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Wallman-Shanin Compactification
218
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Section D: Fairly general properties
Wallman–Shanin Compactification
1. Compactifications A compactification of a topological space X is a compact space Y that contains X as a dense subspace; more formally, a compactification of X is a pair (Y, e) of a compact space Y and a topological embedding e : X → Y such that e(X) is dense in Y . It should be noted that in this article the space Y is not required to be Hausdorff (cf. [E, p. 177]). The Wallman–Shanin compactification is a method for constructing compactifications which may be used for obtaining compactifications of X with specified properties. In the 1960s it was shown that all of the known compactifications could also be obtained as Wallman–Shanin compactifications. It was not until 1977 that Ul’janov [13] exhibited an example of a space X and a Hausdorff compactification Y of X which is not a Wallman–Shanin compactification of X. For the description of the Wallman–Shanin compactification we employ subbases for the closed sets of a topological space. A family B of closed subsets of a topological space X is called a base for the closed sets of X if each closed subset of X is the intersection of some subfamily of B or, equivalently, for each closed subset G of X and each point x of X \ G there exists a B in B such that G ⊆ B and x ∈ / B. A family S of closed subsets of a topological space X is said to be a subbase for the closed subsets of X if the family of all unions of finite subfamilies of S is a base for the closed subsets of X. Suppose that S is a subbase for the closed subsets of the topological space X. A subfamily T of S is called a centered system of S if S1 ∩ · · · ∩ Sn = ∅ for each finite subfamily {S1 , . . . , Sn } of T . Then, from the Alexander Subbase Lemma [E, 3.12.2] it follows that the space X is compact if and only if each centered system of S has nonempty intersection. Let X be a topological space and S a subbase for its closed subsets. The basic idea of the Wallman–Shanin compactification (with respect to S) is forcing the intersection of each centered system of S to be non-empty. This is achieved by adding a new point to the space X whenever a centered system of S happens to have an empty intersection. A more accurate description follows (cf. [12, 2, 7]). By the Teichmuller–Tukey lemma [E, p. 22] each centered system of S is contained in a maximal one. The set of all maximal centered systems of S is denoted by ω(X, S). The set ω(X, S) is the underlying set of the Wallman–Shanin compactification. Its points are denoted by small Greek letters. For each S in S the subset S ∗ of ω(X, S) is defined by S ∗ = {ξ ∈ ω(X, S): S ∈ ξ }. A topology on ω(X, S) is defined by taking S ∗ = {S ∗ : S ∈ S} as a subbase for the closed sets. Thus a subset G of ω(X, S) is closed if and only if for each ξ in ω(X, S) such that ξ ∈ / G there exist S1 , . . . , Sn in S
/ Si∗ for i = 1, . . . , n. The such that G ⊆ S1∗ ∪ · · · ∪ Sn∗ and ξ ∈ key property of ω(X, S) is: for all S1 , . . . , Sn in S one has S1 ∩ · · · ∩ Sn = ∅ if and only if S1∗ ∩ · · · ∩ Sn∗ = ∅. The proof of the compactness of ω(X, S) by the Alexander subbase lemma follows. Suppose that T ∗ is a centered system of S ∗ . We may assume that T ∗ is maximal. From the key property it follows that the family T = {S: S ∗ ∈ T ∗ } is a maximal centered system of S. Consequently T is a point of ω(X, S) that belongs to every set in T ∗ . It follows that T ∗ = ∅, whence ω(X, S) is compact. To obtain a topological embedding e : X → ω(X, S) we define e(x) = {S ∈ S: x ∈ S}. Without any further restriction on S the family e(x) need not be a maximal centered system for any x, and even if e is a map, it need not be an embedding. We shall say that the subbase S is disjunctive if for each S of S and each point x of X such that x ∈ / S there exists a T in S such that x ∈ T ⊆ X \ S. If the subbase S of X is disjunctive, the space X as well as ω(X, S) are T1 -spaces and e is an embedding. Thus for a disjunctive subbase S the space ω(X, S) is a compactification of X. Further it can be shown that S ∗ coincides with the closure of S in ω(X, S). In general, the compactification ω(X, S) need not be Hausdorff. We shall formulate sufficient conditions for ω(X, S) to be Hausdorff. A family V of subsets of a set X is called a ring if for every finite subfamily {V1 , . . . , Vn } of V both the intersection V1 ∩ · · · ∩ Vn and the union V1 ∪ · · · ∪ Vn belong to V. Suppose that a subbase S of the closed subsets of a topological space X is a ring. Note that because of the ring property S is a base for the closed sets. The family S is said to be normal if for any two disjoint elements A and B of S there are elements C and D of S such that A ∩ D = ∅, B ∩ C = ∅ and C ∪ D = X; the pair (C, D) is called a screening the pair (A, B). A base B for the closed subsets of a topological space X is called a Wallman–Shanin base, or WS-base, if (1) B is a ring, (2) B is disjunctive, (3) B is normal. If B is a Wallman–Shanin base the compactification ω(X, B) is called a Wallman–Shanin compactification, or WS-compactification, of X. Note that in view of property (1) distinct points of ω(X, B) are contained in disjoint elements of B ∗ and that by property (3) these disjoint elements of B ∗ have disjoint neighbourhoods in ω(X, B). In other words, the space ω(X, B) is a compact Hausdorff space. Also observe that in view of the ring property of B the maximal centered systems of B are ultrafilters. The WScompactification first appeared in the paper [14] of Wallman. We shall discuss this somewhat more in detail in the next section. In [9, 8, 10] Shanin indicated that the compactification method of Wallman had much wider applications and introduced what is now called the WS-compactification. The work of Shanin has remained unnoticed for a long time.
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That explains why the WS-compactification is often referred to as Wallman compactification. In 1961 Frink [5] rediscovered some of Shanin’s results and posed the question whether each Hausdorff compactification Y of a space X is a WS-compactification, i.e., whether Y can be represented as ω(X, B) for some base B for the closed subsets of X. The question of Frink has triggered many partial answers. We shall discuss this further in the final section of the article.
2. Wallman representation In [14] Wallman developed a theory of representing lattices by bases for the family of all closed subsets of topological spaces. This work is related to Stone’s theory of representing Boolean rings by the family of closed-and-open subsets of compact zero-dimensional spaces. In this section the spaces under discussion are T1 -spaces. Wallman observed that much information about a space X, more specifically ˇ its covering dimension and its Cech homology, is related to the lattice of some base for the closed subsets rather than its topology. A lattice is a nonempty set L with a reflexive partial order such that for each pair (x, y) of elements of L there is a unique smallest element x ∨ y, called the join of x and y, such that x x ∨ y and y x ∨ y and there is a unique largest element x ∧ y, called the meet of x and y, such that x ∧ y x and x ∧ y y [2, 4, 14]. The family of all closed subsets of a topological T1 -space X with the partial order ⊆ is a lattice which is denoted by L(X). The join and meet of two closed subsets are the union and intersection respectively. The lattice L(X) is distributive, has a largest element 1 (the set X) and a smallest element 0 (the empty set). Also the lattice L(X) has the disjunction property: for all x and y of L(X) that are distinct there exists a z in L(X) such that one of x ∧ z and y ∧ z in 0 and the other is not. (For example, z is a one point subset of the symmetric difference of the subsets x and y of X.) In the first part of Wallman’s paper [14] it is shown that each distributive lattice L with 0 and 1 that has the disjunction property can be represented as the lattice of a base for the closed subsets of some compact T1 -space ωL. The construction of the space ωL is similar to the construction of the compactifications in Section 1. A nonempty subset ξ of L is called a dual ideal (or filter) if for all x and y in L we have: (1) if x and y in ξ then x ∧ y ∈ L, (2) if x ∈ ξ and x y then y ∈ ξ . Every dual ideal is contained in a maximal one. Note that if L = L(X) for some T1 -space X then a dual ideal of L is a centered system of the family of closed sets of X and a maximal dual ideal of L is just an ultrafilter. The points of ωL are the maximal dual ideals of L. For each u in L a subset Bu of ω(L) is defined by Bu = {ξ : u ∈ ξ }. The topology on ωL is defined by taking the family F = {Bu : u ∈ L} as a base for the closed subsets. Then ωL is a compact T1 space and is referred to as the Wallman representation of the lattice L. The proof of the compactness is essentially the same as that in Section 1. It can be shown that the closed
219 base F has the following properties: (1) if u and v are distinct elements of L, then Bu = Bv , (2) Bu ∪ Bv = Bu∨v and Bu ∩ Bv = Bu∧v for all u and v of L. In other words, the map of L to F defined by sending u to Bu is an isomorphism of lattices. The space ωL is Hausdorff if and only if for all s and t in L with s ∧ t = 0 there exist u and v in L such that s ∧ v = 0, u ∧ t = 0 and u ∨ v = 1. The topological representation of lattices can be used to construct a compactification of a topological space via the lattice of its closed sets. The Wallman compactification of a T1 -space X is the topological representation of the lattice L(X) of all closed subsets of X. The space is denoted by ωL(X) or, as in Section 1, by ω(X, L(X)). The embedding of X in ω(X, L(X)) is defined by sending the point x to the ultrafilter of all closed sets of X that contain x. The Wallman compactification of a T1 -space X is Hausdorff if and only if X is normal. The Wallman compactification is a maximal compactification in the following sense. If f : X → K is a continuous map of a T1 -space X to a compact Hausdorff space K, then f can be extended to a continuous map f˜ : ω(X, L(X)) → K.
3. Wallman–Shanin compactifications Suppose that B is a WS-base of a topological space X. Then the topological representation of the lattice B coincides with the space ω(X, B) as defined in Section 1. The problem of deciding whether a Hausdorff compactification Y of a space X is a WS-compactification ω(X, B) for some WS-base B was already considered by Shanin. He showed that Y = ω(X, B) for some base B for the closed subsets if and only if (1) {B: B ∈ B} is a base for the closed subsets of Y and (2) B1 ∩ B2 = B1 ∩ B2 for all B1 , B2 in B, where the an upper bar denotes the closure in the space Y . The following more general statement can be found in [1]. Suppose that Y is a Hausdorff compactification of a space X. Then Y = ω(X, S) for some subbase S for the closed sets if and only if the following two conditions hold: (1) {S: S ∈ S} is a subbase for the closed subsets of Y and (2) S1 ∩ · · · ∩ Sn = ∅ if and only if S1 ∩ · · · ∩ Sn = ∅ for all S1 , . . . , Sn in S. And if the conditions (1) and (2) are satisfied, then Y = ω(X, B), where B is the family of all finite unions and intersections of S. This result is related to Steiner’s result [12] that ω(X, B) = ω(X, S) and the following observation. If S is a subbase for the closed sets of a compact Hausdorff space, then the family of all finite unions and intersections of elements of S is a WS-base. In Section 2 we have mentioned the maximality of the ˇ Wallman compactification. As the Cech–Stone compactification is characterized by its maximality [E, 3.6.6], it is a ˇ natural question whether the Cech–Stone compactification is a WS-compactification. Note that the Wallman compactification of a normal T1 -space X is Hausdorff. Then, by maxiˇ mality it must be the Cech–Stone compactification of X. We shall exhibit a WS-base of X. A subset Z of a topological T1 -space X is called a zero set if Z = f −1 [{0}] for some
220 continuous real-valued function f : X → R. The set Z is closed by continuity of f . A T1 -space X is completely regular if the family Z of all zero sets is a base for the closed subsets of X. If X is a completely regular T1 -space then the family Z of all zero sets is a WS-base and the compactifiˇ cation ω(X, Z) is the same as the Cech–Stone compactification [6, Section 6.5]. In connection with the characterization of completely regular T1 -spaces as subspaces of compact Hausdorff spaces [E, 3.5.1], we make the observation that a T1 -space X is completely regular if and only if X has a WS-base [5]. A Hausdorff space X is locally compact if each of its points has an open neighbourhood the closure of which is compact. A non-compact Hausdorff space X is locally compact if and only if X has a Hausdorff one-point compactification α(X) (i.e., α(X) is a Hausdorff compactification such that α(X) \ X consists of one point). It may be seen that the one-point compactification α(X) of a non-compact, locally compact Hausdorff space X is a WS-compactification. An appropriate WS-base consists of all compact subsets of X and all sets of the form X \ K for some compact subset K. WS-compactifications may be used to show the existence of compactifications with specified properties. It can be shown, for example, that for a completely regular T1 -space X with dim X n and a family A of closed subsets such that the cardinality of the family does not exceed the weight w(X) of the space there exists a compactification Y such that w(Y ) = w(X), dim Y dim X and dim A¯ dim A for all A in A. The idea of the proof is that the dimension of a space or of one of its closed subsets can be formulated in terms of finite families from a given base for the closed subsets of the space. Then an appropriate WS-base is built in an inductive way. We refer to [2, Chapter 6] for various examples of applications of WS-compactifications along these lines In investigating Frink’s problem whether every Hausdorff compactification of a space is a WS-compactification we may restrict our attention to compactifications of discrete spaces. This is a consequence of a theorem of Bandt [3]: if all Hausdorff compactifications of the discrete space of cardinality γ are WS-compactifications, then so are all Hausdorff compactifications of spaces of density γ . In addressing Frink’s problem it is natural to study compactifications with increasing complexity. A measure for the complexity is the weight of the compactification. It was shown by Steiner and Steiner [11] and Aarts [1] that all compactifications with weight ℵ0 are WS-compactifications. This statement can be rephrased as: all metrizable compactifications are Wallman. This result was substantially strengthened by Bandt [3] in showing that all compactifications with weight ℵ1 are WS-compactifications. Under the Continuum Hypothesis it follows that every compactification of a separable space is a WS-compactification. In a sense this
Section D: Fairly general properties result is the best possible: Ul’janov [13] showed that for any cardinal τ such that 2τ ℵ2 , there exists a completely regular space S of cardinality τ and a connected Hausdorff compactification νS of weight 2τ which is not a WScompactification.
References [1] J.M. Aarts, Every metric compactification is a Wallman-type compactification, Proc. Internat. Symp. on Topology and Its Applications, Herceg-Novi, Yugoslavia, 1968, D.R. Kurepa, ed. (1969). [2] J.M. Aarts and T. Nishiura, Dimension and Extensions, North-Holland Math. Library, Vol. 48, North-Holland, Amsterdam (1993). [3] C. Bandt, On Wallman–Shanin-compactifications, Math. Nachr. 77 (1977), 333–351. [4] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ., Vol. 25, Amer. Math. Soc., Providence, RI (1973). [5] O. Frink, Compactifications and semi-normal spaces, Amer. J. Math. 86 (1964), 602–907. [6] L. Gillman and M. Jerison, Rings of Continuous Functions, University Series in Higher Mathematics, Van Nostrand, Princeton, NJ (1960). Newer edition: Graduate Texts in Math., Vol. 43, Springer, Berlin (1976). [7] J. van Mill, The Infinite-Dimensional Topology of Function Spaces, North-Holland Math. Library, Vol. 64, North-Holland, Amsterdam (2001). [8] N.A. Shanin, On separation in topological spaces, C. R. (Doklady) Acad. Sci. URSS 38 (1943), 110–113. [9] N.A. Shanin, On special extensions of topological spaces, C. R. (Doklady) Acad. Sci. URSS 38 (1943), 6–9. [10] N.A. Shanin, On the theory of bicompact extensions of topological spaces, C. R. (Doklady) Acad. Sci. URSS 38 (1943), 154–156. [11] A.K. Steiner and E.F. Steiner, Products of compact metric spaces are regular Wallman, Indag. Math. 30 (1968), 428–430. [12] E.F. Steiner, Wallman spaces and compactifications, Fund. Math. 61 (1968), 295–304. [13] V.M. Ul’janov, Solution of a basic problem on compactifications of Wallman type, Soviet Math. Dokl. 18 (1977), 567–571. [14] H. Wallman, Lattices and topological spaces, Ann. of Math. 39 (1938), 112–126.
J.M. Aarts Delft, The Netherlands
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Wallman–Shanin Compactification
1. Compactifications A compactification of a topological space X is a compact space Y that contains X as a dense subspace; more formally, a compactification of X is a pair (Y, e) of a compact space Y and a topological embedding e : X → Y such that e(X) is dense in Y . It should be noted that in this article the space Y is not required to be Hausdorff (cf. [E, p. 177]). The Wallman–Shanin compactification is a method for constructing compactifications which may be used for obtaining compactifications of X with specified properties. In the 1960s it was shown that all of the known compactifications could also be obtained as Wallman–Shanin compactifications. It was not until 1977 that Ul’janov [13] exhibited an example of a space X and a Hausdorff compactification Y of X which is not a Wallman–Shanin compactification of X. For the description of the Wallman–Shanin compactification we employ subbases for the closed sets of a topological space. A family B of closed subsets of a topological space X is called a base for the closed sets of X if each closed subset of X is the intersection of some subfamily of B or, equivalently, for each closed subset G of X and each point x of X \ G there exists a B in B such that G ⊆ B and x ∈ / B. A family S of closed subsets of a topological space X is said to be a subbase for the closed subsets of X if the family of all unions of finite subfamilies of S is a base for the closed subsets of X. Suppose that S is a subbase for the closed subsets of the topological space X. A subfamily T of S is called a centered system of S if S1 ∩ · · · ∩ Sn = ∅ for each finite subfamily {S1 , . . . , Sn } of T . Then, from the Alexander Subbase Lemma [E, 3.12.2] it follows that the space X is compact if and only if each centered system of S has nonempty intersection. Let X be a topological space and S a subbase for its closed subsets. The basic idea of the Wallman–Shanin compactification (with respect to S) is forcing the intersection of each centered system of S to be non-empty. This is achieved by adding a new point to the space X whenever a centered system of S happens to have an empty intersection. A more accurate description follows (cf. [12, 2, 7]). By the Teichmuller–Tukey lemma [E, p. 22] each centered system of S is contained in a maximal one. The set of all maximal centered systems of S is denoted by ω(X, S). The set ω(X, S) is the underlying set of the Wallman–Shanin compactification. Its points are denoted by small Greek letters. For each S in S the subset S ∗ of ω(X, S) is defined by S ∗ = {ξ ∈ ω(X, S): S ∈ ξ }. A topology on ω(X, S) is defined by taking S ∗ = {S ∗ : S ∈ S} as a subbase for the closed sets. Thus a subset G of ω(X, S) is closed if and only if for each ξ in ω(X, S) such that ξ ∈ / G there exist S1 , . . . , Sn in S
/ Si∗ for i = 1, . . . , n. The such that G ⊆ S1∗ ∪ · · · ∪ Sn∗ and ξ ∈ key property of ω(X, S) is: for all S1 , . . . , Sn in S one has S1 ∩ · · · ∩ Sn = ∅ if and only if S1∗ ∩ · · · ∩ Sn∗ = ∅. The proof of the compactness of ω(X, S) by the Alexander subbase lemma follows. Suppose that T ∗ is a centered system of S ∗ . We may assume that T ∗ is maximal. From the key property it follows that the family T = {S: S ∗ ∈ T ∗ } is a maximal centered system of S. Consequently T is a point of ω(X, S) that belongs to every set in T ∗ . It follows that T ∗ = ∅, whence ω(X, S) is compact. To obtain a topological embedding e : X → ω(X, S) we define e(x) = {S ∈ S: x ∈ S}. Without any further restriction on S the family e(x) need not be a maximal centered system for any x, and even if e is a map, it need not be an embedding. We shall say that the subbase S is disjunctive if for each S of S and each point x of X such that x ∈ / S there exists a T in S such that x ∈ T ⊆ X \ S. If the subbase S of X is disjunctive, the space X as well as ω(X, S) are T1 -spaces and e is an embedding. Thus for a disjunctive subbase S the space ω(X, S) is a compactification of X. Further it can be shown that S ∗ coincides with the closure of S in ω(X, S). In general, the compactification ω(X, S) need not be Hausdorff. We shall formulate sufficient conditions for ω(X, S) to be Hausdorff. A family V of subsets of a set X is called a ring if for every finite subfamily {V1 , . . . , Vn } of V both the intersection V1 ∩ · · · ∩ Vn and the union V1 ∪ · · · ∪ Vn belong to V. Suppose that a subbase S of the closed subsets of a topological space X is a ring. Note that because of the ring property S is a base for the closed sets. The family S is said to be normal if for any two disjoint elements A and B of S there are elements C and D of S such that A ∩ D = ∅, B ∩ C = ∅ and C ∪ D = X; the pair (C, D) is called a screening the pair (A, B). A base B for the closed subsets of a topological space X is called a Wallman–Shanin base, or WS-base, if (1) B is a ring, (2) B is disjunctive, (3) B is normal. If B is a Wallman–Shanin base the compactification ω(X, B) is called a Wallman–Shanin compactification, or WS-compactification, of X. Note that in view of property (1) distinct points of ω(X, B) are contained in disjoint elements of B ∗ and that by property (3) these disjoint elements of B ∗ have disjoint neighbourhoods in ω(X, B). In other words, the space ω(X, B) is a compact Hausdorff space. Also observe that in view of the ring property of B the maximal centered systems of B are ultrafilters. The WScompactification first appeared in the paper [14] of Wallman. We shall discuss this somewhat more in detail in the next section. In [9, 8, 10] Shanin indicated that the compactification method of Wallman had much wider applications and introduced what is now called the WS-compactification. The work of Shanin has remained unnoticed for a long time.
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That explains why the WS-compactification is often referred to as Wallman compactification. In 1961 Frink [5] rediscovered some of Shanin’s results and posed the question whether each Hausdorff compactification Y of a space X is a WS-compactification, i.e., whether Y can be represented as ω(X, B) for some base B for the closed subsets of X. The question of Frink has triggered many partial answers. We shall discuss this further in the final section of the article.
2. Wallman representation In [14] Wallman developed a theory of representing lattices by bases for the family of all closed subsets of topological spaces. This work is related to Stone’s theory of representing Boolean rings by the family of closed-and-open subsets of compact zero-dimensional spaces. In this section the spaces under discussion are T1 -spaces. Wallman observed that much information about a space X, more specifically ˇ its covering dimension and its Cech homology, is related to the lattice of some base for the closed subsets rather than its topology. A lattice is a nonempty set L with a reflexive partial order such that for each pair (x, y) of elements of L there is a unique smallest element x ∨ y, called the join of x and y, such that x x ∨ y and y x ∨ y and there is a unique largest element x ∧ y, called the meet of x and y, such that x ∧ y x and x ∧ y y [2, 4, 14]. The family of all closed subsets of a topological T1 -space X with the partial order ⊆ is a lattice which is denoted by L(X). The join and meet of two closed subsets are the union and intersection respectively. The lattice L(X) is distributive, has a largest element 1 (the set X) and a smallest element 0 (the empty set). Also the lattice L(X) has the disjunction property: for all x and y of L(X) that are distinct there exists a z in L(X) such that one of x ∧ z and y ∧ z in 0 and the other is not. (For example, z is a one point subset of the symmetric difference of the subsets x and y of X.) In the first part of Wallman’s paper [14] it is shown that each distributive lattice L with 0 and 1 that has the disjunction property can be represented as the lattice of a base for the closed subsets of some compact T1 -space ωL. The construction of the space ωL is similar to the construction of the compactifications in Section 1. A nonempty subset ξ of L is called a dual ideal (or filter) if for all x and y in L we have: (1) if x and y in ξ then x ∧ y ∈ L, (2) if x ∈ ξ and x y then y ∈ ξ . Every dual ideal is contained in a maximal one. Note that if L = L(X) for some T1 -space X then a dual ideal of L is a centered system of the family of closed sets of X and a maximal dual ideal of L is just an ultrafilter. The points of ωL are the maximal dual ideals of L. For each u in L a subset Bu of ω(L) is defined by Bu = {ξ : u ∈ ξ }. The topology on ωL is defined by taking the family F = {Bu : u ∈ L} as a base for the closed subsets. Then ωL is a compact T1 space and is referred to as the Wallman representation of the lattice L. The proof of the compactness is essentially the same as that in Section 1. It can be shown that the closed
219 base F has the following properties: (1) if u and v are distinct elements of L, then Bu = Bv , (2) Bu ∪ Bv = Bu∨v and Bu ∩ Bv = Bu∧v for all u and v of L. In other words, the map of L to F defined by sending u to Bu is an isomorphism of lattices. The space ωL is Hausdorff if and only if for all s and t in L with s ∧ t = 0 there exist u and v in L such that s ∧ v = 0, u ∧ t = 0 and u ∨ v = 1. The topological representation of lattices can be used to construct a compactification of a topological space via the lattice of its closed sets. The Wallman compactification of a T1 -space X is the topological representation of the lattice L(X) of all closed subsets of X. The space is denoted by ωL(X) or, as in Section 1, by ω(X, L(X)). The embedding of X in ω(X, L(X)) is defined by sending the point x to the ultrafilter of all closed sets of X that contain x. The Wallman compactification of a T1 -space X is Hausdorff if and only if X is normal. The Wallman compactification is a maximal compactification in the following sense. If f : X → K is a continuous map of a T1 -space X to a compact Hausdorff space K, then f can be extended to a continuous map f˜ : ω(X, L(X)) → K.
3. Wallman–Shanin compactifications Suppose that B is a WS-base of a topological space X. Then the topological representation of the lattice B coincides with the space ω(X, B) as defined in Section 1. The problem of deciding whether a Hausdorff compactification Y of a space X is a WS-compactification ω(X, B) for some WS-base B was already considered by Shanin. He showed that Y = ω(X, B) for some base B for the closed subsets if and only if (1) {B: B ∈ B} is a base for the closed subsets of Y and (2) B1 ∩ B2 = B1 ∩ B2 for all B1 , B2 in B, where the an upper bar denotes the closure in the space Y . The following more general statement can be found in [1]. Suppose that Y is a Hausdorff compactification of a space X. Then Y = ω(X, S) for some subbase S for the closed sets if and only if the following two conditions hold: (1) {S: S ∈ S} is a subbase for the closed subsets of Y and (2) S1 ∩ · · · ∩ Sn = ∅ if and only if S1 ∩ · · · ∩ Sn = ∅ for all S1 , . . . , Sn in S. And if the conditions (1) and (2) are satisfied, then Y = ω(X, B), where B is the family of all finite unions and intersections of S. This result is related to Steiner’s result [12] that ω(X, B) = ω(X, S) and the following observation. If S is a subbase for the closed sets of a compact Hausdorff space, then the family of all finite unions and intersections of elements of S is a WS-base. In Section 2 we have mentioned the maximality of the ˇ Wallman compactification. As the Cech–Stone compactification is characterized by its maximality [E, 3.6.6], it is a ˇ natural question whether the Cech–Stone compactification is a WS-compactification. Note that the Wallman compactification of a normal T1 -space X is Hausdorff. Then, by maxiˇ mality it must be the Cech–Stone compactification of X. We shall exhibit a WS-base of X. A subset Z of a topological T1 -space X is called a zero set if Z = f −1 [{0}] for some
220 continuous real-valued function f : X → R. The set Z is closed by continuity of f . A T1 -space X is completely regular if the family Z of all zero sets is a base for the closed subsets of X. If X is a completely regular T1 -space then the family Z of all zero sets is a WS-base and the compactifiˇ cation ω(X, Z) is the same as the Cech–Stone compactification [6, Section 6.5]. In connection with the characterization of completely regular T1 -spaces as subspaces of compact Hausdorff spaces [E, 3.5.1], we make the observation that a T1 -space X is completely regular if and only if X has a WS-base [5]. A Hausdorff space X is locally compact if each of its points has an open neighbourhood the closure of which is compact. A non-compact Hausdorff space X is locally compact if and only if X has a Hausdorff one-point compactification α(X) (i.e., α(X) is a Hausdorff compactification such that α(X) \ X consists of one point). It may be seen that the one-point compactification α(X) of a non-compact, locally compact Hausdorff space X is a WS-compactification. An appropriate WS-base consists of all compact subsets of X and all sets of the form X \ K for some compact subset K. WS-compactifications may be used to show the existence of compactifications with specified properties. It can be shown, for example, that for a completely regular T1 -space X with dim X n and a family A of closed subsets such that the cardinality of the family does not exceed the weight w(X) of the space there exists a compactification Y such that w(Y ) = w(X), dim Y dim X and dim A¯ dim A for all A in A. The idea of the proof is that the dimension of a space or of one of its closed subsets can be formulated in terms of finite families from a given base for the closed subsets of the space. Then an appropriate WS-base is built in an inductive way. We refer to [2, Chapter 6] for various examples of applications of WS-compactifications along these lines In investigating Frink’s problem whether every Hausdorff compactification of a space is a WS-compactification we may restrict our attention to compactifications of discrete spaces. This is a consequence of a theorem of Bandt [3]: if all Hausdorff compactifications of the discrete space of cardinality γ are WS-compactifications, then so are all Hausdorff compactifications of spaces of density γ . In addressing Frink’s problem it is natural to study compactifications with increasing complexity. A measure for the complexity is the weight of the compactification. It was shown by Steiner and Steiner [11] and Aarts [1] that all compactifications with weight ℵ0 are WS-compactifications. This statement can be rephrased as: all metrizable compactifications are Wallman. This result was substantially strengthened by Bandt [3] in showing that all compactifications with weight ℵ1 are WS-compactifications. Under the Continuum Hypothesis it follows that every compactification of a separable space is a WS-compactification. In a sense this
Section D: Fairly general properties result is the best possible: Ul’janov [13] showed that for any cardinal τ such that 2τ ℵ2 , there exists a completely regular space S of cardinality τ and a connected Hausdorff compactification νS of weight 2τ which is not a WScompactification.
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J.M. Aarts Delft, The Netherlands