arXiv:1610.09245v1 [math.GN] 28 Oct 2016 · arXiv:1610.09245v1 [math.GN] 28 Oct 2016 ON THE...

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arXiv:1610.09245v1 [math.GN] 28 Oct 2016 ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES N.A. CARLSON AND J.R. PORTER ABSTRACT. We introduce the cardinal invariant aL (X) and show that |X|≤ 2 aL (X)χ(X) for any Hausdorff space X (a corollary of Theorem 4.4). This invariant has the properties a) aL (X)= 0 if X is H-closed, and b) aL(X) aL (X) aLc(X). Theorem 4.4 then gives a new improvement of the well- known Hausdorff bound 2 L(X)χ(X) from which it follows that |X|≤ 2 ψc(X) if X is H-closed (Dow/Porter [5]). The invariant aL (X) is constructed using convergent open ultrafilters and an operator c : P(X) P(X) with the property clA c(A) cl θ (A) for all A X. As a comparison with this open ultrafilter approach, in §3 we additionally give a κ-filter variation of Hodel’s proof [10] of the Dow-Porter result. Finally, for an infinite cardinal κ, in §5 we introduce κwH-closed spaces, κH -closed spaces, and κH ′′ -closed spaces. The first two notions generalize the H-closed property. Key results in this connection are that a) if κ is an infinite cardinal and X a κwH-closed space with a dense set of isolated points such that χ(X) κ, then |X|≤ 2 κ , and b) if X is κH -closed or κH ′′ -closed then aL (X) κ. This latter result relates these notions to the invariant aL (X) and the operator c. 1. I NTRODUCTION. A space X is H-closed if every open cover V of X has a finite subfamily W such that X = W W clW . In 1982 Dow and Porter [5] used H-closed extensions of discrete spaces to demonstrate that |X|≤ 2 χ(X) (in fact, |X|≤ 2 ψc(X) ) for any H-closed space X. The technique was simplified in Porter [15], and in 2006 Hodel [10] gave a proof of the Dow-Porter result using κ-nets. A natural general question is the following: Question 1.1. Does there exists a strengthening of Arhangel ski˘ ı’s cardinal in- equality |X|≤ 2 L(X)χ(X) [2] for a general Hausdorff space X for which it follows as a corollary that |X|≤ 2 χ(X) if X is H-closed? This question was asked by Angelo Bella in personal communication with the second author. Another way to ask this is, does there exists a property P of a Hausdorff space X that a) generalizes both the H-closed property and the Lindel¨ of property simultaneously, and b) |X|≤ 2 χ(X) for spaces X with property P? As both H-closed spaces and Lindel¨ of spaces are almost-Lindel¨ of (that is, every open cover has a countable subfamily whose closures cover), the property “almost- Lindel¨ of” would seem to be a suitable candidate. However, in 1998 Bella and 2010 Mathematics Subject Classification. 54D20, 54A25, 54D10. Key words and phrases. cardinality bounds, cardinal invariants. 1

Transcript of arXiv:1610.09245v1 [math.GN] 28 Oct 2016 · arXiv:1610.09245v1 [math.GN] 28 Oct 2016 ON THE...

Page 1: arXiv:1610.09245v1 [math.GN] 28 Oct 2016 · arXiv:1610.09245v1 [math.GN] 28 Oct 2016 ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES N.A. CARLSON AND J.R. PORTER ABSTRACT.We

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016 ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED

SPACES

N.A. CARLSON AND J.R. PORTER

ABSTRACT. We introduce the cardinal invariantaL′(X) and show that|X| ≤

2aL′(X)χ(X) for any Hausdorff spaceX (a corollary of Theorem 4.4). This

invariant has the properties a)aL′(X) = ℵ0 if X is H-closed, and b)aL(X) ≤aL′(X) ≤ aLc(X). Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound2L(X)χ(X) from which it follows that|X| ≤ 2ψc(X)

if X is H-closed (Dow/Porter [5]). The invariantaL′(X) is constructed usingconvergent open ultrafilters and an operatorc : P(X) → P(X) with the propertyclA ⊆ c(A) ⊆ clθ(A) for all A ⊆ X. As a comparison with this open ultrafilterapproach, in§3 we additionally give aκ-filter variation of Hodel’s proof [10]of the Dow-Porter result. Finally, for an infinite cardinalκ, in §5 we introduceκwH-closed spaces,κH ′-closed spaces, andκH ′′-closed spaces. The first twonotions generalize the H-closed property. Key results in this connection are thata) if κ is an infinite cardinal andX a κwH-closed space with a dense set ofisolated points such thatχ(X) ≤ κ, then|X| ≤ 2κ, and b) ifX is κH ′-closedor κH ′′-closed thenaL′(X) ≤ κ. This latter result relates these notions to theinvariantaL′(X) and the operatorc.

1. INTRODUCTION.

A spaceX is H-closedif every open coverV of X has a finite subfamilyWsuch thatX =

⋃W∈W clW . In 1982 Dow and Porter [5] used H-closed extensions

of discrete spaces to demonstrate that|X| ≤ 2χ(X) (in fact, |X| ≤ 2ψc(X)) forany H-closed spaceX. The technique was simplified in Porter [15], and in 2006Hodel [10] gave a proof of the Dow-Porter result usingκ-nets.

A natural general question is the following:

Question 1.1. Does there exists a strengthening of Arhangel′skiı’s cardinal in-equality|X| ≤ 2L(X)χ(X) [2] for a general Hausdorff spaceX for which it followsas a corollary that|X| ≤ 2χ(X) if X is H-closed?

This question was asked by Angelo Bella in personal communication with thesecond author. Another way to ask this is, does there exists apropertyP of aHausdorff spaceX that a) generalizes both the H-closed property and the Lindelofproperty simultaneously, and b)|X| ≤ 2χ(X) for spacesX with propertyP?

As both H-closed spaces and Lindelof spaces arealmost-Lindelof (that is, everyopen cover has a countable subfamily whose closures cover),the property “almost-Lindelof” would seem to be a suitable candidate. However, in 1998 Bella and

2010Mathematics Subject Classification.54D20, 54A25, 54D10.Key words and phrases.cardinality bounds, cardinal invariants.

1

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2 N.A. CARLSON AND J.R. PORTER

Yaschenko [4] showed that ifκ is a non-measurable cardinal then there exists analmost-Lindelof, first-countable Hausdorff spaceX such that|X| > κ. Thus,|X| ≤ 2χ(X) does not hold for all almost-Lindelof Hausdorff spacesX. (It does,however, hold if the space is additionally Urysohn [3]). In 1988, Bella and Cam-maroto [3] gave the bound2aLc(X)t(X)ψc(X) for the cardinality of a HausdorffspaceX, whereaLc(X) is defined before Definition 2.6 below. AsaLc(X) ≤L(X), this suggests that “aLc(X) = ℵ0” might be the required propertyP. Yetthe Katetov H-closed extensionκω of the discrete spaceω is an example of anH-closed space for whichaLc(X) = c > ℵ0, demonstrating that the propertyaLc(X) = ℵ0 does not hold for all H-closed spaces.

In this study we construct a cardinal invariantaL′(X) such that a)|X| ≤

2aL′(X)χ(X) for a Hausdorff spaceX (Theorem 4.4 gives a slightly stronger ver-

sion of this statement), b)aL(X) ≤ aL′(X) ≤ aLc(X) (Proposition 2.10), and c)aL′(X) = ℵ0 if X is H-closed (follows from Corollary 3.5). Thus, the property“aL′(X) = ℵ0” is the required propertyP above. Theorem 4.4 then gives a newbound on the cardinality of a Hausdorff space that is strong enough to capture theH-closed bound2χ(X) given by Dow and Porter.

For an open setU in a spaceX, convergent open ultrafilters are used to definea setU (Definition 2.2) such thatU ⊆ U ⊆ clU . Using the setU , we thendefine an operatorc : P(X) → P(X) that satisfiesclA ⊆ c(A) ⊆ clθ(A) forall A ⊆ X. Both U and the functionc have relationships to the Iliadis absoluteEX that are outlined in§2. After this set-up, the invariantaL′(X) is defined as inDefinition 2.6.

In Theorem 3.4, we give the following characterization of H-closed spaces,which is of interest in its own right: a spaceX is H-closed if and only if forevery open coverV of X there existsW ∈ [V]<ω such thatX =

⋃W∈W W .

Given thatW ⊆ clW for every open setW , this characterization is then a logi-cally stronger property that the usual definition of the H-closed property. It followsnaturally to define a cardinal invariantL′(X) as the least infinite cardinalκ suchthat if V is a cover ofX then there existsW ∈ [V]≤κ such thatX =

⋃W∈W W

(Definition 3.1). Theorem 3.4 shows thatL′(X) = ℵ0 if X is H-closed. Wedemonstrate thatL′(X) is hereditary onc-closed subsets (Proposition 3.2), fromwhich it follows thataL′(X) ≤ L′(X) ≤ L(X). Thus, given our main cardinalitybound for general Hausdorff spaces (Theorem 4.4), we see that the following isa sufficient property of both Lindelof and H-closed spacesX from which it fol-lows that|X| ≤ 2χ(X): every open coverV has a countable subfamilyW suchthatX =

⋃W∈W W (that is,L′(X) = ℵ0). As aL′(X) ≤ L′(X), another such

property (albeit weaker) is “aL′(X) = ℵ0”, as mentioned above.In [10], Hodel gave a proof that|X| ≤ 2ψc(X) for H-closed spacesX using the

notion of aκ-net for a cardinalκ. This proof is different than previous proofs ofthis result given by Dow and Porter, and also different than the approach taken inTheorem 4.4 in this study. In§3 we use a filter characterization of H-closed spacesgiven in Theorem 2.21 and thec-adherence of a filter to give another proof that thecardinality of an H-closed spaceX is bounded by2ψc(X). This particular method

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 3

can be seen as a variation of the method used by Hodel [10] for nets. We presenttwo examples at the end of§3.

In §4 we give the proof of our main result, Theorem 4.4, after establishing pre-liminary results in§2 − §4. The proof is fundamentally a standard closing-offargument. We use a theorem of Hodel (re-stated in Theorem 4.3) that gives a set-theoretic generalization of many such arguments. Typically the closure operator isused in a closing-off argument, or occassionally theθ-closure operator. We use theoperatorc referred to above.

In §5 we introduce two notions that generalize the H-closed property and a re-lated third notion. The first is, for an infinite cardinalκ, the concept of aκwH-closed space (Definition 5.5). This notion grows naturally out of recent work ofOsipov in [14]. In Proposition 5.6 we give this characterization of H-closed:Xis H-closed if an only ifX is ℵ0wH-closed. A key result is Theorem 5.10, whichstates that ifκ is an infinite cardinal andX is aκwH-closed space with a denseset of isolated points such thatχ(X) ≤ κ, then |X| ≤ 2κ. The second notionintroduced in§5 is that of aκH ′-closed space (Definition 5.12). Proposition 5.13demonstrates thatX is H-closed if and only ifX is ℵ0H

′-closed. After definingz(X) = inf{κ ≥ ℵ0 : X is κH ′ − closed}, it is shown in Corollary 5.16(a)that aL′(X)+ ≤ z(X), thereby relating the notion ofκH ′-closed to conceptsdefined in previous sections. It follows immediately|X| ≤ 2z(X)χ(X) for anyHausdorff spaceX after applying Theorem 4.4. Finally, we introduce the prop-erty of κH ′′-closed in Definition 5.17 and, for a spaceX, we define the cardinalinvariantz′(X) = inf{κ ≥ ℵ0 : X is κH′′–closed}. While it can be shown thataL′(X) ≤ z′(X) and thus|X| ≤ 2z

′(X)χ(X) for any Hausdorff spaceX (Corol-lary 5.20), it is not guaranteed that aℵ0H

′′-closed space is H-closed. In fact, anycountable space isℵ0H

′′-closed.All spaces are assumed to be Hausdorff. For all undefined notions see Engelk-

ing [6], Juhasz [12], or Porter-Woods [17]. Hodel’s surveypaper [10] also containsthorough discussion of many cardinal invariants and cardinality bounds related tothose discussed in this study.

2. CONSTRUCTION OF THE CARDINAL FUNCTIONaL′(X).

Given a Hausdorff spaceX and an open setU of X, define

0U = {U : U is a convergent open ultrafilter containingU}.

We recall the construction of the Iliadis absoluteEX as the set of convergent openultrafilters onX with the topology generated by the basis{0U : U is open inX}.(See [17], Chapter 6, for example). Under this topologyEX is an extremallydisconnected, zero-dimensional, Tychonoff space. For each U ∈ EX, let k(U) bethe unique convergent point ofU. We have the following basic facts concerningEX and the mapk : EX → X (see [17] 6.6(e)(5), 6.8(d,f) and [16] 1.2(b)). Recalla subsetA of a spaceX is anH-set if for every coverV of A by sets open inXthere existsW ∈ [V]<ω such thatA ⊆

⋃W∈W clW and a spaceX is Katetov ifX

has a coarser H-closed topology.

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4 N.A. CARLSON AND J.R. PORTER

Proposition 2.1. For a open setsU, V ⊆ X of a spaceX, andU ∈ EX,

(a) The mapk : EX → X is aθ-continuous, perfect, irreducible, surjection,(b) U ∈ U iff int(cl(U)) ∈ U iff U ∈ 0U , and thus0U = 0(int(clU)),(c) k[0U ] = cl(U),(d) k←(k(U)) ⊆ 0U iff k(U) ∈ int(cl(U)), and(e) If B ⊆ EX is compact, thenk[B] is Katetov and an H-set.(f) 0(U ∩ V ) = 0U ∩ 0V and0(U ∪ V ) = 0U ∪ 0V .

Let b : X → EX be an injective function such thatk◦b = idX . That is, for allx ∈X, b(x) is an open ultrafilter converging tox. Denote the subspaceb[X] of EXby Xb. The spaceXb is asectionof EX [18] and is an extremally disconnected,Tychonoff space. We observe thatk |Xb : Xb → X is a bijection asX is Hausdorff.

Definition 2.2. For a spaceX, an open setU , and a sectionXb of EX, define

Ub = {x ∈ X : U ∈ b(x)}.

We give several properties ofUb in Proposition 2.3. As is indicated in Proposi-tion 2.3(a),Ub consists of a special set of closure pointsx of U having the strongerproperty thatU is a member of the open ultrafilterb(x). The setUb will play amajor role in the construction of the cardinal invariantaL′(X) and in the proof ofour main theorem, Theorem 4.4. In addition, for a space(X, τ), {Ub : U ∈ τ(X)}forms a basis for a topologyσb onX such that(X,σb) is homeomorphic to the sec-tionXb (Proposition 2.5). Furthermore, Proposition 3.4 gives a characterization ofH-closed spaces using sets of the formUb. It is this characterization that will givethe cardinality bound2ψc(X) for H-closed spaces as an immediate consequence ofthe general Hausdorff bound given in Theorem 4.4.

Proposition 2.3. Let (X, τ) be a space,U, V open sets, andXb be any section ofEX. Then,

(a) U ⊆ Ub ⊆ clU ,(b) Ub = k[0U ∩Xb],

(c)(U ∩ V

)b= Ub ∩ Vb and

(U ∪ V

)b= Ub ∪ Vb ,

(d) X\Ub = (X\clXU)b .

Proof. (a) If x ∈ U , thenU is a member of any open ultrafilter converging tox. Thus,U ∈ b(x) andx ∈ Ub. This showsU ⊆ Ub. Supposex ∈ Ub. ThenU ∈ b(x). If x ∈ X\clU , thenX\clU ∈ b(x) and thus∅ = U ∩(X\clU) ∈ b(x),a contradiction. Thusx ∈ clU andUb ⊆ clU .

(b) If x ∈ Ub, thenb(x) ∈ Xb, U ∈ b(x), andb(x) ∈ 0U . Sincek ◦ b = idX ,we see thatx = k(b(x)), thusx ∈ k[0U ∩Xb]. This showsUb ⊆ k[0U ∩Xb]. Thereverse containment is similar.

(c) follows from (b) above and Proposition 2.1(f).(d) If x ∈ X\Ub, thenU /∈ b(x) and thereforeX\clU ∈ b(x). Thusx ∈

(X\clXU)b andX\Ub ⊆ (X\clXU)b. The reverse containment is identical.�

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 5

Recall that the semiregularizationX(s) of a spaceX is the (Hausdorff) spacewith underlying setX and topology generated by the basis of regular-open sets inX.

Corollary 2.4. LetX be a space,U ∈ τ(X), andb : X → EX be a section. ThenXb = X(s)b.

Proof. Note thatb : X(s) → EX(s) is also a section andEX = E(X(s)).

By 2.3(b), Ub = k[0U ∩ Xb] = k[0(intXclXU ∩ Xb] = ( intXclXU)b. Thus,Xb = X(s)b. �

Proposition 2.5. Let (X, τ) be a space andXb a section ofEX. Then{Ub : U ∈τ(X)} is a clopen base for an extremally disconnected, Tychonoff topologyσb onX such that(X,σb) is homeomorphic toXb.

Proof. The proof follows from 2.3(b) and the fact thatk|Xb : Xb → X is abijection. �

For a spaceX and a sectionXb of EX, define an operatorcb : P(X) → P(X)by

cb(A) = {x ∈ X : Ub ∩A 6= ∅ for all U such thatx ∈ U ∈ τ(X)}.

We say thatA ⊆ X is cb-closedif cb(A) = A.Recall that forA ⊆ X, we defineaL(A,X) as the least infinite cardinalκ such

that if V is a cover ofA by sets open inX then there existsW ∈ [V]≤κ suchthatA ⊆

⋃W∈W clW . Thealmost Lindelof degree ofX, denoted byaL(X), is

aL(X,X). Thealmost Lindelof degree ofX with respect to closed setsis

aLc(X) = sup{aL(C,X) : C is closed}+ ℵ0.

It is straightforward to see thataL(X) ≤ aLc(X) ≤ L(X) and that all three areidentical ifX is regular.

Definition 2.6. For a sectionXb of EX, we define the cardinal invariantaLb(X),by aLb(X) = sup{aL(C,X) : C is cb-closed} + ℵ0. As aLb(X) depends on thechoice of sectionXb, we define the unique cardinal invariantaL′(X) by

aL′(X) = min{aLb(X) : Xb is a section ofEX}.

LetXb′ be any section witnessing thataL′(X) = aLb′

(X) and define the operatorc : P(X) → P(X) by c = cb′ . We then refer toaL′(X) as thealmost Lindelofdegree ofX with respect toc-closed sets.

Notation. For an open setU of X, we let U denoteUb′ . For A ⊆ X, we letA′ = b′[A] ⊆ EX. For a pointx in a spaceX, letUx represent the open ultrafilterb′(x). In general, throughout this study we will reserve the symbol “U” to representa convergent open ultrafilter.

The functionc defined above is the main operator in the closing-off argumentused to prove our main result, Theorem 4.4. We give properties of c below. For asubsetA ⊆ X, we also give a characterization ofc(A) usingEX in Theorem 2.15.

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6 N.A. CARLSON AND J.R. PORTER

Proposition 2.7. LetX be a space, andA,B ⊆ X.

(a) A ⊆ c(A).(b) if A ⊆ B thenc(A) ⊆ c(B).(c) clA ⊆ c(A) ⊆ clθ(A).(d) if U is open, thenclU = c(U) ⊆ c(U).(e) if X is regular thenclA = c(A) = clθ(A).(f) If A is c-closed thenA is closed.

Proof. (a) If x ∈ A andU is an open set containingx, thenx ∈ U ∩ A byProposition 2.3(a).

(b) If x ∈ c(A) then∅ 6= U ∩A ⊆ U ∩B for all open setsU containingx. Thisshowsx ∈ c(B).

(c) If x ∈ clA, then by Proposition 2.3(a),∅ 6= U ∩A ⊆ U ∩A for all open setsU containingx. Thusx ∈ c(A). And if x ∈ c(A), then∅ 6= U ∩ A ⊆ clU ∩ A,also by Proposition 2.3(a). Thus,x ∈ clθ(A).

(d) As clU = clθU for an open setU , the equality follows from (b). Thecontainment follows from (a) and Propositio 2.3(a).

(e) AsclA = clθA for regular spaces, the result follows from (b).(f) if A is c-closed andx ∈ X\A, then there exists an open setU containingx

such that∅ = U ∩A ⊇ U ∩A. ThusA is closed. �

The following example provides a spaceX and a subsetA such thatc(A) 6=clX(A).

Example 2.8. Consider the Katetov H-closed extensionκω of ω with the discretetopology (cf. Ch 7 in [17]). Recall thatκω(s) = βω; that is,βω is the under-lying set ofκω. Also note thatκω\ω is discrete and closed. By 9.11 in [7], theclosed setβω\ω contains a copy ofβω. That is, there a countable discrete sub-spaceA of βω\ω such thatβA = βω, in particular,clβωA = βω. ThenA is aclosed subset ofκω. Let k : βω → κω denote the identity function.βω is an ex-tremally disconnected, Tychonoff space and the bijectionk is perfect, irreducible,andθ−continuous. By 6.7(a) in [17],EX = βω is the absolute ofX = κω withk : EX → X the absolute map. There is only one injective functionb : X → EXsuch thatk ◦ b = idX . It follows that c(A) = clβω(A). By 9.3 in [7], c(A) hascardinality2c; thus,c(A) 6= clX(A) = A. This example also illustrates that therecan be a marked size difference betweenc(A) andcl(A). �

Observe that for the spaceX = κω in Example 2.8, we haveσb ( τ , whereτis the topology onX. However, by 5.1(d) in [18], for a regular spaceX, we havethatτ ⊆ σb. Thus there is no universal containment relationship betweenτ andσb.

Let X be a space,U ∈ τ(X), andb : X → EX be a section. By 2.3(d), itfollows thatUb is also closed inσb. The next example shows thatUb may not bec-closed.

Example 2.9.LetU andV be distinct free open ultrafilters onω, i.e., distinct pointsin βω\ω. Let αω denote the compactification ofω (discrete topology) whereUandV in βω are identified as the pointy. Let X = αω. ThenEX = βω and

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 7

k : EX → X is the identity function onEX\{U,V} andk(U) = k(V) = y.Consider the section defined by the function

b : X → EX : x 7→

{x x 6= yU x = y.

ForA ∈ [ω]ω, let

oαA = A ∪ {W ∈ βω\{U,V} : A ∈ W} ∪

{∅ A 6∈ U ∩ V

{y} A ∈ U ∩ V}.

Then{oαA : A ∈ [ω]ω} ∪ {{n} : n ∈ ω} is a base forτ(αω). In particular, aneighborhood base fory is {oαW :W ∈ U ∩ V}. Also, note that forA ∈ [ω]ω,

oαA = oαA ∪

{∅ A 6∈ U

{y} A ∈ U.

Let T ∈ V\U. Theny 6∈ oαT . ForW ∈ U ∩ V, y ∈ oαW and oαW ∩ oαT ⊇

W ∩ T 6= ∅. That is,c(oαT ) = oαT ∪ {y} 6= oαT . �

In view of Proposition 2.7(f) and the fact that any space isc-closed in itself, wehave the following:

Corollary 2.10. For any spaceX, aL(X) ≤ aL′(X) ≤ aLc(X) ≤ L(X).

For a spaceX, we define a cardinal invarianttc(X) related to the tightnesst(X). While tc(X) andt(X) appear to be incomparable, Proposition 2.12 showsthattc(X) is bounded above by the characterχ(X).

Definition 2.11. For a spaceX, thec-tightnessof X, tc(X), is defined as the leastcardinalκ such that ifx ∈ c(A) for somex ∈ X andA ⊆ X, then there existsB ∈ [A]≤κ such thatx ∈ c(B).

Note thatt(κω) = ℵ0 and tc(κω) = t(βω) = c. This shows thatt(κω) andtc(κω) are not equal.

Proposition 2.12. For any spaceX, tc(X) ≤ χ(X). If X is regular thentc(X) =t(X).

Proof. To showtc(X) ≤ χ(X), letκ = χ(X) and letx ∈ c(A). LetN be an openneighborhood base atx such that|N| = κ. For allN ∈ N there existsaN ∈ N∩A.Let D = {aN : N ∈ N} ∈ [A]≤κ. We showx ∈ c(D). Let U be an open setcontainingx. There existsN ∈ N such thatx ∈ N ⊆ U . As aN ∈ N ∩ A, wehaveN ∈ UaN . AsUaN is an open filter, we have thatU ∈ UaN andaN ∈ U ∩D.This showsx ∈ c(D) andtc(X) ≤ κ. If X is regular thentc(X) = t(X) followsfrom Proposition 2.7(e). �

In Theorem 2.15, we give a characterization ofc(A) for a subsetA ⊆ X interms of the absoluteEX. This is one of several results below that describe howc(A) relates to the broader framework ofEX.

Let K = {k←(p) : p ∈ X}, wherek : EX → X is as in 2.1(a). ForA ⊆ EX,defineclKA = A ∪

⋃{K ∈ K : if K ⊆ 0U for U ∈ τ(X), 0U ∩A 6= ∅}.

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8 N.A. CARLSON AND J.R. PORTER

Lemma 2.13. For A ⊆ EX, clXk[A] ⊆ k[clEXA] ⊆ k[clKA] ⊆ clθk[A].

Proof. As k is a closed function, we immediately have thatclXk[A] ⊆ k[clEXA].To showk[clEXA] ⊆ k[clKA], it suffices to show thatclEXA ⊆ clKA. Let U ∈clEXA andK = k←(k(U) ⊆ 0U for someU ∈ τ(X). ThenU ∈ 0U and0U ∩A 6= ∅ asU ∈ clEXA. Thus,U ∈ K ⊆ clKA.Now, letp ∈ k[clKA] andp ∈ U ∈ τ(X). Then, by Proposition 2.1(d),k←(p) ⊆0U . So,0U ∩ A 6= ∅ and∅ 6= k[0U ∩ A] ⊆ k[0U ] ∩ k[A] ⊆ clX(U) ∩ k[A].Therefore,x ∈ clθk[A]. �

Lemma 2.14. For A ⊆ EX,

clKA =⋃

{K ∈ K : K ∩ clθA 6= ∅} =⋃k←[k[clθA]] = k←[k[clEXA]].

Proof. SupposeK ∩ clθA = ∅ for K ∈ K. Then for eachU ∈ K, there isUU

such that0(UU) ∩ A = ∅ and{0(UU) : U ∈ K} is an open cover of the compactsetK. There existsU1, . . . ,Un ∈ K such that

K ⊆

n⋃

i=1

0(UUi) = 0(

n⋃

i=1

UUi),

by 2.1(f). LetU =⋃ni=1 UUi . As K ⊆ 0(U) and0(U) ∩ A = ∅, we see that

K ∩ clKA = ∅. Conversely, supposeU ∈ K ∩ clθA andK ⊆ 0(U). ThenU ∈ 0(U) and0(U) ∩A 6= ∅. ThusK ⊆ clKA. This shows the first equality.

To show⋃{K ∈ K : K ∩ clθA 6= ∅} =

⋃k←[k[clθA]], it suffices to note

that if U ∈ clθA, thenk←[k(U)] ∩ clθA 6= ∅ andk←[k(U)] ∈ K. The equality⋃k←[k[clθA]] = k←[k[clEXA]] follows asEX is Tychonoff. �

Theorem 2.15. Let X be a space andA ⊆ X. Thenc(A) = k[clKA′] =

k[clEXA′].

Proof. We first show the first equality. Clearly,A ⊆ c(A) ∩ k[clKA′]. Let p ∈

c(A)\A andk←(p) ⊆ 0U whereU ∈ τ(X). By 2.1(d),p ∈ int(cl(U)). Thereis a ∈ A such thatint(cl(U)) ∈ Ua. By 2.1(b),U ∈ Ua ⊆ k←(a). Thus,Ua ∈0(int(cl(U))) = 0(U). That is,Ua ∈ 0(U) ∩A′. This shows thatk←(p) ⊆ clKA

andp ∈ k[clKA′]. Conversely supposep ∈ k[clKA

′]\A. Let p ∈ U ∈ τ(X). Thenk←(p) ⊆ 0(U) and0(U) ∩ A′ 6= ∅. There isa ∈ A such thatUa ∈ 0(U). Thus,U ∈ Ua andp ∈ c(A). This shows thatc(A) = k[clKA

′].To show the second equality, note that by Lemma 2.14, we have

c(A) = k[clKA′] = k[k←[k[clEXA

′]]] = k[clEXA′].

As the mapk : EX → X is always a closed map, we have the followingcorollary to Theorem 2.15.

Corollary 2.16. For any spaceX and everyA ⊆ X, c(A) is closed subset ofX.

By Theorem 2.15 and Proposition 2.7(c), we also have the following corollary.We see that Corollary 2.17 is stronger than Proposition 2.7(c) and demonstrateshow c(A) sits betweenclXA andclθA in terms of the absoluteEX.

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 9

Corollary 2.17. Let X be a space andA ⊆ X. ThenclXA ⊆ k[clEXA′] =

c(A) ⊆ clθA.

Our next corollary to Theorem 2.15 demonstrates that thec−closure of a subsetof an H-closed space is both Katetov and an H-set. This result should be comparedwith Lemma 3.10 which gives different conditions under which a subset of an H-closed space is an H-set and the result from [11] that theθ−closure of a subset ofan H-closed space is an H-set.

Corollary 2.18. If X is H-closed andA ⊆ X, thenc(A) is Katetov and an H-set.

Proof. As c(A) = k[clEXA′] by Theorem 2.15 and since the absoluteEX is com-

pact whenX is H-closed ([17] 6.9(b)(1)), we have thatclEXA′ is compact andc(A) is Katetov and an H-set by 2.1(e). �

Another consequence of Theorem 2.15 are the following properties of the c-closure operator and a new characterization of H-closed spaces.

Proposition 2.19. LetX be a space andA,B subsets ofX.(a) If A ⊆ B, thenc(A) ⊆ c(B).(b) c(A ∩B) ⊆ c(A) ∩ c(B).(c) c(A ∪B) = c(A) ∪ c(B).

Proof. (a) is immediate from 2.15, and (b) follows from (a). For (c),note thatc(A ∪B) = k[clEX(A ∪B)′] = k[clEX(A

′ ∪B′)′ = k[clEX(A′) ∪ clEX(B

′)] =k[clEX(A

′)] ∪ k[clEX(B′)] = c(A) ∪ c(B). �

Definition 2.20. Let F be a filter base on a spaceX. We define the c-adherenceof F, denoted asac(F), as∩{c(F ) : F ∈ F}. By 2.7(c), it follows thata(F) ⊆ac(F) ⊆ aθ(F).

We will use the concept of c-adherence to obtain a new characterization of H-closed spaces in the next result.

Theorem 2.21. LetX be a space. ThenX is H-closed iff for every filter baseFonX, ac(F) 6= ∅.

Proof. Let F be a filter base onX. To showX is H-closed, it suffices to showthat aθ(F) 6= ∅. But c(F ) ⊆ clθ(F ) for eachF ∈ F andac(F) 6= ∅. Thus,aθ(F) 6= ∅. Conversely supposeX is H-closed. LetF be a filter base onX. Then{clEXF

′ : F ∈ F} is a filter base of compact subsets onEX. Thus, there isp ∈ ∩{clEXF

′ : F ∈ F}. It follows thatk(p) ∈ ac(F). �

In the next section we will develop further connections between the H-closedproperty, the operatorc, and the setU for an open setU ⊆ X.

3. H-CLOSED SPACES.

For a spaceX we defineL′(X), a cardinal invariant related toaL′(X), andshow in Proposition 3.2 that it is hereditary onc-closed subsets. The filter charac-terization of H-closed spaces used in Theorem 2.21 usingc-adherence of a filter in

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10 N.A. CARLSON AND J.R. PORTER

conjunction with a variation of a method used by Hodel [10] for nets provides adirect path for proving that the cardinality of an H-closed spaceX is bounded by2ψc(X).

Definition 3.1. For a subsetA ⊆ X, defineL′(A,X) as the least cardinalκ suchthat for every coverV of A by sets open inX there existsW ∈ [V ]≤κ such thatA ⊆

⋃W∈W W . SetL′(X) = L′(X,X).

Proposition 3.2. LetX be a space. IfA ⊆ X is c-closed, thenL′(A,X) ≤ L′(X).

Proof. Let κ = L′(X) and letV be a cover ofA by sets open inX. As A isc-closed, for allx ∈ X\A, there exists an open setWx containingx such thata /∈ Wx for all a ∈ A. LetW = {Wx : x ∈ X\A}. ThenW ∪ V is an open coverof X. AsL′(X) = κ, there existsW′ ∈ [W]≤κ andV′ ∈ [V]≤κ such that

X =⋃

W∈W′

W ∪⋃

V ∈V′

V .

Suppose there existsa ∈ A ∩⋃W∈W′ W . Then there existsW ∈ W′ such that

a ∈ W , a contradiction. ThusA∩⋃W∈W′ W = ∅ andA ⊆

⋃V ∈V′ V . This shows

L′(A,X) ≤ κ. �

Corollary 3.3. For any spaceX, aL′(X) ≤ L′(X) ≤ L(X).

Proof. To show the first inequality, letκ = L′(X), letA be ac-closed subset ofX,and letV be a cover ofA by sets open inX. AsL′(A,X) ≤ κ by Propostion 3.2,there existsW ∈ [V]≤κ such thatA ⊆

⋃W∈W W . By Proposition 2.3(a), we

see thatA ⊆⋃W∈W W ⊆

⋃W∈W clW . This showsaL′(X) ≤ κ. To see that

L′(X) ≤ L(X), just observe again by Proposition 2.3a thatV ⊆ V for everymemberV of an open cover ofX. �

In addition to Theorem 2.21, we obtain another new characterization of H-closedspaces.

Theorem 3.4. A spaceX is H-closed if and only if for every open coverV of Xthere exists a finite familyW ⊆ V such thatX =

⋃W∈W W .

Proof. Let V be an open cover ofX and suppose there exists a finite familyW ⊆

V such thatX =⋃W∈W W . Then, by Proposition 2.3(a),X =

⋃W∈W clW ,

showingX H-closed.Suppose now thatX is H-closed and letV be an open cover ofX. AsX is H-

closed, there is a finite familyW ∈ V such thatX =⋃W∈W clW . Suppose by way

of contradiction that there existsx ∈ X\(⋃W∈W W ). Then, by Proposition 2.3(d),

x ∈ X\(⋃

W∈W

W ) =⋂

W∈W

(X\W ) =⋂

W∈W

(X\clW ).

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 11

Then,X\clW is a member of the open ultrafilterUx for all W ∈ W. It follows bythe finite intersection property that

∅ =⋂

W∈W

(X\clW ) ∈ Ux.

As this is a contradiction, we seeX =⋃W∈W W . �

We have the following immediate corollary of Theorem 3.4.

Corollary 3.5. If X is H-closed thenL′(X) = ℵ0.

For the spaceX = κω in Example 2.8, we note by 3.5 thatL′(X) = ℵ0. YetL(X) = 2c. Furthermore, sinceXb = X(s)b = βω for the sectionXb in thatexample, it follows thatL(X(s)) = ℵ0.

We now present an example of an H-closed spaceX and a subsetA such thatclX(A) 6= c(A) 6= clθ(A) showing that 2.7(c) is the best general result.

Example 3.6.We use Urysohn’s spaceU defined in 1925 to show that the converseof Proposition 2.7(c) is not true in the setting of H-closed spaces. LetZ denote theset of all integers with the discrete topology andN denote the subspace of positiveintegers. For the setU = N × Z ∪ {±∞}, a subsetU ⊆ U is defined to be openif +∞ ∈ U (resp. −∞ ∈ U ) implies for somek ∈ N, {(n,m) : n ≥ k,m ∈N} ⊆ U (resp.{(n,−m) : n ≥ k,m ∈ N} ⊆ U) and if (n, 0) ∈ U implies forsomek ∈ N, {(n,±m) : m ≥ k} ⊆ U). The spaceU is first countable, minimalHausdorff (H-closed and semiregular) but is not compact asA = {(n, 0) : n ∈ N}is an infinite, closed discrete subset. Letk : EU → U be the absolute map fromthe absoluteEU to U. Let U ∈ k←(∞) such thatN × {2} ∈ U; thus,U → ∞.Let V ∈ k←(−∞) such thatN × {−2} ∈ V; thus,V → −∞. For n ∈ N, letUn ∈ k←((n, 0)) such that{n}×N ∈ Un; thus,Un → (n, 0). Defineb : U → EUby b(∞) = U, b(−∞) = V, b((n, 0)) = Un, and for(n,m) ∈ N × Z\N × {0},b(n,m) = {U ∈ τ(U) : (n,m) ∈ U}. It follows thatclU(A) = A andclθ(A) =A∪{±∞}. By 2.7(c), it followsA ⊆ c(A) ⊆ A∪{±∞}. To show that∞ ∈ c(A),for n ∈ N, let Tn = (N\{1, 2, · · · , n}) × N. A basic open set containing∞ isTn ∪ {∞}. As {n+ 1} ×N ∈ Un+1 = b(n+ 1, 0), Tn ∈ Un+1 andTn ∩A 6= ∅.A similar argument shows that−∞ 6∈ c(A). Thus,c(A) = A ∪ {∞} and thisshows thatclX(A) 6= c(A) 6= clθ(A). Also, note that bothc(A) andclθ(A) areH-sets. �

Definition 3.7. Let X be a space,κ an infinite cardinal, andA ⊆ X. A is κ-H-closedif for each open (inX) coverC of A such that|C| ≤ κ, there is a finitesubfamilyD ⊆ C such thatA ⊆

⋃D clX(U ∩A)).

We note that in particular, a Hausdorff spaceX is ω-H-closed iffX is feeblycompact.

We prove the following lemmas. The key lemma is Lemma 3.10, which is ofinterest on its own.

Lemma 3.8. LetX be a space,κ an infinite cardinal, andA ⊆ X. If for eachfilter baseF ∈ [[A]≤κ]≤κ, ac(F) ∩A 6= ∅, thenA is aκ-H-closed.

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12 N.A. CARLSON AND J.R. PORTER

Proof. Let C be an open cover ofA by sets open inX and suppose thatC is closedunder finite unions and suppose|C| ≤ κ. For eachV ∈ C, assume there ispV ∈A\cl(V ∩ A). LetBV = {pU : V ⊆ U ∈ C}. ForT, S ∈ C, BT ∩ BS ⊇ BT∪S .ThenF = {BV : V ∈ C} is a filter base onA. Thus, there is a pointp ∈ ac(F)∩A.There isT ∈ C such thatp ∈ T . Now,BT ⊆ A\cl(T ∩ A) ⊆ X\cl(T ∩ A). ByPropositions 2.7(d) and 2.19(a), using thatX\cl(T ∩ A) is open, we have thatp ∈ c(BT ) ⊆ c(X\cl(T )) = cl(X\cl(T )) ⊆ X\T , a contradiction asp ∈ T . �

The small filter base method presented in the above lemma stands in contrast tothe open ultrafilter techniques frequently used in H-closedsettings. An immediateconsequence is this corollary.

Corollary 3.9. LetX be a space andA ⊆ X. If for every filter baseF on A,ac(F) ∩A 6= ∅, then the subspaceA is H-closed.

Lemma 3.10. LetX be an H-closed space,κ an infinite cardinal,A ⊆ X, andψc(X) ≤ κ. If for every filter baseF on [[A≤κ]≤κ], ac(F) ∩A 6= ∅. ThenA is anH-set.

Proof. Let G be an open filter that meetsA. We can assume thatG is maximal withrespect to meetingA. AsX is H-closed, there isp ∈ a(G). The goal is to showthatp ∈ A. Assume thatp 6∈ A. There is a familyV = {Vα : α < κ} of openneighborhoods ofp such that∩κcl(Vα) = {p}. For eachVα, asp 6∈ cl(X\cl(Vα)),X\cl(Vα) 6∈ G. There is someGα ∈ G such that(X\cl(Vα))∩Gα∩A = ∅. Thus,Gα ∩ A ⊆ cl(Vα) andcl(Gα ∩ A) ⊆ cl(Vα). Assume, by way of contradiction,thatA ∩

⋂cl(Gα ∩ A) = ∅. Thus,{X\cl(Gα ∩ A) : α < κ} is open cover of

A. As A is κ-H-closed by Lemma 3.8, there is a finite setF ∈ [κ]<ω such thatA ⊆

⋃F cl((X\cl(Gα ∩ A)) ∩ A). There is aG ∈ G such thatG ⊆

⋂F Gα. For

α ∈ F , G ∩ A ⊆ Gα ∩ A implying thatX\cl(Gα ∩ A) ⊆ X\cl(G ∩ A) and(X\cl(Gα∩A))∩A ⊆ (X\cl(G∩A))∩A. ThusA ⊆ cl((X\cl(G∩A))∩A) =cl(A\clA(G∩A)) implying thatA ⊆ clA(A\clA(G∩A)) = A\intAclA(G∩A),a contradiction asintAclA(G ∩A) ∩A 6= ∅. �

Theorem 3.11. LetX be H-closed,κ an infinite cardinal, andψc(X) ≤ κ. Then|X| ≤ 2κ.

Proof. For eachx ∈ X, let {V (α, x) : α ∈ κ} be a family of open sets containingx such that

⋂κ cl(V (α, x)) = {x}. Let L : P(X) → X be a choice function.

Using transfinite induction, we will construct a sequence{Hα : 0 ≤ α < κ+} ofsubsets ofX such that for0 ≤ α < κ+:

(a)H0 = {L(∅)};(b) if Hβ is defined forβ < α, defineHα as follows:

f((⋃

β<α

Hβ) ∪ {L(X\⋃

x∈A

cl(V (x, g(x))) : A ∈ [⋃

β<α

Hβ]<ω, g : A→ κ}).

Note that|Hα| ≤ 2κ for 0 ≤ α < κ+. LetH =⋃{Hα : α < κ+}. It follows that

|H| ≤ 2κ andf(H) ⊆ H. Thus,H = f(H) and ifF ∈ [[H]κ]κ, aHF(F) 6= ∅. By

Lemmas 3.8 and 3.10,H is an H-set.

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 13

To show thatH = X, assume thatq /∈ H. Sinceψc(X) ≤ κ, for eachx ∈ H,there isαx < κ such thatq 6∈ clV (αx, x). Using thatH is H-set, there a finitesubsetA ∈ [H]<ω such thatH ⊆

⋃A cl(V (αx, x)) ⊆ X\{q}. Now choose

α < κ+ such thatA ∈ [⋃β<αHβ]

<ω. By (b), L(X\⋃x∈A cl(V (αx, x)) ∈ Hα

and it follows thatHα\⋃x∈A cl(V (αx, x)) 6= ∅. This is a contradiction asHα ⊆

H ⊆⋃x∈A cl(V (αx, x)). �

It follows from Theorem 3.11 that the cardinality of an H-closed space is at most2χ(X). The Dow-Porter result given in Corollary 4.6 now follows, using a proofsimilar to the proof of that corollary. We see then two very different proofs of thisresult, one using open ultrafilters (which generalizes to a result for all Hausdorffspaces, Theorem 4.4) and the other usingκ-nets [10] which can be reframed interms ofκ-filters as in Theorem 3.11. We note that in [15], Porter used adifferenttype of open ultrafilter approach.

We present several examples.

Example 3.12.This example demonstrates that the converse of Lemma 3.8 is false,i.e., anω-H-closed spaceX with a filter baseF ∈ [[X]≤ω ]≤ω suchac(F) = ∅.The spaceX is Tychonoff; so,ac(F) = aθ(F) = a(F).

Consider the partition{An : n ∈ ω} of ω where eachAn is infinite. Pickone point, sayan ∈ clβωAn\An. Let B = clβω{an : n ∈ ω}\{an : n ∈ ω}.We will show that the subspaceX = βω\B is ω-H-closed but has a filter baseF ∈ [[X]≤ω ]≤ω suchac(F) = ∅. For n ∈ ω, let Fn = {am : m ≥ n}; thus,F = {Fn : n ∈ ω} ∈ [[X]≤ω ]≤ω.

Let x ∈ X. Then, inβω, B ∪ {an : n ∈ ω}\{x} is compact and there aredisjoint open setsU, V in βω such thatB ∪ {an : n ∈ ω}\{x} ⊆ U andx ∈ V .U\B andV \B are disjoint open sets inX such that{an : n ∈ ω}\{x} ⊆ U\Bandx ∈ V \B. If x = am for somem ∈ ω, thenFm+1 ∩ cl(X(V \B) = ∅ andx 6∈ ac(F). If x 6∈ {an : n ∈ ω}, thenF0 ∩ clX(V \B) = ∅ andx 6∈ ac(F). So inboth cases,x 6∈ ac(F) andac(F) = ∅.

To show thatX is ω-H-closed (= feebly compact), it suffices to show that theTychonoff spaceX is pseudocompact by 1.10(d)(2) in [17]. It suffices, by 1Q(6)in [17] to show that every infinite subset ofω is not closed inX. LetC = {bn :n ∈ ω} be infinite subset ofω. As clβωAn∩B = ∅ for eachn ∈ ω andclβω(An∩C)\ω 6= ∅ wheneverAn ∩ C is infinite, it follows thatAn ∩ C is finite for eachn ∈ ω. Thus, by 4B(6) in [7],{an : n ∈ ω} andC are contained in disjointcozero-sets (in an extremely disconnected space) and henceB ∩ clβωC = ∅. Itfollows thatclβωC\C ⊆ X. �

Example 3.13.Theψ spaceX is an example of a first countable, Tychonoff, pseu-docompact space with a filter baseF ∈ [[X]≤c]≤c such thatac(F) = ∅. LetX = ω ∪ M whereM is a maximal family of almost disjoint infinite subsets ofω andU ⊆ X is open ifA ∈ M ∩ U implies there is aF ∈ ω<ω such thatA\F ⊆ U . It is well-known thatX is first countable, locally compact, Tychonoff,pseudocompact space that is not countably compact and|M| = c.

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14 N.A. CARLSON AND J.R. PORTER

ForB ∈ [M]<ω, let FB = M\B andF = {FB : B ∈ [M]<ω}. We will showthat suchac(F) = ∅. Let x ∈ X. If x ∈ ω, then{x} is a clopen set disjoint fromM ∈ F. If x = A ∈ M, then ifM\{A} ∈ F, then asclX({A} ∪ A) = {A} ∪ A,clX({A} ∪A) ∩ (M\{A}) = ∅. Thus,ac(F) = ∅. �

For the spaceU constructed in Example 3.6, the subspace{(n, 0) : n ∈ N} ∪{∞} is anω-H-set but notω-H-closed.

4. A NEW CARDINALITY BOUND FOR HAUSDORFF SPACES.

Proposition 4.1. If X is Hausdorff andψc(X) ≤ κ, then for allx ∈ X there existsa familyV of open sets such that|V| ≤ κ and

{x} =⋂

V =⋂

V ∈V

clV =⋂

V ∈V

c(V ).

Proof. Fix x ∈ X. As ψc(X) ≤ κ, there exists a familyV of open sets such that{x} =

⋂V =

⋂V ∈V clV and|V| ≤ κ. Supposey 6= x. There existsV ∈ V such

thaty ∈ X\clV . LetW = X\clV and supposey ∈ c(V ). Then,

∅ 6= W ∩ V = W ∩ V = ∅ = ∅,

a contradiction. Thusy /∈ c(V ) and {x} =⋂V ∈V c(V ). As

⋂V ∈V clV ⊆⋂

V ∈V c(V ) by Proposition 2.7(c), it follows that

{x} =⋂

V =⋂

V ∈V

clV =⋂

V ∈V

c(V ).

Proposition 4.2. If X is Hausdorff andA ⊆ X, then|c(A)| ≤ |A|tc(X)ψc(X).

Proof. Let κ = tc(X)ψc(X). For eachx ∈ c(A), by Proposition 4.1 there exists afamily Vx of open sets such that|Vx| ≤ κ and

{x} =⋂

Vx =⋂

V ∈Vx

clV =⋂

V ∈Vx

c(V ).

As tc(X) ≤ κ, for all x ∈ c(A) there existsA(x) ∈ [A]≤κ such thatx ∈ c(A(x)).Defineφ : c(A) →

[[A]≤κ

]≤κby

φ(x) = {V ∩A(x) : V ∈ Vx}.

Observe thatφ(x) ∈[[A]≤κ

]≤κ. Fixx ∈ c(A). We will show thatx ∈ c(V ∩A(x))

for all V ∈ Vx. Let V ∈ Vx and letU be any open set containingx. As x ∈

c(A(x)), there existsa ∈ A(x) such thatU ∩V ∈ Ua. Thus,a ∈ U ∩ V = U ∩ V

and it follows thatU ∩ V ∩ a(X) 6= ∅. This showsx ∈ c(V ∩A(x)). Thus,

{x} ⊆⋂

V ∈Vx

c(V ∩A(x)) ⊆⋂

V ∈Vx

c(V ) = {x},

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 15

where the second containment above follows from Proposition 2.7(a). Then{x} =⋂V ∈Vx

c(V ∩ A(x)). Thus if x 6= y thenφ(x) 6= φ(y), andφ is one-to-one.Therefore,|c(A)| ≤ |A|κ. �

We turn now to our main result, a new bound for the cardinalityof a HausdorffspaceX. To establish this bound, we use the set-theoretic Theorem 3.1 from [10].This theorem generalizes many closing-off arguments needed to prove cardinalitybounds on topological spaces. For reference, we re-state the particular case of thistheorem that is used here.

Theorem 4.3(Hodel). LetX be a set,κ be an infinite cardinal,d : P(X) → P(X)an operator onX, and for eachx ∈ X let {V (α, x) : α < κ} be a collection ofsubsets ofX. Assume the following:

(T) (tightness condition) ifx ∈ d(H) then there existsA ⊆ H with |A| ≤ κsuch thatx ∈ d(A);

(C) (cardinality condition) ifA ⊆ X with |A| ≤ κ, then|d(A)| ≤ 2κ;(C-S) (cover-separation condition) ifH 6= ∅, d(H) ⊆ H, and q /∈ H, then

there existsA ⊆ H with |A| ≤ κ and a functionf : A → κ such thatH ⊆

⋃x∈A V (f(x), x) andq /∈

⋃x∈A V (f(x), x).

Then|X| ≤ 2κ.

Typically, the operatord used in Theorem 4.3 is either the standard closure op-eratorcl, or in some instances theθ-closureclθ. We use the operatorc.

Theorem 4.4. If X is Hausdorff then|X| ≤ 2aL′(X)tc(X)ψc(X).

Proof. Let κ = aL′(X)tc(X)ψc(X). As ψc(X) ≤ κ, for all x ∈ X there existsa family Wx = {W (α, x) : α < κ} of open sets such that{x} =

⋂Wx =⋂

W∈WxclW . For allx ∈ X andα < κ, setV (α, x) = cl(W (α, x)). We verify

the three conditions in Theorem 4.3, where the operatord is c. The (T) conditionfollows immediately astc(X) ≤ κ, and (C) follows from Proposition 4.2. To verify(C-S), supposeH 6= ∅ satisfiesc(H) ⊆ H. Then,c(H) = H, asH ⊆ c(H)by Proposition 2.7(a), andH is c-closed. Letq /∈ H. For all a ∈ H, there existαa < κ such thatq /∈ cl(W (αa, a)) = V (αa, a). Definef : A→ κ by f(a) = αa.Then{W (f(a), a) : a ∈ H} is a cover ofH by sets open inX. As aL′(X) ≤ κandH is c-closed, there existsA ∈ [H]≤κ such thatH ⊆

⋃a∈A V (f(a), a). Since

q /∈⋃a∈A V (f(a), a), we see that (C-S) is satisfied. By Theorem 4.3,|X| ≤

2κ. �

AsaL′(X) ≤ aLc(X) by Proposition 2.10 andtc(X)ψc(X) ≤ χ(X) by Propo-sition 2.12, we obtain the following Corollary 4.5. This is aslight weakening of theBella-Cammaroto bound2aLc(X)t(X)ψc(X) for Hausdorff spaces. WhileaL′(X) ≤aLc(X), it is unclear whethert(X) andtc(X) are comparable for a non-regularspaceX, making it unclear whether2aL

′(X)tc(X)ψc(X) and2aLc(X)t(X)ψc(X) arecomparable.

Corollary 4.5. [Bella/Cammaroto] IfX is Hausdorff then|X| ≤ 2aLc(X)χ(X).

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16 N.A. CARLSON AND J.R. PORTER

Corollary 4.6. [Dow/Porter] If X is H-closed then|X| ≤ 2ψc(X).

Proof. The semiregularizationX(s) ofX is also H-closed, and so by Corollary 3.3and Corollary 3.5 it follows thataL′(Xs) = ℵ0. Thus, by Theorem 4.4, we havethat

|X| = |Xs| ≤ 2tc(X(s))ψc(X(s)) ≤ 2χ(X(s)) = 2ψc(X(s)) ≤ 2ψc(X),

where the second equality follows asX(s) is minimal Hausdorff. �

We see then that Theorem 4.4 leads to a common proof of the cardinality bound2χ(X) for both H-closed spaces and Lindelof spaces simultaneously. We can isolatethe precise propertyP that both H-closed spaces and Lindelof spacesX sharefrom which it follows from Theorem 4.4 that|X| ≤ 2χ(X). PropertyP is theproperty that every open coverV of X has a countable subfamilyW such thatX =

⋃W∈W W . That is,L′(X) = ℵ0. In fact, the weaker propertyaL′(X) = ℵ0

also suffices.

5. GENERALIZED H-CLOSED SPACES

The standard method of generalizing the concept of H-closedis to use the well-known cardinality invariant of almost Lindelof – whenaL(X) ≤ ℵ0, the spaceXis a generalized H-closed space. One of the main goals in thispaper is seek gen-eralized H-closed spaces for which it is possible to obtain acardinality bound ofX. A spaceX satisfyinga′L(X) ≤ ℵ0 is another generalized H-closed space forwhich it is possible to obtain a cardinality bound ofX (see Theorem 4.4). We usedthe concept ofκ-H-closed, another generalized H-closed space, to obtain acardi-nality bound of H-closed spaces (see Theorem 3.11). In this section, we examinethree new generalized H-closed concepts with the common goal of obtaining a car-dinality bound of a space.

Approach I.

The roots of our first generalized H-closed space can be traced back to the fa-mous 1929 memoir (the Russian version of [1]) and uses a recent characterizationof H-closed spaces by Osipov [14]. Alexandroff and Urysohn proved this propertyof H-closed spaces: IfX is an H-closed space andA ⊆ X is an infinite subset,there is a pointp ∈ X such that|A| = |A ∩ cl(U)| wheneverp ∈ U ∈ τ(X).

Definition 5.1. Let X be a space andA ⊆ X. A point p ∈ X is aΘ-completeaccumulation point ofA (we writep ∈ ΘCAP(A)) if wheneverp ∈ U ∈ τ(X),|cl(U) ∩A| = |A|. In particular,clθA = (ΘCAP(A))∪(clθA\ΘCAP(A)).

An exciting new characterization of H-closed spaces using the concept ofΘ-complete accumulation points was established in 2013 by Osipov [14].

Theorem 5.2. [Osipov]LetX be a Hausdorff space andA ⊆ X an infinite subset,

(a) X is H-closed iff for each open coverC of ΘCAP(A), there is a finitesubfamilyF ⊆ C such that|A\int(cl(∪F))| < |A|.

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 17

(b) If X is H-closed space , thenΘCAP(A) is an H-set.

We start the process of generalizing H-closed spaces by expanding the notationΘCAP.

Definition 5.3. LetX be an H-closed space,κ an infinite cardinal, andA an infinitesubset. LetclκθA denote{p ∈ X : |cl(U) ∩A| ≥ κ for p ∈ U ∈ τ(X)}. Note that

cl|A|θ A = ΘCAP(A). If σ is an infinite cardinal andκ ≥ σ, thenclκθ (A) ⊆ clσθ (A)

andclωθ (A) is the set of accumulation points ofA.

Using a techniques similar to the proof of Theorem 5.2, it is possible to obtainthis result at theκ level.:

Proposition 5.4. LetX be an H-closed space,κ an infinite cardinal,A an infinitesubset, andC an open cover ofclκθA. There is a finite subfamilyD ⊆ C such that|A\int(cl(∪D))| < κ.

Proof. Let C be an open cover ofclκθA. Forp 6∈ clκθA, there is an open setUp suchthatp ∈ Up and|cl(Up) ∩ A| < κ. LetO = {Up : p 6∈ clκθA}. AsX is H-closed,there are finite subfamiliesD ⊆ C andU ⊆ O such thatX = cl(∪D) ∪ cl(∪U).So, cl(X\cl(∪D)) ⊆ cl(∪U) implying thatX\int(cl(∪D)) ⊆ cl(∪U). Since|(cl(∪U)) ∩A| < κ, it follows that|A\int(cl(∪D))| < κ. �

Definition 5.5. LetX be a space andκ be infinite cardinal. A filter baseF onXis said to beκ−wide if |clθ(A)| ≥ κ for eachA ∈ F. A spaceX is κwH-closediffor eachκ−wide filter baseF onX, aθF 6= ∅.

Proposition 5.6. LetX be a space andκ be an infinite cardinal.

(a) The spaceX is H-closed iffX is ℵ0wH-closed.(b) LetX be a space andκ be infinite cardinal.X is κwH-closed iff for each

κ−wide open filter baseF, aF 6= ∅.

Proof. The proof of (a) is immediate. The proof of (b) follows the known result thatif F is a filter base on a spaceX, then the open filter baseG = {U ∈ τ(X) : F ⊆ Ufor someF ∈ F} has the propertyaG = aθF. �

Theorem 5.7. LetX be space andκ an infinite cardinal. The spaceX is κwH-closed iff for every subsetA ⊆ X whereκ ≤ |A| andC is an open cover ofclκθA,there is a finite subfamilyB ⊆ C such that|A\int(cl(∪B))| < κ.

Proof. SupposeX is κwH-closed andA ⊆ X whereκ ≤ |A| andC is an opencover ofclκθA. For eachp 6∈ clκθA, there isp ∈ Up ∈ τ(X) such that|cl(Up)∩A| <κ. LetE = {Up : p 6∈ clκθA}. Assume, by way of contradiction, that for each finitesubfamilyB of C, |A\int(cl(∪B))| ≥ κ.

Claim: F = {X\cl(∪A) : A ∈ [E ∪ C]<ω} is anκ-wide filter base such that|cl(V )| ≥ κ for eachV ∈ F.

Proof of Claim. Let A ∈ [C ∪ E]<ω. Then there are finite subfamiliesB ⊆ C andD ⊆ E such thatA = (∪B) ∪ (∪D). It suffices to show that|cl(X\cl(∪A))| =

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18 N.A. CARLSON AND J.R. PORTER

|cl(X\cl((∪B)∪(∪D)))| ≥ κ. Note that|cl(∪D)∩A| < κ implies|(X\cl(∪D))∩A| = |A\cl(∪D)| = |A| ≥ κ. Also, note that|cl(X\cl(∪B))| ≥ κ by theassumption. We have:

cl(X\cl((∪B ∪ (∪D))) = cl((X\cl(∪B)) ∩ (X\cl(∪D)))

= cl(cl(X\cl(∪B)) ∩ (X\cl(∪D)))

= cl((X\int(cl(∪B))) ∩ (X\cl(∪D)))

⊇ cl((X\int(cl(∪B))) ∩ (A\cl(∪D)))

= cl((A\int(cl(∪B)))\cl(∪D))).

This shows that|cl(X\cl(∪A))| ≥ |cl((A\int(cl(∪B)))\cl(∪D)))| ≥ κ andF isaκ−wide filter base. �

AsX is κwH-closed, there is somep ∈ aF. If p ∈ clκθA, there isU ∈ C suchthat p ∈ U . Thus,X\cl(U) ∈ F andp 6∈ cl(X\cl(U)); so, p 6∈ a(F). On theother hand, ifp 6∈ aF, there isU ∈ E such thatp ∈ U . Again,X\cl(U) ∈ F, p 6∈cl(X\cl(U)), andp 6∈ a(F). Hence,a(F) = ∅. This contradicts the hypothesis.

To show the converse, letF be a freeκ−wide open filter base onX. LetU ∈ F

such |cl(U)| is minimum. We will apply the condition in the statement of thetheorem to the setcl(U). In particular,|cl(U)| ≥ κ. If p ∈ clκθ (cl(U)), there isVp ∈ F such thatp 6∈ cl(Vp) andVp ⊆ U . Thenp ∈ X\cl(Vp) and|cl(X\cl(Vp))∩cl(U)| ≥ κ. Let C = {X\cl(Vp) : p ∈ clκθ (cl(U))}. By the hypothesis of theconverse, there is a finite subfamilyB ⊆ C such that|cl(U)\int(cl(∪B))| < κ.For V = ∩{Vp : X\cl(Vp) ∈ B} ∩ U , note thatV ∈ F, cl(V ) ⊆ cl(U) andV ∩ (∪B) = ∅. It follows thatclV ∩ int(cl(∪B)) = ∅. This shows thatclV ⊆clU\int(cl(∪B)). That is,V ∈ F and|clV | < κ, a contradiction. �

As corollaries of Theorems 5.2 and 5.7, we have the followingresults.

Corollary 5.8. LetX be a space andκ be an infinite cardinal.

(a) The spaceX is H-closed iffX is κwH-closed for all infiniteκ ≤ |X|.(b) If X is aκwH-closed space,A ⊆ X such that|A| ≥ κ|, andF is aκ−wide

open filter base onX that meetsclκθA, thena(F) ∩ clκθA 6= ∅.(c) If X is κwH-closed andA ⊆ X such that|A| ≥ κ, thenclκθA 6= ∅.

Proof. The proof of (a) is straightforward. To prove (b), letF be aκ−wide openfilter base onX that meetsclκθA. Assume thata(F) ∩ clκθA = ∅. Let U ∈F such |cl(U) ∩ A| is minimum. Note that|cl(U) ∩ A| ≥ κ. If p ∈ clκθ (A),there isVp ∈ F such thatp 6∈ cl(Vp) andVp ⊆ U . Thenp ∈ X\cl(Vp) and|cl(X\cl(Vp)) ∩ A| ≥ κ. LetC = {X\cl(Vp) : p ∈ clκθA}. AsX is κwH-closed,there is a finite subfamilyB ⊆ C such that|A\int(cl(∪B))| < κ. Now V =∩{Vp : X\cl(Vp) ∈ B}∩U ∈ F. Also, clV ⊆ clU andV ∩ (∪B) = ∅. It followsthat clV ∩ int(cl(∪B)) = ∅. This shows thatclV ⊆ clU\int(cl(∪B)). Thus,clV ∩A ⊆ (clU\int(cl(∪B)))∩A = (clU ∩A)\int(cl(∪B)) ⊆ A\int(cl(∪B)).This implies there is aV ∈ F such that|clV ∩A| < κ, a contradiction.

To show (c), assume thatclκθA = ∅. For eachp ∈ X, there is an open setp ∈ Up ∈ τ(X) such that|clUp ∩ A| < κ. ThenU = {Up : p ∈ X} is an open

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 19

cover ofX (andclκθA). There is a finiteV ⊆ U such that|A\int(cl(∪V))| < κ.LetB = A\int(cl(∪V)). ThenA ⊆ B ∪ int(cl(∪V)) ⊆ B ∪ cl(∪V). It followsthatA ⊆ B ∪ (∪V ∈Vcl(V )∩A). Thus,|A| ≤ |B|+ΣV ∈V|cl(V )∩A)| < κ. Thisis a contradiction. �

The study ofκwH-closed spaces is a new approach to understanding the theoryof H-closed spaces by using the width of a filter base. The width is a measure ofthe size of the closure of the elements of a filter base.

There is still the question of obtaining a cardinality boundof κwH-closed spaces.We are able to obtain such a result only forκwH-closed spaces with a dense subsetof isolated points. We start by statinga well-known result that is similar to 4.2.

Lemma 5.9. Letκ be an infinite cardinal andχ(X) ≤ κ. For A ⊆ X, |cl(A)| ≤|A|κ.

Theorem 5.10. Let κ be an infinite cardinal andX a κwH-closed space with adense set of isolated points andχ(X) ≤ κ. Then|X| ≤ 2κ.

Proof. For eachp ∈ X, let V (p) = {V (α, p) : α < κ} be an open neighborhoodbase atp, and forB ⊆ X andf : B → κ, letV (f,B) =

⋃p∈B V (f(p), p). LetD

be the set of isolated points ofX. If p ∈ D, we letV (α, p) = {p} for all α ∈ κ.LetH : P(D) → D be a choice function andA0 = H(∅). We will inductively

defineAα for α < κ+. Forα < κ+, supposeAβ is defined forβ < α. Let

Aβ =⋃

α<β

Aα ∪⋃

{H(D\V (g,B)) : B ∈ [cl(⋃

α<β

Aα)]≤κ, g : B → κ}.

By induction, |Aα| ≤ 2κ; let C =⋃α<κ+ Aα. It follows that |C| ≤ 2κ. By

Lemma 5.9, forA = cl(C), |A| ≤ 2κ. AsC ⊆ D and is open, we also have thatA = clθ(C). Also, note that asC is an increasing chain overκ+ andχ(X) ≤ κ,A = cl(C) =

⋃α<κ+ cl(Aα).

To apply Theorem 5.7, we need to show that|C| ≥ κ. Suppose that|C| ≤ κ. AsC =

⋃α<κ+ Aα, there isα < κ such thatC ⊆ Aα. Let g : C → κ : p 7→ 0. Then

C = V (g,C) and it follows thatD\V (C, g) = ∅. Thus,D ⊆ C, |D| ≤ κ, and itfollows that|X| ≤ 2κ and we are done. We are reduced the case when|C| ≥ κ.

To finish the proof of the theorem, we will prove thatX = cl(C) = A byshowing thatD ⊆ C. Let d ∈ D\C = D\A. For eachp ∈ A, there is someUp = V (αp, p) ∈ V (p) such thatd 6∈ cl(Up). ThenC = {Up : p ∈ A} is anopen cover ofA = clθC ⊇ clκθC; note thatclκθC ⊆ X\D. By Theorem 5.7,there is a finite subfamilyB of C such that|C\int(cl(∪B))| < κ. There is afinite subsetF ⊆ A such thatB = {Up : p ∈ F}. For U = ∪p∈FUp ∩ D,C\int(cl(∪B)) = C\U . Defineh : C\U ∪ F → κ : p 7→ αp; it follows thatC ⊆ V (h,C\U ∪ F ).

Now, as|C\U ∪ F | < κ andcl(C) =⋃α<κ+ cl(Aα), there is someβ < κ+,

such thatC\U ∪ F ⊆ cl(Aβ). Asd 6∈ V (h,C\U ∪ F ),D\V (h,C\U ∪ F ) 6= ∅.Thus,H(D\V (C\U ∪ F, h)) ∈ Aβ+1 ⊆ C, a contradiction. This completes theproof thatD ⊆ C and finishes the proof. �

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20 N.A. CARLSON AND J.R. PORTER

We ask whether the above Theorem 5.10 is true without the hypothesis thatXhas a dense set of isolated points.

Question 5.11. If κ is an infinite cardinal andX a κwH-closed space such thatχ(X) ≤ κ, is |X| ≤ 2κ?

Approach II.

The application ofκ-H-closed spaces in Theorem 3.11 to obtain a cardinalitybound of H-closed spaces provides another approach to studying H-closed spacesby using “thin” filter bases where a filter base is a member of[[X]≤κ]≤κ. Thistechnique is another way of measuring the width of a filter base and provides oursecond path in defining generalized H-closed.

Definition 5.12. Let X be a space,κ an infinite cardinal, andA ⊆ X such that|A| ≥ κ. Definecκ(A) = {x ∈ X : if x ∈ U ∈ τ(X), then|U ∩ A| ≥ κ}. AspaceX is κH ′-closedif A ⊆ X, |A| ≥ κ, andU is an open cover ofcκ(A), thereis a finite subfamilyV ⊆ U such that|A\∪V ∈V V | < κ. If |X| < κ, it follows thatX is κH′-closed.

Proposition 5.13. A spaceX is ℵ0H′-closed iffX is H-closed.

Proof. SupposeX is ℵ0H′-closed. LetU be open cover ofX. We can assumethat |X| ≥ ℵ0. ThenU coverscℵ0(X). Then there is a finiteV ⊆ U such that|X\ ∪V V | < ℵ0. By Theorem 3.4,X is H-closed. Conversely, supposeX isH-closed. LetA ⊆ X such that|A| ≥ ℵ0. Let U be open cover ofcℵ0(A). Foreachp 6∈ cℵ0(A), there isp ∈ Up ∈ τ(X) such that|Up ∩ A| < ℵ0. Now,{Up : p 6∈ cℵ0(A)} ∪ U is open cover ofX. There is a finiteB ⊆ X\cℵ0(A) anda finiteV ⊆ U such thatX = ∪p∈BUp ∪ ∪V ∈VV . Thus,X\ ∪V ∈V V ⊆ ∪p∈BUpand|A\ ∪V ∈V V | ≤ | ∪p∈B (Up ∩ A)| ≤

∑p∈B |Up ∩ A| < ℵ0. This shows that

X is ℵ0H′-closed. �

Proposition 5.14.Letκ be infinite cardinal andX beκH′-closed. ThenaL′(X) <κ.

Proof. Let A be c-closed. If |A| < κ, thenaL′(A,X) < κ. So, suppose that|A| ≥ κ. Let U be open cover ofA. For eachp 6∈ A, there isp ∈ Up ∈ τ(X)

such thatUp ∩ A = ∅. Now, {Up : p 6∈ A} ∪ U is open cover ofX. There is afiniteB ⊆ X\A and a finiteV ⊆ U such that|X\(∪p∈BUp∪∪V∈VV )| < κ. Now,|A\(∪p∈BUp ∪ ∪V ∈VV )| = |A\(∪V ∈VV )| < κ. Thus, there isW ⊆ U such that|W| < κ andA\(∪V ∈VV ) ⊆ ∪WW ) ⊆ ∪WW ). Therefore,A ⊆ ∪VV ∪ ∪WWand|V ∪W| < κ. So,aL′(A,X) < κ. �

Definition 5.15. For a spaceX, definez(X) = inf{κ ≥ ℵ0 : X is κH′–closed}.

By Proposition 5.14 and Theorem 4.4, we have the following two results.

Corollary 5.16. For a spaceX,

(a) aL′(X)+ ≤ z(X) and

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ON THE CARDINALITY OF HAUSDORFF SPACES AND H-CLOSED SPACES 21

(b) |X| ≤ 2z(X)tc(X)ψc(X).

Remark.This last corollary is an indication that the concept ofκH′-closed is sub-summed by the theory usingc-closure andaL′.

Approach III.

In this third approach to generalized H-closed spaces, we modify the conceptdefined in Definition 5.12 in Path II.

Definition 5.17. Let X be a space,κ an infinite cardinal, andA ⊆ X such that|A| ≥ κ. Definecκ(A) = {x ∈ X : if x ∈ U ∈ τ(X), then|U ∩ A| ≥ κ}. AspaceX isκH ′′-closedif A ⊆ X, |A| ≥ κ, andU is an open cover ofcκ(A), thereis a subfamilyV ⊆ U such that|V| ≤ κ and|A\ ∪V ∈V V | < κ. In particular, if|X| < κ, thenX is κH′′-closed.

Using essentially the same proof as the proof of Proposition5.14, we obtain thefollowing result.

Proposition 5.18.Letκ be infinite cardinal andX beκH′′-closed. ThenaL′(X) ≤κ.

Definition 5.19. For a spaceX, definez′′(X) = inf{κ ≥ ℵ0 : X is κH′′–closed}.

By Proposition 5.18 and Theorem 4.4, we have the following two results.

Corollary 5.20. For a spaceX,

(a) aL′(X) ≤ z′′(X) and(b) |X| ≤ 2z

′′(X)tc(X)ψc(X).

Remark.Using Approach III, 5.20(a) is sharper than 5.16(a). However, the price isthat the counterpart to Proposition 5.13 is not true; that is, in the case whenκ = ℵ0,we do not necessarily get H-closed. In fact, any countable space isℵ0H′′-closed.

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