210 Section D: Fairly general properties

d-17 The Cech–Stone Compactification

1. Constructions

In [11] Tychonoff not only proved the Tychonoff ProductTheorem (for powers of the unit interval), he also showedthat every completely regular space of weight κ can be em-bedded into the space κ [0,1]. This inspired Cech, in [1], toconstruct for every completely regular space S a compactHausdorff space β(S) that contains S as a dense subspace –a compactification of S – and such that every bounded real-valued continuous function on S can be extended to a contin-uous function on β(S). The construction amounts to takingthe family C of all continuous functions from S to [0,1],the corresponding diagonal map e = �f∈Cf :S→ [0,1]Cand obtaining β(S) as the closure of e[S] in [0,1]C . Cechalso proved that for any other compactification B of S thereis a continuous map h :β(S)→ B with h(x) = x for x ∈ S

and h[β(S) \ S] = B \ S. From this he deduced that if B

is a compactification of S in which functionally separatedsubsets in S have disjoint closures in B then the map h is ahomeomorphism, whence B = β(S).

Somewhat earlier, in [10], M.H. Stone applied his the-ory of representations of Boolean algebras to various to-pological problems. One of the major applications is theconstruction, using the ring C∗(X) of bounded real-valuedcontinuous functions, of a compactification X of X withthe same extension property as the compactification thatCech would construct. The construction proceeds by takingthe Boolean algebra B generated by all cozero sets and allnowhere dense subsets of X. As a first step Stone took therepresenting space S(B) – the Stone space – of B. Next, toevery maximal ideal m of C∗(X) he associated the ideal Imof B consisting of those sets B in B for which there aref ∈ C∗(X) and a, b, c ∈ R such that A⊆ f←[(a, b)]; f ≡ c

(mod m) and c < a or b < c. Finally Fm is the closed sub-set of S(B) determined by the filter that is dual to Im. Thespace X is the quotient space of S(B) by the decompo-sition consisting of the sets Fm. One obtains an embed-ding of X into X by associating x with the maximal idealmx = {f : f (x)= 0}. Stone also proved that every continu-ous map on X with a compact co-domain can be extendedto X .

The compactification constructed by Cech and Stoneis nowadays called the Cech–Stone compactification;1

Cech’s β is still used, we write βX (without Cech’s paren-theses). The properties of βX that Cech and Stone es-tablished each characterize it among all compactificationsof X: (1) it is the maximal compactification; (2) function-ally separated subsets of X have disjoint closures in βX;

1In Europe; elsewhere one speaks of the Stone–Cech compactification.

and (3) every continuous map from X to a compact spaceextends to all of βX (the extension of f :X→K is denotedβf ). Of the many other constructions of βX that have beendevised we mention two. First, in [6], Gel’fand and Kol-mogoroff showed that the hull-kernel topology on the setof maximal ideals of C∗(X) immediately gives us βX andthat one may also use the ring C(X) of all real-valued con-tinuous functions on X. Second, in [7], Gillman and Jerisongave what for many is the definitive construction of βX: theWallman compactification of X with respect to the familyZ(X) of all zero sets of X. This means βX is the set of all ul-trafilters on the family Z(X) – the z-ultrafilters or zero-setultrafilters – with the family {�Z: Z ∈ Z(X)} as a base forthe closed sets, where �Z = {u ∈ βX: Z ∈ u}. One identifiesa point x of X with the z-ultrafilter ux = {Z: x ∈Z}.

Property 2 above is usually reformulated as: (2′) disjointzero sets of X have disjoint closures in βX. For normalspaces one can obtain βX as the Wallman compactificationof X, i.e., using the family of all closed sets; property (2′)then becomes (2′′) disjoint closed sets in X have disjoint clo-sures in βX. The equality βX=K should be taken to meanthat K is compact and there is an embedding f :X→K , forwhich f [X] is dense in K and for which the extension βf

is a homeomorphism – especially if X is dense in K . In thissense the notation βf is unambiguous: the graph of βf is theCech–Stone compactification of the graph of f .

The assignment X �→ βX is a covariant functor fromthe category of Tychonoff spaces to the category of com-pact Hausdorff spaces, both with continuous maps as mor-phisms. It is in fact the adjoint of the forgetful functor fromcompact Hausdorff spaces to Tychonoff spaces. This givesanother way of proving that “βX exists”: because the cate-gory of compact Hausdorff spaces is closed under productsand closed subsets. In fact Cech’s construction of βX maybe construed as a forerunner of the Adjoint Functor Theo-rem.

2. Properties

We say that a subspace A is C-embedded (C∗-embed-ded) in a space X if every (bounded) real-valued continuousfunction on A can be extended to a continuous real-valuedfunction on X. Thus a completely regular space X is C∗-embedded in its Cech–Stone compactification βX and anycompactification of X in which X is C∗-embedded mustbe βX. These remarks help us to recognize some Cech–Stone compactifications: if A⊆X then clβX A= βA iff A isC∗-embedded in X and βY = βX whenever X ⊆ Y ⊆ βX.If X is normal then clβX A= βA whenever A is closed in X

(by the Tietze–Urysohn theorem).

d-17 The Cech–Stone compactification 211

One can use C∗-embedding to calculate βX explicitly fora few X, by which we mean that βX is an already famil-iar space. The well-known fact that every continuous real-valued function on the ordinal space ω1 is constant on a tailimplies that βω1 = ω1+1 – the same holds for every ordinalof uncountable cofinality. Other examples are provided byΣ-products: if κ is uncountable and X = {x: {α: xα �= 0} iscountable} as a subspace of κ [0,1] (or κ2) then every con-tinuous real-valued function on X depends on countablymany coordinates and hence can be extended to the ambientproduct so that βX = κ [0,1] (or βX = κ2). Note that thesespaces, with an easily identifiable Cech–Stone compactifica-tion, are pseudocompact. In fact if X is not pseudocompactthen it contains a C-embedded copy of N, whence βX con-tains a copy of βN. The space βN is, in essence, hard todescribe: if u is a point in βN \ N and so a free ultrafilteron N then the set {∑n∈U 2−n: U ∈ u} is a non-Lebesguemeasurable set of reals. This illustrates that the constructionof βX requires a certain amount of Choice, indeed, the exis-tence of βX is equivalent to the Tychonoff Product Theoremfor compact Hausdorff spaces, which, in turn, is equivalentto the Boolean Prime Ideal Theorem.

The map f �→ βf from C∗(X) to C(βX) is an isomor-phism of rings (or lattices, or Banach spaces . . . ); this ex-plains why very often investigations into C∗(X) assume thatX is compact; this gives the advantage that ideals are fixed,i.e., if I is an ideal of C∗(X) then there is a point x withf (x)= 0 for all f ∈ I . Furthermore the maximal ideals areprecisely the ideals of the form {f : f (x)= 0} for some x .

A perfect map is one which is continuous, closed andwith compact fibers. A map f :X→ Y between completelyregular spaces is perfect iff its Cech-extension βf satisfiesβf [βX \ X] ⊆ βY \ Y or equivalently X = βf−1[Y ]. Onecan use this, for instance, to show that complete metrizabil-ity is preserved by perfect maps (also inversely if the domainis metrizable). A metrizable space is completely metrizableiff it is a Gδ-set in its Cech–Stone compactification; the lat-ter property is then easily seen to be preserved both waysby perfect maps. One calls a space a Cech-complete space(sometimes topologically complete) if it is a Gδ-set in itsCech–Stone compactification. This is an example of βX pro-viding a natural setting for defining or characterizing topo-logical properties of X – the best-known example being ofcourse local compactness: a space is locally compact iff itis open in its Cech–Stone compactification. Other propertiesthat can be characterized via βX are: the Lindelöf property(X is normally placed in βX, which means that for everyopen set U ⊇X there is an Fσ -set F with X ⊆ F ⊆U ) andparacompactness (X× βX is normal).

The product βN× βN is not β(N×N): the characteristicfunction of the diagonal ofN witnesses thatN×N is not C∗-embedded in βN× βN. The definitive answer to the ques-tion when β

∏=∏β was given in [8]: if both X and Y areinfinite then βX×βY = β(X×Y ) iff X×Y is pseudocom-pact and the same holds for arbitrary products, with a simi-lar proviso:

∏i βXi = β

∏i Xi iff

∏i Xi is pseudocompact,

provided∏

i �=i0 Xi is never finite. If the product can be fac-tored into two subproducts without isolated points then even

the homeomorphy of∏

i βXi and β∏

i Xi implies∏

i Xi ispseudocompact.

If X is not compact then no point of X∗ = βX\X is a Gδ-set, in fact a Gδ-set of βX that is a subset of X∗ contains acopy of N∗. This implies that nice properties like metrizabil-ity, and second- or first-countability do not carry over to βX.Of course separability carries over from X to βX, but notconversely: a Σ-product in c2 is not separable but c2 is.

Properties that are carried over both ways are usually ofa global nature. Examples are connectedness, extremal dis-connectedness, basic disconnectedness and the values of thelarge inductive dimension (for normal spaces) and coveringdimension. These properties have in common that they canbe formulated using the families of (co)zero sets and/or thering C∗(X), which makes it almost automatic for each thatX satisfies it iff βX does. Interestingly βX is locally con-nected iff X is locally connected and pseudocompact; so,e.g., βR is connected but not locally connected.

A particularly interesting class of spaces in this context isthat of the F -spaces; it can be defined topologically (cozerosets are C∗-embedded) or algebraically (every finitely gener-ated ideal in C∗(X) is principal). The algebraic formulationshows that X is an F -space iff βX is, because C(βX) andC∗(X) are isomorphic. Also, X∗ is an F -space wheneverX is locally compact and σ -compact. Neither property byitself guarantees that X∗ is F -space: Q∗ is not an F -space,nor is (ω1 × [0,1])∗ (which happens to be [0,1]).

This result shows that F -spaces are quite ubiquitous;for example, N∗, R∗, and (Rn)∗ are F -spaces, as well as(⊕

n Xn)∗ for any topological sum of countably many com-

pact spaces. An F -space imposes some rigidity on mapshaving it for its range: if f :Xκ → Z is a continuous mapfrom a power of the compact space X to an F -space Z thenXκ can be covered by finitely many clopen sets such thatf depends on one coordinate on each of them. This im-plies, e.g., that a continuous map from a power of [0,∞)∗to [0,∞)∗ itself depends on one coordinate only.

Some of the properties mentioned above have relation-ships beyond the implications between them. Every P -spaceis basically disconnected. But an extremally disconnectedP -space that is not of a measurable cardinal number is dis-crete (the converse is clearly also true). As noted above X isextremally (or basically) disconnected iff βX is but thereis more: if X is extremally disconnected or a P -space thenβX can be embedded into βD for a large discrete space D.Every compact subset of βD is an F -space but a charac-terization of the compact subspaces of βD is not known.In the special case of βN there is a characterization underthe assumption of the Continuum Hypothesis, but it is alsoconsistent that not all basically disconnected spaces embedinto βN and that not every F -space embeds into a basicallydisconnected space.

As seen above, pseudocompactness is a property thathelps give positive structural results about βX; this happensagain in the context of topological groups. If X is a topolo-gical group then the operations can be extended to βX, mak-ing βX into a topological group, if and only if X is pseudo-compact.

212 Section D: Fairly general properties

3. Special points

It is a general theorem that X∗ = βX \ X is not homoge-neous whenever X is not pseudocompact [5]. This in itselfvery satisfactory result prompted further investigation intothe structure of remainders and a search for more reasons forthis nonhomogeneity. Many special points were defined thatwould exhibit different topological behaviour in X∗ or βX.The best known are the remote points: a point p of X∗ isa remote point of X if p /∈ clβX N for all nowhere densesubsets N of X. If p is a remote point of X then βX is ex-tremally disconnected at p.

Many spaces have remote points, e.g., spaces of count-able π -weight (or even with a σ -locally finite π -base) andω × κ2. If X is nowhere locally compact, i.e., when X∗ isdense in βX, then X∗ is extremally disconnected at every re-mote point of X. This gives another reason for the nonhomo-geneity of, for example, Q∗, as this space is not extremallydisconnected.

Under CH all separable spaces have remote points but inthe side-by-side Sacks model there is a separable space with-out remote points. Many spaces with the countable chaincondition have remote points and it is unknown whetherthere is such a space without remote points. Proofs that cer-tain spaces have remote points have generated interestingcombinatorics; the proof for ω× κ2 contains a crucial ingre-dient for one proof of the consistency of the Normal MooreSpace Conjecture.

Further types of points are obtained by varying on thetheme of ‘not in the closure of a small set’. Thus one obtainsfar points: not in the closure of any closed discrete subsetof X; requiring this only for countable discrete sets definesω-far points – a near point is a point that is not far.

Of earlier vintage are P -points, points for which the fam-ily of neighbourhoods is closed under countable intersec-tions. They occured in the algebraic context: a point is aP -point iff every continuous function is constant on a neigh-bourhood of it. This means that for a P -point x the ideals{f : f (x) = 0} and {f : x ∈ intf←(f (x))} coincide. Un-der CH or even MA one can prove that many spaces of theform X∗ have P -points, thus obtaining witnesses to the non-homogeneity of X∗. Many of these results turned out to beindependent of ZFC. The search for a general theorem thatwould, once and for all, establish nonhomogeneity of X∗ bymeans of a ‘simple’ topological property of some-but-not-allpoints lead to weak P -points. A weak P -point is one that isnot an accumulation point of any countable set. Their advan-tage over P -points is that their existence is provable in ZFC,

first for N∗, later for more spaces. The final word has notbeen said, however. The weakest property that has not beenruled out by counterexamples is ‘not an accumulation pointof a countable discrete set’.

Further readingWalker’s book [12] gives a good survey of work on βX upto the mid 1970s. Van Douwen’s [4, 2, 3] and van Mill’s [9]laid the foundations for the work on βX in more recentyears.

References

[1] E. Cech, On bicompact spaces, Ann. of Math. 38(1937), 823–844.

[2] E.K. van Douwen, Remote points, Dissertationes Math.(Rozprawy Mat.) 188 (1981), 1–45.

[3] E.K. van Douwen, Prime mappings, number of fac-tors and binary operations, Dissertationes Math.(Rozprawy Mat.) 199 (1981), 1–35.

[4] E.K. van Douwen, Martin’s Axiom and pathologicalpoints in βX \X, Topology Appl. 34 (1990), 3–33.

[5] Z. Frolík, Non-homogeneity of βP − P , Comment.Math. Univ. Carolin. 8 (1967), 705–709.

[6] I.M. Gel’fand and A.N. Kolmogoroff, On rings of con-tinuous functions on topological spaces, C. R. (Dok-lady) Acad. Sci. URSS 22 (1939), 11–15.

[7] L. Gillman and M. Jerison, Rings of Continuous Func-tions, University Series in Higher Mathematics, VanNostrand, Princeton, NJ (1960), Newer edition: Grad-uate Texts in Math., Vol. 43, Springer, Berlin (1976).

[8] I. Glicksberg, The Stone–Cech compactification ofproducts, Trans. Amer. Math. Soc. 90 (1959), 369–382.

[9] J. van Mill, Weak P -points in Cech–Stone compactifi-cations, Trans. Amer. Math. Soc. 273 (1982), 657–678.

[10] M.H. Stone, Applications of the theory of Booleanrings to general topology, Trans. Amer. Math. Soc. 41(1937), 375–481.

[11] A. Tychonoff, Über die topologische Erweiterung vonRaümen, Math. Ann. 102 (1930), 544–561.

[12] R.C. Walker, The Stone–Cech Compactification, Sprin-ger, Berlin (1974).

Alan Dow and Klaas Pieter HartCharlotte, NC, USA and Delft, The Netherlands