Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

WALLMAN-FRINK PROXIMITIES

MARLON C. RAYBURN

Abstract. This is a survey of compactification extension results and problems for a special class of proximities.

The compatible Efremovič proximities on a Tichonov space are ordered by δ ≤ ρ if for all A, B ⊆ X, AδB → AρB. It is well known that the Smirnov completions of the Efremovič proximities give a one-to-one, order reversing correspondence between the Hausdorff compactifications and the proximities. This merely means that the proximities share the compactification lattice order problem (“find a necessary and sufficient condition on X that the Hausdorff compactifications form a lattice”).

While it is not practical to work on the order problems by embedding the Hausdorff compactifications in the larger family of T_{1} compactifications, there simply being no end to the latter, the Efremovič proximities generalize nicely to the complete lattice of Ladato proximities. When Gagrat and Naimpally showed that the compatible separated proximities on a space complete to T_{1} compactifications, it seemed a solution to the order problem was at last in sight.

Unfortunately, troubles remain. The Gagrat-Naimpally compactifications are among the T_{1} compactifications, yes, but which ones are they? Moreover, the corresponding order between the completions breaks down. Even a p-map between proximity spaces will only lift, in general, to a continuous extension from the Gagrat-Naimpally completion of the domain to the “bunch space” of the range — something rich and strange.

What follows is an attempt to solve these problems for the special case of Lodato proximities in the style of Wallman-Frink.

Definition 1. Let X be a T_{1} space with at least two points. Let B be any base for the closed sets such that (1) B is a network (i.e. x ∈ G, open, implies there is some B ∈ B with

2000 Mathematics Subject Classification. 54E05.

Key words and phrases. Proximity structures.

Marlon C. Rayburn, Wallman-Frink proximities, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 265–270, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204138.

x ∈ B ⊆ G), and

(2) B is a ring of sets (i.e. B is closed under finite unions and finite

intersections). Then B is a normal base for the closed sets.

N. b. since x has at least two points, ∅ ∈ B.

Definition 2. The Wallman-Frink proximity δ_{b} associated with normal base B is given by:

A δ̸_{b}B ≡ ∃F_{1}, F_{2} ∈ B with A ⊆ F_{1}, B ⊆ F_{2} and F_{1} ∩ F_{2} = ∅.

Hence Aδ_{b}B ⟺ ∀F_{1}, F_{2} ∈ B, A ⊆ F_{1} and B ⊆ F_{2} ⇒ F_{1} ∩ F_{2} ≠ ∅.

Theorem 3. δ_{b} is a compatible, separated Lodato proximity on X.

Proof. Since ∅ ∈ B, clearly ∅ δ̸_{b}A for any A ⊆ X. It is also clear that A δ̸_{b}B ⇒ B δ̸_{b}A, and A δ̸_{b}B ⇒ A ∩ B = ∅.

Since B is a ring of sets, if A δ̸_{b}B and A δ̸_{b}C, then A δ̸_{b}(B ∪ C).

Since B is a network, x ≠ y ⇒ x δ̸_{b}Y, and x ∉ cl(A) ⟺ x δ̸_{b}A. Thus A δ̸_{b}C and B ⊆ cl(C) ⇒δ̸_{b}B. □

Corollary 4. δ_{b} is Efremovič ⟺ whenever F_{1}, F_{2} ∈ B such that F_{1}∩F_{2} = ∅, then there exists some C, D ∈ B such that F_{1} ⊆ X\C, F_{2} ⊆ X\D, and (X\C) ∩ (X\D) = ∅.

A b-filter is a filterbase F of sets from B such that whenever B ∈ F and B ⊆ F ∈ B, then F ∈ F. A b-ultrafilter is a maximal b-filter. By Zorn’s Lemma, every b-filter is contained in at least one b-ultrafilter. N.b., for any ultrafilter μ on X, μ ∩ B is a b-filter.

Definition 5. An ultrafilter μ on X is an ultrafilter of type b if μ ∩ B is a b-ultrafilter.

Lemma 6. For any normal base B on T_{1}-space X and each x ∈ X, the point ultrafilter μ_{x} = {A: x ∈ A} is an ultrafilter of type b.

Proof. We need only check that the b-filter μ_{x} ∩ B is a maximal among the b-filters. Suppose φ is a b-filter with μ_{x} ∩ B ⊆ φ. Then each F ∈ φ ⊆ B, F ∩B ≠ ∅ for all B ∈ μ_{x}∩B. Since B is a network, we see that x ∈ cl(F) = F for each F ∈ φ. Hence μ_{x} ∩ B = φ. □

Lemma 7. If μ is any b-ultrafilter, then there is an ultrafilter α of type b such that μ = α ∩ B.

Proof. μ is a filterbase of sets, so it is contained an ultrafilter α. Thus μ ⊆ α ∩ B. Since μ is maximal, μ = α ∩ B. □

Lemma 8. Let δ be any compatible Lodato proximity with δ_{b} ≤ δ. For every ultrafilter μ on X, there is an ultrafilter α of type b such that μδα.

Proof. Since μ∩B is a b-filter, we must have μ∩B ⊆ φ for some b-ultrafilter φ. By the last lemma, there is an ultrafilter α of type b with φ = α ∩ B. Suppose μ δ̸α. Then μ δ̸_{b}α, so we must have an M ∈ μ, and A ∈ α and some F_{1}, F_{2} ∈ B with M ⊆ F_{1}, A ⊆ F_{2} and F_{1} ∩ F_{2} = ∅. But since μ ∩ B ⊆ α ∩ B, we have F_{1} and F_{2} ⊆ α, contradiction to filter. □

Lemma 9. On (X, δ_{b}), given any ultrafilter μ and any ultrafilter α of type b, μδ_{b}α if and only if μ ∩ B ⊆ α ∩ B. Hence any two ultrafilters near an ultrafilter of type b are near each other. In particular, δ_{b} is transitive over ultrafilters of type b.

Proof. Suppose μδ_{b}α. Then for each B ∈ μ∩B, B∩F ≠ ∅ or every F ∈ α∩B. Since α ∩ B is a b-ultrafilter, we have that B ∈ α ∩ B. The converse is clear from the definition of δ_{b}. □

Definition 10. Let γ be any filterbase on proximity space (X, δ) and set Π_{δ}(γ) = ⋃{υ: υ is an ultrafilter and υδγ}.

Definition 11.

a) A grill γ on a proximity space (X, δ) is a precluster if whenever A ⊆ X and {A}δγ, then A ∈ γ.

b) A pre-cluster σ on (X, δ) is a cluster if it is a clan.

Example 12. Let α be any filterbase on proximity space (X, δ). Then Π(φ) is a pre-cluster.

Theorem 13. On (X, δ_{b}), if α is an ultrafilter of type b, then Π(α) is a cluster.

Proof. By Example 12, Π(α) is a pre-cluster and by lemma 9, Π(α) is a clan. □

Theorem 14. The subspace T_{b}X of the Gagrat-Naimpally completion α_{δ_{b}}X given by the set of all maximal clans is a T_{1} compactification on X.

Proof. By Theorem 11,

T_{b}X = {Π(α): α is an ultrafilter of type b}

is a subset of α_{δ}X, the set of all maximal clans on X. Hence T_{b}X is a T_{1} space, which by Lemma 6 contains a dense copy of X. It remains to show that T_{b}X is compact.

For each F ∈ B, let

F′ = {Π(α): F ∈ Π(α)}.

These will be the basic closed sets of the topology of T_{b}X. Let L = {F′_{j}: j ∈ Γ} be a family of basic closed sets with the finite intersection property. Let

F = {F_{j}: j ∈ Γ} ⊆ B.

Let

F′ = {⋂_{j∈∀} F_{j}: Λ a non-empty, finite subset of Γ}.

Then F′ is a filterbase of sets from B since B is a ring, and F ⊆ F′. By Zorn’s Lemma, F′ is contained in some b-ultrafilter, which by Lemma 7 we may write as α ∩ B of some ultrafilter α of type b. But then F ⊆ Π(α), so for each j ∈ Γ, Π(α) ∈ F′_{j}, and thus ⋂ L ≠ ∅. □

If B is a normal base of closed sets for T_{1} space X, we may constuct a “Wallman-Frink” compactification bX of X in the usual way (See Willard’s “General Topology”, Exercise 19K p 142):

a) Let bX be the set of all b-ultrafilters on X.

b) For each B ∈ B, let B′ = {φ ∈ bX: B ∈ φ}. This is a base for closed sets for a topology on bX. Call it the “absorption” topology for B.

c) Under the absorption topology, bX is compact and T_{1}.

d) τ: X → bX by t(x) = μ_{x} ∩ B is a homeomorphism from X onto a dense subset of bX.

Theorem 15. There is a homeomorphism between the T_{1} compactification T_{b}X of (X, δ_{b}) and the Wallman-Frink compactification bX, which fixes X point-wise.

Proof. Let f be the bijection assigning maximal b-clan σ = Π(α) to the b-ultrafilter σ ∩ B. Set B ∈ B gives us both basic closed set

B∗ = {σ ∈ T_{b}X: B ∈ σ} and B′ = {φ ∈ bX: B ∈ φ}.

But clearly B ∈ φ if and only if Bδα for every ultrafilter α of type b for which φ = α∩B, and F ∈ α if and only if F ∈ σ_{α} for σ_{α} = ⋃{μ: ultrafilter μδ_{b}α}. Hence f is one-to-one between basic closed sets of bX and T_{b}X, and is therefore a homeomorphism. That f is 1 − 1 on X follows from Lemma 6. □

Problem. This identifies the Wallman compactification of a T_{1} space with a subspace of the Gagrat-Naimpally compactification of (X, δ_{w}). Conceivably this is possible, yet it does not rule out the equality of the two extensions. This would amount to showing the reverse of Theorem 11, that every maximal clan is of form Π(α) for some ultrafilter α of type b.

Specifically, let σ be any maximal clan on (X, δ_{b}) and consider

σ ∩ B = ⋂{μ ∩ B: ultrafilter μ ⊆ σ}.

Let F ∈ σ∩B and F ⊆ B ∈ B. Then there is some ultrafilter μ ⊆ σ such that F ∈ μ. Therefore B ∈ μ, so B ∈ σ ∩ B. Now let F_{1} and F_{2} ∈ σ ∩ B. Then F_{1}δ_{b}F_{2} since σ is a clan, so F_{1}∩F_{2} ≠ ∅. Suppose F_{1}∩F_{2} ∈ σ. Then σ∩B is a b-filter, so there is an ultra-filter α of type b such that σ ∩ B ⊆ α ∩ B. Hence for every μ ⊆ σ, μδ_{b}α, so σ ⊆ Π(α). By the maximality of σ, σ = Π(α). Thus

Theorem 16. Let σ be a maximal clan on (X, δ_{b}). Then σ = Π(α) for some ultrafilter α of type b ⟺ [F_{1}, F_{2} ∈ σ ⇒ F_{1} ∩ F_{2} ∈ σ].

Now suppose X to be a T_{1}-space for which B and D are normal bases with D ⊆ B. [Hence δ_{b} ≤ δ_{d}.]

Lemma 17. Let σ be any element of T_{d}. Then there is an ultrafilter β of both types b and d such that σ = Π_{d}(β).

Proof. Since σ is an element of T_{d}, there is an ultrafilter γ of type d such that σ = Π_{d}(γ). But γ is an ultrafilter in (X, δ_{b}), so there exists an ultrafilter β of type b such that γδ_{b}β. This means that γ ∩ B ⊆ β. But D ⊆ B implies γ ∩ D ⊆ B ∩ D. By maximality of γ ∩ D, we have γ ∩ D = β ∩ D. Thus β is also an ultrafilter of type d.

Now ultrafilter μ ⊆ Π_{d}(γ) if and only if μ∩D ⊆ γ ∩D = β ∩D. Therefore μ ⊆ Π_{d}(γ) if and only if μ ⊆ Π_{d}(β), and σ = Π_{d}(β). □

Notation 18. An ultrafilter β of both types b and d will be called an ultrafilter of joint type.

Lemma 19. Let α be any ultrafilter of type b. Then there is an ultrafilter β of joint type such that αδ_{d}β.

Proof. There is an ultrafilter γ of type d such that σδ_{d}γ. By the last lemma, there is an ultrafilter β of joint type with γδ_{d}β. Hence αδ_{d}β. □

Definition 20. Let β be an ultrafilter of joint type and put

Γ_{β} = {Π_{b}(α): α is an ultrafilter of type b and αδ_{d}β}.

Theorem 21. Γ_{β} is a δ_{d}-clan, and is contained in a unique maximal δ_{d}- clan, namely Π_{d}(β).

Proof. First, Γ_{β} is a d-clan. Let S and T be sets in Γ_{β}. By definition of Γ_{β}, we can find ultrafilters α_{1} and α_{2} so that α_{1}δ_{d}β, α_{2}δ_{d}β and S ∈ α_{1}, T ∈ α_{2}. Suppose S δ̸_{d}T. Then there exist D_{1}, D_{2} ∈ D such that S ⊆ D_{1}, T ⊆ D_{2} and D_{1} ∩ D_{2} = ∅. Since α_{1} and α_{2} are ultrafilters, we must have D_{1} ∈ α_{1} and D_{2} ∈ α_{2}. Now α_{1} ∩ D ⊆ β ∩ D and α_{2} ∩ D ⊆ β ∩ D. Thus D_{1} and D_{2} are in β ∩ D, contradiction to D_{1} ∩ D_{2} = ∅. Clearly Γ_{β} ⊆ Π_{d}(β), which is a maximal d-clan. Suppose Π is a maximal d-clan with Γ_{β} ⊆ Π ≠ Π_{d}(β). Then there must an A ∈ Π and some G ∈ β for which A δ̸_{d}G. But Aδ_{d}B for all B ∈ Γ_{β}, and, because β ⊆ Π_{b}(β) ⊆ Γ_{β}, G ∈ Γ, contradiction. Thus Π_{d}(γ) is unique. □

Theorem 22. The following are clearly equivalent:

a) For each β of joint type, Γ_{β} = Π_{d}(β).

b) If α is of type b and β_{1}, β_{2} are of joint type such that β_{1}δ_{d}α and β_{2}δ_{d}α, then β_{1}δ_{d}β_{2}.

c) {Γ_{β}: β is of joint type} partitions T_{b}X.

Theorem 23. If any (hence, all) of the conditions of 22 are met, then there exists a continuous function f: T_{b}X → T_{d}X which is the identity on X.

Proof. Define f: T_{b}X → T_{d}X by the following: For Π_{b}(α) ∈ T_{b}X, there is some β of joint type such that αδ_{d}β. Let f[Π_{b}(α)] = Π_{d}(β). By 19, f is a well defined function.

To show that f is continuous, let A ⊆ T_{b}X and suppose σ_{0} ∈ T_{b}X with f(σ_{0}) ∉ cl_{d}f[A]. We shall show σ_{0} ∉ cl_{b}(A). Now

cl_{d}f[A] = ∩{D′: f[A] ⊆ D′}.

In particular, there is some D_{0} ∈ D with f[A] ⊆ D′_{0}, yet f(σ_{0}) ∉ D′_{0}. Now

D′_{0} = {Π_{d}(γ): D_{0} ∈ Π_{d}(γ)}.

So σ ∈ A implies f(σ) ∈ D′_{0}, whence D_{0} ∈ f(σ) and f^{←}(D_{0}) ∈ σ. Therefore σ ∈ [f^{←}(D_{0})]′. Hence A ⊆ [f^{←}(D_{0})]′, closed in T_{b}X, so cl_{b}(A) ⊆ [f^{←}(D_{0})]′. But f(σ_{0}) ∉ D′_{0}. Therefore D_{0} ∉ f(σ_{0}) and σ_{0} ∉ [f^{←}(D_{0})]′. □

References

[1] M. S. Gagrat and S. A. Naimpally, Proximity approach to extension problems, Fund. Math. 71 (1971), no. 1, 63–76. (errata insert). MR 45 #2653

[2] , Proximity approach to extension problems. II, Math. Japon. 16 (1971), 35–43. MR 46 #4483

[3] S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge University Press, London, 1970. MR 43 #3992

[4] Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.- London-Don Mills, Ont., 1970. MR 41 #9173

Department of Mathematics, The University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada

   E-mail address: rayburn@cc.umanitoba.ca