D-17 Theˇcech–Stone Compactification

210 Section D: Fairly general properties d-17 The Cech–Stoneˇ Compactiﬁcation

1. Constructions all of β X (the extension of f : X → K is denoted β f ). Of the many other constructions of β X that have been devised In [11] Tychonoff not only proved the Tychonoff Product we mention two. First, in [6], Gel’fand and Kolmogoroff Theorem (for powers of the unit interval), he also showed showed that the hull-kernel topology on the set of maximal that every completely regular space of weight κ can be em- ideals of C∗(X) immediately gives us β X and that one may bedded into the space κ [0, 1]. This inspired Cech,ˇ in [1], to also use the ring C(X) of all real-valued continuous func- construct for every completely regular space S a compact tions on X. Second, in [7], Gillman and Jerison gave what Hausdorff space β(S) that contains S as a dense subspace – for many is the definitive construction of β X: the Wallman a compactification of S – and such that every bounded real- compactification of X with respect to the family Z(X) of all valued continuous function on S can be extended to a contin- zero sets of X. This means β X is the set of all ultrafilters on uous function on β(S). The construction amounts to taking the family Z(X) – the z-ultrafilters or zero-set ultrafilters the family C of all continuous functions from S to [0, 1], – with the family {SZ: Z ∈ Z(X)} as a base for the closed C the corresponding diagonal map e = 4 f ∈C f : S → [0, 1] sets, where SZ = {u ∈ β X: Z ∈ u}. One identifies a point x C and obtaining β(S) as the closure of e[S] in [0, 1] . Cechˇ of X with the z-ultrafilter ux = {Z: x ∈ Z}. also proved that for any other compactification B of S there Property 2 above is usually reformulated as: (20) disjoint is a continuous map h : β(S) → B with h(x) = x for x ∈ S zero sets of X have disjoint closures in β X. For normal and h[β(S) \ S] = B \ S. From this he deduced that if B spaces one can obtain β X as the Wallman compactification is a compactification of S in which functionally separated of X, i.e., using the family of all closed sets; property (20) subsets in S have disjoint closures in B then the map h is a then becomes (200) disjoint closed sets in X have disjoint clo- homeomorphism, whence B = β(S). sures in β X. The equality β X = K should be taken to mean Somewhat earlier, in [10], M.H. Stone applied his the- that K is compact and there is an embedding f : X → K , for ory of representations of Boolean algebras to various to- which f [X] is dense in K and for which the extension β f pological problems. One of the major applications is the is a homeomorphism – especially if X is dense in K . In this ∗ construction, using the ring C (X) of bounded real-valued sense the notation β f is unambiguous: the graph of β f is the continuous functions, of a compactification X of X with Cech–Stoneˇ compactification of the graph of f . the same extension property as the compactification that The assignment X 7→ β X is a covariant functor from Cechˇ would construct. The construction proceeds by taking the category of Tychonoff spaces to the category of com- the Boolean algebra B generated by all cozero sets and all pact Hausdorff spaces, both with continuous maps as mor- nowhere dense subsets of X. As a first step Stone took the phisms. It is in fact the adjoint of the forgetful functor from representing space S(B) – the Stone space – of B. Next, to compact Hausdorff spaces to Tychonoff spaces. This gives ∗ every maximal ideal m of C (X) he associated the ideal Im another way of proving that “β X exists”: because the cate- of B consisting of those sets B in B for which there are gory of compact Hausdorff spaces is closed under products ∗ ← f ∈ C (X) and a, b, c ∈ R such that A ⊆ f [(a, b)]; f ≡ c and closed subsets. In fact Cech’sˇ construction of β X may (mod m) and c < a or b < c. Finally Fm is the closed sub- be construed as a forerunner of the Adjoint Functor Theo- set of S(B) determined by the filter that is dual to Im. The rem. space X is the quotient space of S(B) by the decompo- sition consisting of the sets Fm. One obtains an embedding of X into X by associating x with the maximal ideal 2. Properties mx = { f : f (x) = 0}. Stone also proved that every continuous map on X with a compact co-domain can be extended We say that a subspace A is C-embedded (C∗-embed- X to . ded) in a space X if every (bounded) real-valued continuous The compactification constructed by Cechˇ and Stone function on A can be extended to a continuous real-valued is nowadays called the Cech–Stoneˇ compactification;1 function on X. Thus a completely regular space X is C∗- Cech’sˇ β is still used, we write β X (without Cech’sˇ paren- embedded in its Cech–Stoneˇ compactification β X and any theses). The properties of β X that Cechˇ and Stone estab- compactification of X in which X is C∗-embedded must lished each characterize it among all compactifications of X: be β X. These remarks help us to recognize some Cech–ˇ (1) it is the maximal compactification; (2) functionally sepa- Stone compactifications: if A ⊆ X then cl A = β A iff A is rated subsets of X have disjoint closures in β X; and (3) ev- β X C∗-embedded in X and βY = β X whenever X ⊆ Y ⊆ β X. ery continuous map from X to a compact space extends to If X is normal then clβ X A = β A whenever A is closed in X 1In Europe; elsewhere one speaks of the Stone–Cechˇ compactification. (by the Tietze–Urysohn theorem).

One can use C∗-embedding to calculate β X explicitly for can be factored into two subproducts without isolated points Q Q a few X, by which we mean that β X is an already famil- then even the homeomorphy of β Xi and β Xi implies Q i i iar space. The well-known fact that every continuous real- i Xi is pseudocompact. ∗ valued function on the ordinal space ω1 is constant on a tail If X is not compact then no point of X = β X \ X is a Gδ- ∗ implies that βω1 = ω1 +1 – the same holds for every ordinal set, in fact a Gδ-set of β X that is a subset of X contains a ∗ of uncountable cofinality. Other examples are provided by copy of N . This implies that nice properties like metrizabil- Σ-products: if κ is uncountable and X = {x:{α: xα 6= 0} is ity, and second- or first-countability do not carry over to β X. countable} as a subspace of κ [0, 1] (or κ 2) then every con- Of course separability carries over from X to β X, but not tinuous real-valued function on X depends on countably conversely: a Σ-product in c2 is not separable but c2 is. many coordinates and hence can be extended to the ambient Properties that are carried over both ways are usually of product so that β X = κ [0, 1] (or β X = κ 2). Note that these a global nature. Examples are connectedness, extremal dis- spaces, with an easily identifiable Cech–Stoneˇ compactifica- connectedness, basic disconnectedness and the values of the tion, are pseudocompact. In fact if X is not pseudocompact large inductive dimension (for normal spaces) and covering then it contains a C-embedded copy of N, whence β X con- dimension. These properties have in common that they can tains a copy of βN. The space βN is, in essence, hard to be formulated using the families of (co)zero sets and/or the ∗ describe: if u is a point in βN \ N and so a free ultrafilter ring C (X), which makes it almost automatic for each that P −n on N then the set { n∈U 2 : U ∈ u} is a non-Lebesgue X satisfies it iff β X does. Interestingly β X is locally con- measurable set of reals. This illustrates that the construction nected iff X is locally connected and pseudocompact; so, of β X requires a certain amount of Choice, indeed, the exis- e.g., βR is connected but not locally connected. tence of β X is equivalent to the Tychonoff Product Theorem A particularly interesting class of spaces in this context is for compact Hausdorff spaces, which, in turn, is equivalent that of the F-spaces; it can be defined topologically (cozero to the Boolean Prime Ideal Theorem. sets are C∗-embedded) or algebraically (every finitely gener- The map f 7→ β f from C∗(X) to C(β X) is an isomor- ated ideal in C∗(X) is principal). The algebraic formulation phism of rings (or lattices, or Banach spaces . . . ); this ex- shows that X is an F-space iff β X is, because C(β X) and plains why very often investigations into C∗(X) assume that C∗(X) are isomorphic. Also, X ∗ is an F-space whenever X is compact; this gives the advantage that ideals are fixed, X is locally compact and σ -compact. Neither property by ∗ ∗ ∗ i.e., if I is an ideal of C (X) then there is a point x with itself guarantees that X is F-space: Q is not an F-space, ∗ f (x) = 0 for all f ∈ I . Furthermore the maximal ideals are nor is (ω1 × [0, 1]) (which happens to be [0, 1]). precisely the ideals of the form { f : f (x) = 0} for some x. This result shows that F-spaces are quite ubiquitous; ∗ ∗ ∗ A perfect map is one which is continuous, closed and for example, N , R , and (Rn) are F-spaces, as well as L ∗ with compact fibers. A map f : X → Y between completely ( n Xn) for any topological sum of countably many com- regular spaces is perfect iff its Cech-extensionˇ β f satisfies pact spaces. An F-space imposes some rigidity on maps β f [β X \ X] ⊆ βY \ Y or equivalently X = β f −1[Y ]. One having it for its range: if f : X κ → Z is a continuous map can use this, for instance, to show that complete metrizabil- from a power of the compact space X to an F-space Z then ity is preserved by perfect maps (also inversely if the domain X κ can be covered by finitely many clopen sets such that is metrizable). A metrizable space is completely metrizable f depends on one coordinate on each of them. This im- ∗ iff it is a Gδ-set in its Cech–Stoneˇ compactification; the lat- plies, e.g., that a continuous map from a power of [0, ∞) ter property is then easily seen to be preserved both ways to [0, ∞)∗ itself depends on one coordinate only. by perfect maps. One calls a space a Cech-completeˇ space Some of the properties mentioned above have relation- (sometimes topologically complete) if it is a Gδ-set in its ships beyond the implications between them. Every P-space Cech–Stoneˇ compactification. This is an example of β X pro- is basically disconnected. But an extremally disconnected viding a natural setting for defining or characterizing topo- P-space that is not of a measurable cardinal number is dis- logical properties of X – the best-known example being of crete (the converse is clearly also true). As noted above X is course local compactness: a space is locally compact iff it extremally (or basically) disconnected iff β X is but there is open in its Cech–Stoneˇ compactification. Other properties is more: if X is extremally disconnected or a P-space then that can be characterized via β X are: the Lindelöf property β X can be embedded into β D for a large discrete space D. (X is normally placed in β X, which means that for every Every compact subset of β D is an F-space but a charac- open set U ⊇ X there is an Fσ -set F with X ⊆ F ⊆ U) and terization of the compact subspaces of β D is not known. paracompactness (X × β X is normal). In the special case of βN there is a characterization under The product βN × βN is not β(N × N): the characteris- the assumption of the Continuum Hypothesis, but it is also tic function of the diagonal of N witnesses that N × N is consistent that not all basically disconnected spaces embed ∗ not C -embedded in βN × βN. The definitive answer to the into βN and that not every F-space embeds into a basically question when β Q = Q β was given in [8]: if both X and Y disconnected space. are infinite then β X × βY = β(X × Y ) iff X × Y is pseu- As seen above, pseudocompactness is a property that docompact and the same holds for arbitrary products, with helps give positive structural results about β X; this happens Q Q Q a similar proviso: i β Xi = β i Xi iff i Xi is pseudo- again in the context of topological groups. If X is a topo- compact, provided Q X is never finite. If the product logical group then the operations can be extended to β X, i6=i0 i 212 Section D: Fairly general properties making β X into a topological group, if and only if X is pseu- not an accumulation point of any countable set. Their advan- docompact. tage over P-points is that their existence is provable in ZFC, ∗ first for N , later for more spaces. The final word has not been said, however. The weakest property that has not been 3. Special points ruled out by counterexamples is ‘not an accumulation point of a countable discrete set’. It is a general theorem that X ∗ = β X \ X is not homoge- Further reading neous whenever X is not pseudocompact [5]. This in itself Walker’s book [12] gives a good survey of work on β X up very satisfactory result prompted further investigation into to the mid 1970s. Van Douwen’s [4, 2, 3] and van Mill’s [9] the structure of remainders and a search for more reasons for laid the foundations for the work on β X in more recent this nonhomogeneity. Many special points were defined that years. would exhibit different topological behaviour in X ∗ or β X. The best known are the remote points: a point p of X ∗ is a remote point of X if p ∈/ clβ X N for all nowhere dense subsets N of X. If p is a remote point of X then β X is ex- References tremally disconnected at p. Many spaces have remote points, e.g., spaces of count- [1] E. Cech,ˇ On bicompact spaces, Ann. of Math. 38 able π-weight (or even with a σ -locally finite π-base) and (1937), 823–844. ω × κ 2. If X is nowhere locally compact, i.e., when X ∗ is [2] E.K. van Douwen, Remote points, Dissertationes Math. dense in β X, then X ∗ is extremally disconnected at every re- (Rozprawy Mat.) 188 (1981), 1–45. mote point of X. This gives another reason for the nonhomo- [3] E.K. van Douwen, Prime mappings, number of fac- ∗ geneity of, for example, Q , as this space is not extremally tors and binary operations, Dissertationes Math. disconnected. (Rozprawy Mat.) 199 (1981), 1–35. Under CH all separable spaces have remote points but in [4] E.K. van Douwen, Martin’s Axiom and pathological the side-by-side Sacks model there is a separable space with- points in β X \ X, Topology Appl. 34 (1990), 3–33. out remote points. Many spaces with the countable chain [5] Z. Frolík, Non-homogeneity of β P − P, Comment. condition have remote points and it is unknown whether Math. Univ. Carolin. 8 (1967), 705–709. there is such a space without remote points. Proofs that cer- [6] I.M. Gel’fand and A.N. Kolmogoroff, On rings of con- tain spaces have remote points have generated interesting tinuous functions on topological spaces, C. R. (Dok- × κ combinatorics; the proof for ω 2 contains a crucial ingre- lady) Acad. Sci. URSS 22 (1939), 11–15. dient for one proof of the consistency of the Normal Moore [7] L. Gillman and M. Jerison, Rings of Continuous Func- Space Conjecture. tions, University Series in Higher Mathematics, Van Further types of points are obtained by varying on the Nostrand, Princeton, NJ (1960), Newer edition: Grad- theme of ‘not in the closure of a small set’. Thus one obtains uate Texts in Math., Vol. 43, Springer, Berlin (1976). far points: not in the closure of any closed discrete subset [8] I. Glicksberg, The Stone–Cechˇ compactification of of X; requiring this only for countable discrete sets defines products, Trans. Amer. Math. Soc. 90 (1959), 369–382. ω-far points – a near point is a point that is not far. [9] J. van Mill, Weak P-points in Cech–Stoneˇ compactifi- Of earlier vintage are P-points, points for which the fam- cations, Trans. Amer. Math. Soc. 273 (1982), 657–678. ily of neighbourhoods is closed under countable intersec- [10] M.H. Stone, Applications of the theory of Boolean tions. They occured in the algebraic context: a point is a rings to general topology, Trans. Amer. Math. Soc. 41 P-point iff every continuous function is constant on a neigh- bourhood of it. This means that for a P-point x the ideals (1937), 375–481. { f : f (x) = 0} and { f : x ∈ int f ←( f (x))} coincide. Un- [11] A. Tychonoff, Über die topologische Erweiterung von der CH or even MA one can prove that many spaces of the Raümen, Math. Ann. 102 (1930), 544–561. ˇ form X ∗ have P-points, thus obtaining witnesses to the non- [12] R.C. Walker, The Stone–Cech Compactification, Sprin- homogeneity of X ∗. Many of these results turned out to be ger, Berlin (1974). independent of ZFC. The search for a general theorem that would, once and for all, establish nonhomogeneity of X ∗ by means of a ‘simple’ topological property of some-but-not-all Alan Dow and Klaas Pieter Hart points lead to weak P-points. A weak P-point is one that is Charlotte, NC, USA and Delft, The Netherlands