is c ∈ C such that x ⊆∗ c ⊆∗ y. It follows that such y is not inG′C . Thus x 6∈ GC and hence x 6∈ ∩G.

The following proposition implies that, in the most of the cases, the right half of Cichón’sdiagram obtains the value 2ℵ0 under the weak Freese-Nation property of P(ω):

Proposition 2.5.5. Suppose that χ is a sufficiently large regular cardinal and M ≺ H(χ) be suchthat

(i) [M ]ℵ0 ∩M is cofinal in [M ]ℵ0 with respect to ⊆;(ii) for h ∈ ωω, there is f ∈ ωω ∩M such that h(n) 6= f(n) for every n ∈ ω;(iii) R \M is not empty.

Then P(ω) ∩M is not a σ-subordering of P(ω).

Proof. Towards a contradiction, assume that P(ω) ∩M ≤σ P(ω). Fix x ∈ R \M .

Claim 2.5.5.1.There is a countable set C ∈M of infinite closed subsets of R such that, for anyclosed set c ∈M containing x, there is c′ ∈M such that x ∈ c′ ⊆ c .

Proof of the Claim. Let Q be as in the proof of Proposition 2.5.4, (1). Let s = (q, r) ∈Q : x ≤ q ∨ r ≤ x. Then, by assumption, there is a countable D ⊆ P(Q) ∩ M cofinal iny ∈ P(Q) : y ⊆ s with respect to ⊆. By (i), let D′ ∈ [P(Q)]ℵ0 ∩M be such that D ⊆ D′. ThenC = R \ o(d) : d ∈ D′, R \ o(d) is infinite is as desired where o(d) is defined as in the proof ofProposition 2.5.4, (1).

Let C be as in Claim 1 and let 〈cn : n ∈ ω〉 be an enumeration of C in M and let umn : m ∈ω, n ∈ ω ∈M be a family of open subsets of R such that for each n ∈ ω, umn , m ∈ ω are pairwisedisjoint and, for every m ∈ ω, umn ∩ cn 6= ∅. Let f ∈ ωω be such that x ∈ umn implies f(n) = m.This is possible as umn , m ∈ ω are disjoint. By (ii), there is a g ∈ ωω ∩M such that f(n) 6= g(n)for every n ∈ ω. Let

c∗ =⋂

n∈ωR \ ug(n)

n .

Then x ∈ c∗ and by definition of c∗, c 6⊆ c∗ for all c ∈ C. This is a contradiction to the choice ofC.

Using Borel coding, we can also prove the variant of Proposition above for sufficiently absoluteinner models M of models of some fragment of ZFC.

The following corollary together with Proposition 2.5.4 establishes that, under the conditionsas in the corollary, the weak Freese-Nation property of P(ω) implies that the values of cardinalinvariants of the reals are just the same as their values in a Cohen model with the same value of2ℵ0 .

Corollary 2.5.6. Suppose that κ < 2ℵ0 : cf([κ]ℵ0 ,⊆) = κ is cofinal in κ < 2ℵ0 : cf(κ) > ω(This is the case if e.g. 2ℵ0 ≤ ℵω+1 or if ¬0# holds). If P(ω) has the weak Freese-Nation propertythen cov(meager) = 2ℵ0 and hence also r = u = i = 2ℵ0 .

Proof. Suppose that cov(meager) < 2ℵ0 . Then, by Lemma 2.5.3 and the assumption, there is anM ≺ H(χ) satisfying (i), (ii), (iii) of Proposition 2.5.5. By Proposition 2.5.5, P(ω)∩M 6≤σ P(ω).Hence by Theorem 2.3.4, P(ω) does not have the weak Freese-Nation property.

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2.5.3. LCS∗ spaces. W. Just showed that there is no LCS∗ space X with cardinal sequence〈ω〉ω2

and 〈ω〉ω1

_〈ω2〉 in a Cohen model. The following theorem generalizes this non-existencetheorem.

Theorem 2.5.7. Suppose that P(ω) has the weak Freese-Nation property. Then:(1) There are no LCS∗ spaceX with height ω1 + 1, such that | I0(X)| = ℵ0, | Iα(X)| ≤ ℵ1 for all

α < ω1 and | Iω1(X)| ≥ ℵ2.(2) There are no LCS∗ space X with height ω2, such that | I0(X)| = ℵ0, and | Iα(X)| ≤ ℵ1 for all

α < ω2.

Proof. (1): Towards a contradiction, assume that there is an LCS∗ X with height ω1 + 1, suchthat | I0(X)| = ℵ0, | Iα(X)| ≤ ℵ1 for all α < ω1 and | Iω1(X)| ≥ ℵ2. Write Xν for Iν(X).

Without loss of generality we may assume that X0 = ω.For each x ∈ X let u(x) be a clopen neighbor of x such that, for α = ht(x), u(x)\X<α = x.

LetS = u ∩ ω : u ⊆ X is closed and covered by finitely many

u(x)’s such that ht(x) < ω1and

L = A ⊆ ω : supht(x) : x ∈ A = ω1.

Claim 2.5.7.1. For A ∈ L, A ∩Xω1 6= ∅.

Proof of the Claim. Suppose that A ∩Xω1 = ∅. Then, by compactness there is S ∈ [ω1]<ℵ0 ,such that A ⊆

⋃α∈S X<α.

Claim 2.5.7.2. For any A ∈ L there exists at most one y ∈ Xω1 such that A ⊆ u(y).

Proof of the Claim. By the previous claim A∩Xω1 6= ∅. If A ⊆ u(y) then A ⊆ u(y) and henceA ∩Xω1 = y.

Now let χ be sufficiently large and let M ≺ H(χ) be such that X ∈ M , |M | = ℵ1 andω1 ⊆ M . Note that Xα ⊆ M for all α < ω1. Let F ∈ M be a weak Freese-Nation mapping on(P(ω),⊆).

By the claim above, and since |M | = ℵ1 and |Xω2 | ≥ ℵ2, there exists an x∗ ∈ Xω1 suchthat, for any A ∈ P(ω) ∩M , if A ⊆ u(x∗) ∩ ω, then ht[A] is bounded. Let 〈An : n ∈ ω〉 be anenumeration of A ∈ P(ω) ∩M : A ∈ F (u(x∗) ∩ ω), A ⊆ u(x∗) ∩ ω. Then βn = sup ht[An] isless than ω1 for each n ∈ ω. Let β = supn∈ω βn + 1 and let y ∈ Xβ ∩ u(x∗). Then there is aB ∈ [X<β ]<ℵ0 such that V = u(y) \

⋃t∈B u(t) ⊆ u(x∗). Since V ∈ M there should be an n ∈ ω

such that V ∩ ω ⊆ An. But this is a contradiction as sup ht[V ] = β > sup ht[An].

(2): Assume that there is a LCS∗ X with height ω2, such that | I0(X)| = ℵ0, and | Iα(X)| ≤ ℵ1

for all α < ω2. Write Xν for Iν(X).Let χ be sufficiently large and let M ≺ H(χ) be such that X ∈ M , ω1 ⊆ M , ω2 ∩M ∈ ω2

and cof (γ) = ω1 for γ = ω2 ∩M .Let F ∈M be a weak Freese-Nation mapping on (P(ω),⊆). Take x∗ ∈ Xγ and let 〈An : n ∈

ω〉 be an enumeration of A ∈ P(ω) ∩M : A ∈ F (u(x∗) ∩ ω), A ⊆ u(x∗) ∩ ω. By elementaritysup ht[An] < γ. Hence the same argument as in the proof of (1) leads to a contradiction.

2.5.4. Closed covers of the real reals. This is a recent application. In [2] we investigatedthe following question: Assume that H is a κ-fold cover of a set X. Under which assumption canyou partition H into κ subcover?

Definition 2.5.8. If X is a set, H ⊂ P(X), and x ∈ X, let H(x) = H ∈ H : x ∈ H.Let κ be a cardinal. A partial function c : H → On is a κ-good coloring of H if for each

x ∈ Xif |H(x)| ≥ κ then κ ⊆ c[H(x)].

The coloring c is a [κ,∞)-good coloring of H over Y if it is λ-good for each cardinal λ ≥ κ.

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Theorem 2.5.9 ([2]).Assume that 〈P(ω),⊆〉 has the weak Freese-Nation property. Let (X, τ) bea topological space which has a countable base, and let H be a cover of X by closed sets. Thenthere exists an [ω1,∞)-good coloring of H.

Corollary 2.5.10 ([2]).Assume that 〈P(ω),⊆〉 has the weak Freese-Nation property. Then anyω1-fold cover of the real line by closed sets can be partition into ω1-many disjoint subcover.

Let us remark that we also proved that under MAω1 there is an ω1-cover of the real line bythe translations of a single perfect set which can not be partitioned into two disjoint subcover.

2.6. Stick and clubs

(This section is based on [6] and [5])

In this section, we study combinatorial principles known as ‘stick’ and ‘club’, and their diversevariants which are all weakenings of ♦. Hence some of the consequences of ♦ still hold under theseprinciples. On the other hand, they are weak enough to be consistent with the negation of thecontinuum hypothesis or even with a weak version of Martin’s axiom in addition. We introducea new kind of side-by-side product of partial orderings which we call pseudo-product. Using suchproducts, we give several generic extensions where some of these principles hold together with¬CH and Martin’s Axiom for countable p.o.-sets. See e.g. [31], [5], [89] for applications of theseprinciples.

( |• ) (read “stick”) is the following principle introduced in S. Broverman, J. Ginsburg, K.Kunen and F. Tall [31]:

( |• ): There exists a sequence (xα)α<ω1 of countable subsets of ω1 such that for any y ∈ [ω1]ℵ1

there exists α < ω1 such that xα ⊆ y.Note that ( |• ) follows from CH.The principle ( |• ) suggests the following cardinal number:

|• = min | X | : X ⊆ [ω1]ℵ0 , ∀y ∈ [ω1]ℵ1 ∃x ∈ X x ⊆ y .

We have ℵ1 ≤ |• ≤ 2ℵ0 and ( |• ) holds if and only if |• = ℵ1.The principle (♣) (‘club’), a strengthening of ( |• ), was first formulated in Ostaszewski [89].

Let Lim(ω1) = γ < ω1 : γ is a limit . For a stationary E ⊆ Lim(ω1),♣(E): There exists a sequence (xγ)γ∈E of countable subsets of ω1 such that for every γ ∈ E,

xγ is a cofinal subset of γ with otp(xγ) = ω and for every y ∈ [ω1]ℵ1 there is γ ∈ Esuch that xγ ⊆ y.

Let us call (xγ)γ∈E as above a ♣(E)-sequence. For E = Lim(ω1) we shall simply write (♣) inplace of ♣(Lim(ω1)). Clearly ( |• ) follows from (♣). Unlike ( |• ), (♣) does not follow from CHsince (♣) + CH is known to be equivalent to ♦ (K. Devlin, see [89]). This equivalence holds alsoin the version argumented with a stationary E ⊆ Lim(ω1).

Fact* 2.6.1 ([89]). For any stationary E ⊆ Lim(ω1), ♣(E) + CH is equivalent to ♦(E).

S. Shelah [96] proved the consistency of ¬CH + (♣) in a model obtained from a model ofGCH by making the size of ℘(ω1) to be ℵ3 by countable conditions and then collapsing ℵ1 to becountable. Soon after that, in an unpublished note, J. Baumgartner gave a model of ¬CH + ♣where collapsing of cardinals is not involved: his model was obtained from a model of V = L byadding many Sacks reals by side by side product. I. Juhász then proved in an unpublished notethat “¬CH + MA(countable) + (♣)” is consistent. Here MA(countable) stands for Martin’s axiomrestricted to countable partial orderings. Later P. Komjáth [67] cited a remark by Baumgartnerthat Shelah’s model mentioned above also satisfies ¬CH + MA(countable) + (♣).

These results are rather optimal in the sense that a slight strengthening of MA(countable)implies the negation of (♣). Let MA(Cohen) denote Martin’s axiom restricted to the partialorderings of the form Fn(κ, 2) for some κ where, as in [77], Fn(κ, 2) is the p.o.-set for adding κCohen reals, i.e. the set of functions from some finite subset of κ to 2 ordered by reverse inclusion.

Fact 2.6.2.MA for the partial ordering Fn(ω1, 2) implies |• = 2ℵ0 .

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2.6. STICK AND CLUBS 121

2.6.1. Pseudo product of partial orderings. In this subsection, we introduce a new kindof side-by-side product of p.o.’s which will be used in the next section to prove various consistencyresults. LetX be any set and (Pi)i∈X be a family of partial orderings. For p ∈ Πi∈XPi the supportof p is defined by supp(p) = i ∈ X : p(i) 6= 1Pi . For a cardinal κ, let Π∗κ,i∈XPi be the set

p ∈ Πi∈XPi : | supp(p) | < κ

with the partial ordering

p ≤ q ⇔ p(i) ≤ q(i) for all i ∈ X and i ∈ X : p(i) <

6= q(i) <6= 1Pi is finite .

For κ = ℵ0 this is just a finite support product. We are mainly interested in the case whereκ = ℵ1. In this case we shall drop the subscript ℵ1 and write simply Π∗i∈XPi. Further, if Pi = Pfor some partial ordering P for every x ∈ X, we shall write Π∗κ,XP (or even Π∗XP when κ = ℵ1)to denote this partial ordering.

For p, q ∈ Π∗κ,i∈XPi the relation p ≤ q can be represented as a combination of the twoother distinct relations which we shall call horizontal and vertical, and denote by ≤h and ≤vrespectively:

p ≤h q ⇔ supp(p) ⊇ supp(q) and p | supp(q) ⊆ q;p ≤v q ⇔ supp(p) = supp(q), p(i) ≤ q(i) for all i ∈ X and

i ∈ X : p(i) <6= q(i) <

6= 1Pi is finite .For p ∈ Π∗κ,i∈XPi and Y ⊆ X let pdY denote the element of Π∗κ,i∈XPi defined by pdY (i) = 1Pifor every i ∈ X \ Y and pdY (i) = p(i) for i ∈ Y .

The following is immediate from definition:

Lemma 2.6.3. For p, q ∈ Π∗κ,i∈XPi, the following are equivalent:

a) p ≤ q;b) There is an r ∈ Π∗κ,i∈XPi such that p ≤h r ≤v q;c) There is an s ∈ Π∗κ,i∈XPi such that p ≤v s ≤h q.

Proof. b)⇒ a) and c)⇒ a) are clear. For a)⇒ b), let r = pdsupp q; for a)⇒ c), s = q | supp(q)∪p | (X \ supp(q)).

Lemma 2.6.4.1) If Pi has the property K for all i ∈ X then P = Π∗i∈XPi preserves ℵ1.2) Suppose that λ ≤ κ. If Pi has the strong λ-cc (i.e. for every C ∈ [Pi]λ there is pairwisecompatible D ∈ [C]λ), then P = Π∗κ,i∈XPi preserves λ.

Proof. This proof is a prototype of the arguments we are going to apply repeatedly. 1) and 2)can be proved similarly. For 1), assume that there would be p ∈ P and a P -name f such that

*) p ‖–P “ f : (ω1)V → ω and f is 1-1 ”.

Then, let (pα)α<ω1 and (qα)α<ω1 be sequences of elements of P such that

a) p0 ≤ p and (pα)α<ω1 is a descending sequence with respect to ≤h ;b) qα ≤v pα and qα decides f(α) for all α < ω1;c) pα | Sα = qα | Sα for every α < ω1 where

Sα = supp(qα) \ (supp(p) ∪⋃β<α

supp(qβ)).

For α < ω1 let dα =⋃β<α supp(qβ). Then (dα)α<ω1 is a continuously increasing sequence in

[X]<ω1 . Let uα = β ∈ supp(qα) : qα(β) 6= pα(β) for α < ω1. By b), uα is finite and by c) wehave uα ⊆ dα. Hence by Fodor’s lemma, there exists an uncountable (actually even stationary)Y ⊆ ω1 such that uα = u∗ for all α ∈ Y , for some fixed u∗ ∈ [X]<ℵ0 . Since Πi∈u∗Pi hasthe property K, there exists an uncountable Y ′ ⊆ Y such that qα | u∗ : α ∈ Y ′ is pairwisecompatible. It follows that qα, α ∈ Y ′ are pairwise compatible. For each α ∈ Y ′ there exists an

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nα ∈ ω such that qα ‖–P “ nα = f(α) ” by b). By *), nα, α ∈ Y ′ must be pairwise distinct. Butthis is impossible as Y ′ is uncountable.

For 2), essentially the same proof works with sequences of elements of P of length λ, usingthe ∆-system lemma argument in place of Fodor’s lemma.

Lemma 2.6.5. If | Pi | ≤ 2<κ for all i ∈ X, then Π∗κ,i∈XPi has the (2<κ)+-cc.

Proof. By the usual ∆-system lemma argument.

Corollary 2.6.6. a) Under CH, if Pi satisfies the property K and | Pi | ≤ ℵ1 for every i ∈ X,then P = Π∗i∈XPi preserves ℵ1 and has the ℵ2-cc. In particular P preserves every cardinals.b) Suppose that 2<κ = κ. If Pi satisfies the strong λ-cc for every ℵ1 ≤ λ ≤ κ and | Pi | ≤ κ thenΠ∗κ,i∈XPi preserves every cardinalities ≤ κ and has the κ+-cc. In particular Π∗κ,i∈XPi preservesevery cardinals.

Proof. By Lemmas 2.6.4, 2.6.5.

Lemma 2.6.7. For any Y ⊆ X and x ∈ X \ Y , we have

Π∗κ,i∈XPi ∼= Π∗κ,i∈Y Pi × Px ×Π∗κ,i∈X\(Y ∪x)Pi.

Proof. The mapping from Π∗κ,i∈XPi to Π∗κ,i∈Y Pi × Px ×Π∗κ,i∈X\(Y ∪x)Pi defined by

p 7→ (p | Y, p(x), p | (X \ (Y ∪ x)))

is an isomorphism.

In the following we mainly use the partial orderings of the form Fn(λ, 2) for some λ as Pi inΠ∗κ,i∈XPi. Note that Fn(λ, 2) has the property K and strong κ-cc in the sense above for everyregular κ.

For a pseudo product of the form Π∗i∈XFn(κi, 2), Lemma 2.6.4 can be still improved:

Theorem* 2.6.8. (T. Miyamoto) For any set X, and sequence (κi)i∈X , the partial orderingP = Π∗i∈XFn(κi, 2) satisfies the Axiom A.

Lemma 2.6.9. Suppose that | Pi | ≤ κ for every i ∈ X and P = Π∗κ+,i∈XPi. Then

1) If x is a P -name with ‖–P “ x ∈ V ”, then for any p ∈ P there is q ∈ P such that q ≤h p and(†) for any r ≤ q, if r decides x then rdsupp(q) already decides x.

2) Let G be P -generic. If u ∈ V [G] is a subset of V of cardinality < κ+, then there is a groundmodel set X ′ ⊆ X of cardinality ≤ κ (in the sense of V ) such that u ∈ V [G ∩ (Π∗κ+,i∈X′Pi)].

Proof. 1): Let Φ : κ→ κ× κ; α 7→ (ϕ1(α), ϕ2(α)) be a surjection such that ϕ1(α) ≤ α for everyα < κ. Let (pα)α<κ, (p′α)α<κ and (rα,β)α<κ,β<κ be sequences of elements of P defined inductivelyby:

a) p0 = p; (pα)α<κ is a descending sequence with respect to ≤h;b) for a limit γ < κ, pγ is such that supp(pγ) =

⋃α<γ supp(pα) and, for i ∈ supp(pγ), pγ(i) =

pα(i) for some α < γ such that i ∈ supp(pα);c) (rα,β)β<κ is an enumeration of r ∈ P : r ≤v pα ;d) let r = rϕ1(α),ϕ2(α) and

p′α = r | supp(r) ∪ pα | (X \ supp(r)).

If there is s ≤h p′α such that s decides x, then let

pα+1 = pα | supp(pα) ∪ s | (X \ supp(pα)).

Otherwise let pα+1 = pα.

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Let q ∈ Π∗κ,i∈XPi be defined by supp(q) =⋃α<κ supp(pα) and, for i ∈ supp(q), q(i) = pα(i)

for some α < κ such that i ∈ supp(Pα). We show that this q is as desired: suppose that r ≤ qdecides x. Then there is some α < κ such that

rdsupp(q) = p′α | supp(p′α) ∪ q | (X \ supp(p′α)).

By d), it follows that rdsupp(q) ≤ rdsupp(pα+1) decides x.

2): Let u be a P -name for u and let xα, α < κ be P -names such that ‖–P “ xα ∈ V ” for everyα < κ and ‖–P “ u = xα : α < κ ”. By 1), for each p ∈ P , we can build a sequence (pα)α<κ ofelements of P decreasing with respect to ≤h such that p0 ≤h p and

(†)α for any r ≤ pα, if r decides xα, then rdsupp(pα) already decides xα.

Let q ∈ P be defined by supp(q) =⋃α<κ supp(pα) and, for i ∈ supp(q), q(i) = pα(i) for some

α < κ such that i ∈ supp(pα). Then q satisfies:

(††) for any r ≤ q, if r decides xα for some α < κ, then rdsupp(q) already decides xα.

The argument above shows that q’s with the property (††) are dense in P . Hence, by genericity,there is such q ∈ G. Clearly, G ∩ Π∗κ,i∈supp(q)Pi contains every information needed to constructu.

2.6.2. Some Consistency results.

Proposition 2.6.10. (CH) For any infinite cardinal λ, let P = Π∗λFn(ω1, 2). Then ‖–P “ |• = λ ”.

Proof.

Claim 2. ‖–P “ |• ≥ λ ”.

Proof of the Claim. If λ = ℵ1 this is clear. So assume that λ ≥ ℵ2. For ξ < λ, let fξ be theP -name of the generic function from ω1 to 2 added by the ξ-th copy of Fn(ω1, 2) in P . Let G be aP -generic filter over V . In V [G] let X ⊆ [ω1]ℵ0 be such that | X | < λ. Then by ℵ2-cc of P thereexists ξ < λ such that X ∈ V [G′] for G′ = G∩Π∗λ\ξFn(ω1, 2). Since (fξ)[G] is Fn(ω1, 2)-genericover V [G′] by Lemma 2.6.7, we have x 6⊆ ((fξ)[G])−10 for every x ∈ X.

Claim 3. ‖–P “ |• ≤ λ ”.

Proof of the Claim. For u ∈ [λ]<ℵ0 , let Pu be a P -name such that

‖–P “ Pu = ([ω1]ℵ0)V [(fξ)ξ∈u] ”

where fξ is as in the proof of the previous claim. Let P be a P -name such that

‖–P “ P =⋃ Pu : u ∈ [λ]<ℵ0 ”.

For each u ∈ [λ]<ℵ0 , (fξ[G])ξ∈u corresponds to a generic filter over Π∗uFn(ω1, 2) ≈ Fn(ω1, 2).Hence, by CH, we have ‖–P “ | Pu | = ℵ1 ”. It follows that ‖–P “ | P | = λ ”. Thus it is enough toshow that ‖–P “ P is a |• -set ”.

Let p ∈ P and A be a P -name such that p ‖–P “ A ∈ [ω1]ℵ1 ”. We show that there is an r ≤ psuch that r ‖–P “ ∃x ∈ P x ⊆ A ”.

Now we proceed as in the proof of Lemma 2.6.4. Let (pα)α<ω1 , (qα)α<ω1 be sequences ofelements of P and (ξα)α<ω1 be a strictly increasing sequence of ordinals < ω1 such that

a) p0 ≤ p and (pα)α<ω1 is a descending sequence with respect to ≤h;b) qα ≤v pα and qα ‖–P “ ξα ∈ A ” for all α < ω1;c) pα | Sα = qα | Sα for every α < ω1 where

Sα = supp(qα) \ (supp(p) ∪⋃β<α

supp(qβ)).

For α < ω1 let uα = β ∈ supp(qα) : qα(β) 6= pα(β) . As in the proof of Lemma 2.6.4,there exists u∗ ∈ [λ]<ℵ0 such that S = α ∈ ω1 : uα = u∗ is stationary. Now (qα | u)α∈S

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is an infinite sequence of elements of Pu∗ = Πu∗Fn(ω1, 2). Since Pu∗ satisfies the ccc, thereexists an ε ∈ S and ζ < ω1 such that qε | u∗ ‖–Pu∗ “ ξ ∈ S ∩ ζ : pξ | u∗ ∈ G is infinite ”. Letr = qε ∪ pζ | (supp(pζ) \ supp(pε)). Let b be a P -name such that

r ‖–P “ b = ξ ∈ S ∩ ζ : qξ | u∗ ∈ p | u∗ : p ∈ G ”.

Let x be a P -name such that r ‖–P “ x = ξα : α ∈ b ”. Then r ‖–P “ | x | = ℵ0 ”. Since b canbe computed in V [(fξ[G])ξ∈u∗ ] we have r ‖–P “ x ∈ Pu∗ ”. It is also clear by definition of x thatr ‖–P “ x ⊆ A ”.

Proposition above shows that |• can be practically every thing. In particular we obtain:

Corollary 2.6.11.The assertion ‘cof( |• ) = ω’ is consistent with ZFC.

The following Lemmas 2.6.12 and 2.6.14 show that, in spite of typographical similarity,Π∗λFn(ω1, 2) and Π∗λFn(ω, 2) are quite different forcing notions: while the first one destroys (♣)or even ( |• ) by Lemma 2.6.10, the second one not only preserves a (♣)-sequence in the groundmodel but also creates such a sequence generically.

Lemma 2.6.12. Let S = (xγ)γ∈E be a ♣(E)-sequence for a stationary E ⊆ Lim(ω1). LetP = Π∗κFn(ω, 2) for arbitrary κ. Then we have ‖–P “ S is a ♣(E)-sequence ”.

Proof. Let p ∈ P and A be a P -name such that p ‖–P “ A ∈ [ω1]ℵ1 ”. We show that thereis q ≤ p and γ ∈ E such that q ‖–P “ xγ ⊆ A ”. Let f be a P -name such that p ‖–P “ f :ω1 → A and f is 1-1 ”. Let (pα)α<ω1 and (qα)α<ω1 be sequence of elements of P satisfying theconditions a) – c) in the proof of Lemma 2.6.4. Also, let uα, α < ω1 be as in the proof of Lemma2.6.4. As there, we can find an uncountable Y ⊆ ω1 and u∗ ∈ [κ]<ℵ0 such that uα = u∗ for allα ∈ Y . Since Πu∗Fn(ω, 2) is countable we may assume that qα | u∗ are all the same for α ∈ Y .Now for each α ∈ Y let βα be such that qα ‖–P “ f(α) = βα ” and let Z = βα : α ∈ Y . Sinceqα, α ∈ Y are pairwise compatible, βα, α ∈ Y are pairwise distinct and so Z is uncountable. Notethat Z is a ground model set. Hence there exists γ ∈ E such that xγ ⊆ Z. Let q =

⋃α∈Y ∩γ qα.

Then q ≤ p. Since supβα : α < γ ≥ γ and ‖–P “ βα : α < γ is an initial segment of Z ”,we have q ‖–P “ Z ∩ γ ⊆ A ”. Hence q ‖–P “ xγ ⊆ A ”.

Theorem 2.6.13. “¬CH+ MA(countable) + there exists a constructible ♣ -sequence” is consis-tent.

Proof. We can obtain a model of the statement by starting from a model of V = L and forcewith P = Π∗κFn(ω, 2) for a regular κ. By Corollary 2.6.6, every cardinal of V is preservedin V [G]. Since P adds κ many Cohen reals over V while | P | = κ and P has the ℵ2-cc, wehave V [G] |= “ 2ℵ0 = κ ”. By Lemma 2.6.7, V [G] |= “ MA(countable) ”. By Lemma 2.6.12, the♦-sequence in V remains a ♣ -sequence in V [G].

In fact, we do not need a ♣-sequence in the ground model to get (♣) in the generic extension byΠ∗κFn(ω, 2) :

Lemma 2.6.14. Let κ be uncountable and P = Π∗κFn(ω, 2). Then for any stationary E ⊆Lim(ω1) we have ‖–P “ ♣(E) holds ”.

Proof. For γ ∈ E letfγ : [γ, γ + ω)→ γ

be a bijection and let

Sγ = x ⊆ γ : x is a cofinal subset of γ, otp(x) = ω .

For each x ∈ Sγ let px ∈ P be defined by

px = (γ + n, (0, i) ) : n ∈ ω, i ∈ 2, i = 1 ⇔ fγ(γ + n) ∈ x .

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For distinct x, x′ ∈ Sγ , px and px′ are incompatible. Hence, for each γ ∈ E, we can find a P -namexγ such that

‖–P “ xγ is a cofinal subset of γ and otp(xγ) = ω ”and

px ‖–P “ xγ = x ” for each x ∈ Sγ .We show that ‖–P “ (xγ)γ∈E is a ♣(E)-sequence ”. For this, it is enough to show that, for anyp ∈ P and a P -name A, if p ‖–P “ A ∈ [ω1]ℵ1 ”, then there is q ≤ p and γ ∈ E such thatq ‖–P “ xγ ⊆ A ”. Let f be such that

p ‖–P “ f : ω1 → A and f is 1-1 ”.

Now let (pα)α<ω1 , (qα)α<ω1 , (uα)α∈ω1 , Y and u∗ be as in the proof of Lemma 2.6.4. For eachα ∈ Y let βα be such that qα ‖–P “ f(α) = βα ” and let Z = βα : α ∈ Y . Let

C = γ ∈ Lim(ω1) :⋃α∈Y ∩γ(supp(qα) ∩ ω1) ⊆ γ

and Z ∩ γ is unbounded in γ .Then C is closed unbounded in ω1 and hence there exists a γ∗ ∈ C ∩ E. Let q′ =

⋃α∈Y ∩γ∗ qα.

Then we have q′ ≤ q and q′ ‖–P “ Z ∩ γ∗ ⊆ A ”. Now let x ∈ Sγ∗ be such that x ⊆ Z ∩ γ∗. Finallylet q = q′ ∪ qx. Then we have q ≤ p and q ‖–P “ xα = x ⊆ Z ∩ γ∗ ⊆ A ”.

Note that E’s in Lemmas 2.6.12 and 2.6.14 are ground model sets. To force ♣(E) for everystationary E ⊆ Lim(ω1) which may be also added generically, we need a sort of iteration describedin [6].

2.6.3. On a theorem of Shapiro. Finally we mention one more application of our newtype of product.

Theorem* 2.6.15 (L.B. Shapiro). If MA(Cohen) holds then for any compact Hausdorff space Xof weight < 2ℵ0 and ℵ0 ≤ τ < 2ℵ0 the following assertions are equivalent:(i) There exists a continuous surjection from X onto τII;(ii) There exists a continuous injection from τ2 into X;(iii) There exists a closed subset Y ⊆ X such that χ(y, Y ) ≥ τ for every y ∈ Y .

In [5] we proved that MA(countable) is not enough to get the equivalence because we provedthe following statement:

Theorem 2.6.16. If principle ( |• ) holds, then there exists a Boolean algebra B of cardinality ω1,such that Frω1 is embeddable into B but there is no surjective Boolean mapping from B ontoFrω1.

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Publications of the author

[1] Mirna Dzamonja and Lajos Soukup. Some note on banach spaces, 2010. manuscript.[2] Márton Elekes, Tamás Mátrai, and Lajos Soukup. On splitting infinite-fold covers. Fund. Math., 2010. to

appear.[3] Sakaé Fuchino, Stefan Geschke, Saharon Shelah, and Lajos Soukup. On the weak Freese-Nation property of

complete Boolean algebras. Ann. Pure Appl. Logic, 110(1-3):89–105, 2001.[4] Sakaé Fuchino, Stefan Geschke, and Lajos Soukup. On the weak Freese-Nation property of P(ω). Arch. Math.

Logic, 40(6):425–435, 2001.[5] Sakaé Fuchino, Saharon Shelah, and Lajos Soukup. On a theorem of Shapiro. Math. Japon., 40(2):199–206,

1994.[6] Sakaé Fuchino, Saharon Shelah, and Lajos Soukup. Sticks and clubs. Ann. Pure Appl. Logic, 90(1-3):57–77,

1997.[7] Sakaé Fuchino and Lajos Soukup. More set-theory around the weak Freese-Nation property. Fund. Math.,

154(2):159–176, 1997. European Summer Meeting of the Association for Symbolic Logic (Haifa, 1995).[8] István Juhász, Piotr Koszmider, and Lajos Soukup. A first countable, initially ω1-compact but non-compact

space. Topology Appl., 156(10):1863–1879, 2009.[9] István Juhász, Saharon Shelah, Lajos Soukup, and Zoltán Szentmiklóssy. A tall space with a small bottom.

Proc. Amer. Math. Soc., 131(6):1907–1916 (electronic), 2003.[10] István Juhász, Saharon Shelah, Lajos Soukup, and Zoltán Szentmiklóssy. Cardinal sequences and Cohen real

extensions. Fund. Math., 181(1):75–88, 2004.[11] István Juhász and Lajos Soukup. How to force a countably tight, initially ω1-compact and noncompact space?

Topology Appl., 69(3):227–250, 1996.[12] István Juhász, Lajos Soukup, and Zoltán Szentmiklóssy. Combinatorial principles from adding Cohen reals.

In Logic Colloquium ’95 (Haifa), volume 11 of Lecture Notes Logic, pages 79–103. Springer, Berlin, 1998.[13] István Juhász, Lajos Soukup, and Zoltán Szentmiklóssy. What is left of CH after you add Cohen reals?

Topology Appl., 85(1-3):165–174, 1998. 8th Prague Topological Symposium on General Topology and ItsRelations to Modern Analysis and Algebra (1996).

[14] István Juhász, Lajos Soukup, and William Weiss. Cardinal sequences of length < ω2 under GCH. Fund.Math., 189(1):35–52, 2006.

[15] Juan Carlos Martinez and Lajos Soukup. Cardinal sequences of lcs spaces under gch. Ann. Pure Appl. Logic,161:1180–1193, 2010.

[16] Juan Carlos Martinez and Lajos Soukup. Universal locally compact scattered spaces. Topology Proc., 35:19–36,2010.

[17] Juan Carlos Martinez and Lajos Soukup. Superatomic boolean algebras constructed from strongly unboundedfunctions. Mathematical Logic Quarterly, 2011. to appear.

[18] Lajos Soukup. A lifting theorem on forcing LCS spaces. In More sets, graphs and numbers, volume 15 ofBolyai Soc. Math. Stud., pages 341–358. Springer, Berlin, 2006.

[19] Lajos Soukup. Wide scattered spaces and morasses. Topology Appl., 2011. submitted.

127

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Other publications

[20] Alexander Arhangelskii. On countably compact and initially ω1-compact topological spaces and groups. Math.Japon., 40(1):39–53, 1994.

[21] Joan Bagaria. Locally-generic Boolean algebras and cardinal sequences. Algebra Universalis, 47(3):283–302,2002.

[22] B. Balcar and F. Franěk. Independent families in complete Boolean algebras. Trans. Amer. Math. Soc.,274(2):607–618, 1982.

[23] Z. Balogh, A. Dow, D. H. Fremlin, and P. J. Nyikos. Countable tightness and proper forcing. Bull. Amer.Math. Soc. (N.S.), 19(1):295–298, 1988.

[24] Tomek Bartoszyński and Haim Judah. Set theory. A K Peters Ltd., Wellesley, MA, 1995. On the structure ofthe real line.

[25] James E. Baumgartner and Saharon Shelah. Remarks on superatomic Boolean algebras. Ann. Pure Appl.Logic, 33(2):109–129, 1987.

[26] M. Bekkali. Topics in set theory, volume 1476 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991.Lebesgue measurability, large cardinals, forcing axioms, rho-functions, Notes on lectures by Stevo Todorčević.

[27] Shai Ben-David and Menachem Magidor. The weak ∗ is really weaker than the full . J. Symbolic Logic,51(4):1029–1033, 1986.

[28] A. Blass. Combinatorial cardinal characteristics of the continuum.[29] Andreas Blass. Applications of superperfect forcing and its relatives. In Set theory and its applications

(Toronto, ON, 1987), volume 1401 of Lecture Notes in Math., pages 18–40. Springer, Berlin, 1989.[30] Jörg Brendle and Sakaé Fuchino. Coloring ordinals by reals. Fund. Math., 196(2):151–195, 2007.[31] S. Broverman, J. Ginsburg, K. Kunen, and F. D. Tall. Topologies determined by σ-ideals on ω1. Canad. J.

Math., 30(6):1306–1312, 1978.[32] G. Cantor. Über die ausdehnung eines stazes aus der theorie der trigonometrischen reihen. Math. Ann.,

5:123–132, 1872.[33] George W. Day. Superatomic Boolean algebras. Pacific J. Math., 23:479–489, 1967.[34] Alan Dow. On initially κ-compact spaces. In Rings of continuous functions (Cincinnati, Ohio, 1982), vol-

ume 95 of Lecture Notes in Pure and Appl. Math., pages 103–108. Dekker, New York, 1985.[35] Alan Dow. Compact spaces of countable tightness in the Cohen model. In Set theory and its applications

(Toronto, ON, 1987), volume 1401 of Lecture Notes in Math., pages 55–67. Springer, Berlin, 1989.[36] Alan Dow. Large compact separable spaces may all contain βN. Proc. Amer. Math. Soc., 109(1):275–279,

1990.[37] Alan Dow and Istvan Juhász. Are initially ω1-compact separable regular spaces compact? Fund. Math.,

154(2):123–132, 1997. European Summer Meeting of the Association for Symbolic Logic (Haifa, 1995).[38] Mirna Džamonja and Saharon Shelah. Similar but not the same: various versions of ♣ do not coincide. J.

Symbolic Logic, 64(1):180–198, 1999.[39] Katsuya Eda, Masaru Kada, and Yoshifumi Yuasa. The tightness about sequential fans and combinatorial

properties. J. Math. Soc. Japan, 49(1):181–187, 1997.[40] Ryszard Engelking. General topology, volume 6 of Sigma Series in Pure Mathematics. Heldermann Verlag,

Berlin, second edition, 1989. Translated from the Polish by the author.[41] K. Er-rhaimini and B. Veličkovic. Pcf structure of height less than ω3. The Journal of Symbolic Logic.[42] M. Foreman, M. Magidor, and S. Shelah. Correction to: “Martin’s maximum, saturated ideals, and nonregular

ultrafilters. I” [Ann. of Math. (2) 127 (1988), no. 1, 1–47; MR0924672 (89f:03043)]. Ann. of Math. (2),129(3):651, 1989.

[43] Matthew Foreman and Menachem Magidor. A very weak square principle. J. Symbolic Logic, 62(1):175–196,1997.

[44] Ralph Freese and J. B. Nation. Projective lattices. Pacific J. Math., 75(1):93–106, 1978.[45] Sakaé Fuchino, Sabine Koppelberg, and Saharon Shelah. A game on partial orderings. In Proceedings of the

International Conference on Set-theoretic Topology and its Applications (Matsuyama, 1994), volume 74,pages 141–148, 1996.

[46] Sakaé Fuchino, Sabine Koppelberg, and Saharon Shelah. Partial orderings with the weak Freese-Nation prop-erty. Ann. Pure Appl. Logic, 80(1):35–54, 1996.

[47] Stefan Geschke. On tightly κ-filtered Boolean algebras. Algebra Universalis, 47(1):69–93, 2002.[48] Martin Goldstern. Tools for your forcing construction. In Set theory of the reals (Ramat Gan, 1991), volume 6

of Israel Math. Conf. Proc., pages 305–360. Bar-Ilan Univ., Ramat Gan, 1993.

129

dc_69_10

130 OTHER PUBLICATIONS

[49] A. Hajnal and I. Juhász. Intersection properties of open sets. Topology Appl., 19(3):201–209, 1985.[50] A. Hajnal, I. Juhász, and S. Shelah. Splitting strongly almost disjoint families. Trans. Amer. Math. Soc.,

295(1):369–387, 1986.[51] Lutz Heindorf and Leonid B. Shapiro. Nearly projective Boolean algebras, volume 1596 of Lecture Notes in

Mathematics. Springer-Verlag, Berlin, 1994. With an appendix by Sakaé Fuchino.[52] Tetsuya Ishiu. α-properness and Axiom A. Fund. Math., 186(1):25–37, 2005.[53] R. Björn Jensen. The fine structure of the constructible hierarchy. Ann. Math. Logic, 4:229–308; erratum,

ibid. 4 (1972), 443, 1972. With a section by Jack Silver.[54] I. Juhász. Cardinal functions. In Recent progress in general topology (Prague, 1991), pages 417–441. North-

Holland, Amsterdam, 1992.[55] I. Juhász. On the minimum character of points in compact spaces. In Topology. Theory and applications,

II (Pécs, 1989), volume 55 of Colloq. Math. Soc. János Bolyai, pages 365–371. North-Holland, Amsterdam,1993.

[56] I. Juhász, Zs. Nagy, L. Soukup, and Z. Szentmiklóssy. Intersection properties of open sets. II. In Paperson general topology and applications (Amsterdam, 1994), volume 788 of Ann. New York Acad. Sci., pages147–159. New York Acad. Sci., New York, 1996.

[57] I. Juhász, L. Soukup, and Z. Szentmiklóssy. What makes a space have large weight? Topology Appl., 57(2-3):271–285, 1994.

[58] I. Juhász and Z. Szentmiklóssy. Convergent free sequences in compact spaces. Proc. Amer. Math. Soc.,116(4):1153–1160, 1992.

[59] I. Juhász and W. Weiss. On thin-tall scattered spaces. Colloq. Math., 40(1):63–68, 1978/79.[60] I. Juhász and W. Weiss. Omitting the cardinality of the continuum in scattered spaces. Top. Appl., 31:19–27,

1989.[61] I. Juhász and W. Weiss. Cardinal sequences. Ann. Pure Appl. Logic, 144(1-3):96–106, 2006.[62] István Juhász. Cardinal functions in topology. TLaen years later, volume 123 of Mathematical Centre Tracts.

Mathematisch Centrum, Amsterdam, second edition, 1980.[63] István Juhász and Kenneth Kunen. The power set of ω. Elementary submodels and weakenings of CH. Fund.

Math., 170(3):257–265, 2001.[64] Winfried Just. Two consistency results concerning thin-tall Boolean algebras. Algebra Universalis, 20(2):135–

142, 1985.[65] Masaru Kada and Yoshifumi Yuasa. Cardinal invariants about shrinkability of unbounded sets. In Proceed-

ings of the International Conference on Set-theoretic Topology and its Applications (Matsuyama, 1994),volume 74, pages 215–223, 1996.

[66] Peter Koepke and Juan Carlos Martínez. Superatomic Boolean algebras constructed from morasses. J. Sym-bolic Logic, 60(3):940–951, 1995.

[67] Péter Komjáth. Set systems with finite chromatic number. European J. Combin., 10(6):543–549, 1989.[68] S. Koppelberg. General theory of boolean algebras. In J.D. Monk and R. Bonnet, editors, Handbook of Boolean

algebras, VOL I. North-Holland, 1989.[69] Sabine Koppelberg. Applications of σ-filtered Boolean algebras. In Advances in algebra and model theory (Es-

sen, 1994; Dresden, 1995), volume 9 of Algebra Logic Appl., pages 199–213. Gordon and Breach, Amsterdam,1997.

[70] Sabine Koppelberg and Saharon Shelah. Subalgebras of Cohen algebras need not be Cohen. In Logic: fromfoundations to applications (Staffordshire, 1993), Oxford Sci. Publ., pages 261–275. Oxford Univ. Press, NewYork, 1996.

[71] P. Koszmider. Splitting ultrafilters of the thin-very tall algebra and initially ω1-compactness. preprint, 1995.[72] Piotr Koszmider. Fact about velemen’s simplifiead morasses. note.[73] Piotr Koszmider. Semimorasses and nonreflection at singular cardinals. Ann. Pure Appl. Logic, 72(1):1–23,

1995.[74] Piotr Koszmider. Forcing minimal extensions of Boolean algebras. Trans. Amer. Math. Soc., 351(8):3073–

3117, 1999.[75] Piotr Koszmider. Universal matrices and strongly unbounded functions. Math. Res. Lett., 9(4):549–566, 2002.[76] Kenneth Kunen. Inaccessibility Properties of Cardinals. PhD thesis, Stanford, 1968.[77] Kenneth Kunen. Set theory, volume 102 of Studies in Logic and the Foundations of Mathematics. North-

Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs, Reprint of the 1980original.

[78] Kenneth Kunen and Franklin D. Tall. Between Martin’s axiom and Souslin’s hypothesis. Fund. Math.,102(3):173–181, 1979.

[79] Robert LaGrange. Concerning the cardinal sequence of a Boolean algebra. Algebra Universalis, 7(3):307–312,1977.

[80] Jean-Pierre Levinski, Menachem Magidor, and Saharon Shelah. Chang’s conjecture for ℵω . Israel J. Math.,69(2):161–172, 1990.

[81] Juan Carlos Martínez. A consistency result on thin-tall superatomic Boolean algebras. Proc. Amer. Math.Soc., 115(2):473–477, 1992.

[82] Juan Carlos Martínez. On uncountable cardinal sequences for superatomic Boolean algebras. Arch. Math.Logic, 34(4):257–261, 1995.

dc_69_10

OTHER PUBLICATIONS 131

[83] Juan Carlos Martínez. On cardinal sequences of scattered spaces. Topology Appl., 90(1-3):187–196, 1998.[84] Juan Carlos Martínez. A forcing construction of thin-tall boolean algebras. Fundamenta Mathematicae,

159(2):99–113, 1999.[85] Juan Carlos Martínez. Some open questions for superatomic Boolean algebras. Notre Dame J. Formal Logic,

46(3):353–356 (electronic), 2005.[86] S. Mazurkiewicz and W. Sierpinski. Contributions a la topologie des ensembles denombrables. Fund. Math.,

1:17–27, 1920.[87] A. Mostowski and A. Tarski. Boolesche ringe mit geordneter basis. Fund. Math., 32:69–86, 1939.[88] S. Mrówka, M. Rajagopalan, and T. Soundararajan. A characterization of compact scattered spaces through

chain limits (chain compact spaces). In TOPO 72—general topology and its applications (Proc. Second Pitts-burgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), pages288–297. Lecture Notes in Math., Vol. 378. Springer, Berlin, 1974.

[89] A. J. Ostaszewski. On countably compact, perfectly normal spaces. J. London Math. Soc. (2), 14(3):505–516,1976.

[90] Mariusz Rabus. An ω2-minimal Boolean algebra. Trans. Amer. Math. Soc., 348(8):3235–3244, 1996.[91] M. Rajagopalan. A chain compact space which is not strongly scattered. Israel J. Math., 23(2):117–125, 1976.[92] J. Roitman. Superatomic boolean algebras. In J.D. Monk and R. Bonnet, editors, Handbook of Boolean

algebras, Vol 3. North-Holland, 1989.[93] Judy Roitman. Height and width of superatomic Boolean algebras. Proc. Amer. Math. Soc., 94(1):9–14, 1985.[94] Judy Roitman. A very thin thick superatomic Boolean algebra. Algebra Universalis, 21(2-3):137–142, 1985.[95] Judy Roitman. Superatomic Boolean algebras. In Handbook of Boolean algebras, Vol. 3, pages 719–740.

North-Holland, Amsterdam, 1989.[96] Saharon Shelah. Whitehead groups may not be free, even assuming CH. II. Israel J. Math., 35(4):257–285,

1980.[97] Saharon Shelah. Proper forcing, volume 940 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1982.[98] Saharon Shelah. Advances in cardinal arithmetic. In Finite and infinite combinatorics in sets and logic (Banff,

AB, 1991), volume 411 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 355–383. Kluwer Acad. Publ.,Dordrecht, 1993.

[99] Saharon Shelah. More on cardinal arithmetic. Arch. Math. Logic, 32(6):399–428, 1993.[100] Saharon Shelah. Further cardinal arithmetic. Israel J. Math., 95:61–114, 1996.[101] Saharon Shelah. On what i do not understand (and have something to say) part i. Fundamenta Mathematicae,

166(1–2):1–82, 2000.[102] Saharon Shelah. PCF and infinite free subsets in an algebra. Arch. Math. Logic, 41(4):321–359, 2002.[103] Stevo Todorcevic. Walks on ordinals and their characteristics, volume 263 of Progress in Mathematics.

Birkhäuser Verlag, Basel, 2007.[104] Dan Velleman. Simplified morasses. J. Symbolic Logic, 49(1):257–271, 1984.[105] Martin Weese. On cardinal sequences of Boolean algebras. Algebra Universalis, 23(1):85–97, 1986.[106] Yoshifumi Yuasa. Shrinkability of unbounded sets in the Cohen extension. Surikaisekikenkyusho Kokyuroku,

(930):55–58, 1995. Foundations of mathematics and its applications (Japanese) (Kyoto, 1995).

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