@ Applied General Topology c© Universidad Politécnica de Valencia Volume 4, No. 2, 2003 pp. 421–444 Bombay hypertopologies Giuseppe Di Maio, Enrico Meccariello and Somashekhar Naimpally Dedicated by the first two authors to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. Recently it was shown that, in a metric space, the upper Wijsman convergence can be topologized with the introduction of a new far-miss topology. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. It leads easily to the generalized or g-Wijsman topology on the hyperspace of any topological space with a compatible LO-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Further generalization involving a topological space with two compatible LO-proximities and a cobase results in a new hypertopology which we call the Bombay topology. The generalized locally finite Bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. We show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences. 2000 AMS Classification: 54B20, 54A10, 54C35, 54D30, 54E05, 54E15. Keywords: Hyperspace, Wijsman topology, ball topology, proximal ball topology, far-miss topology, hit-and-miss topology, Bombay topology, Vietoris topology, Fell topology, proximal locally finite topology, ∆-topology, proximal locally finite ∆-topology. 1. Introduction. The main purpose of this work is to give a unified treatment to the problem of comparing various hypertopologies with one another. We accomplish this by representing known hypertopologies as special cases of just one general hyper- topology the IL- locally finite Bombay hypertopology. We prove one result, with a simple proof, giving necessary and sufficient conditions for one upper 422 G. Di Maio, E. Meccariello and S. Naimpally Bombay topology to be coarser than another. We then show how this one re- sult includes, as special cases, many known results scattered throughout the li- terature. Moreover, our approach gives simple transparent proofs in compari- son with those in the original articles which involve intricate calculations. Hypertopologies, which were born during the early part of the last century, have proven to be useful in Continua Theory, Topologies on Spaces of Functions, Optimization, Convex Analysis, Game Theory, Differential Equations, Image Analysis and Fractal Geometry, etc. Two early discoveries where: (a) the Vietoris topology that can be defined in any topological space, and (b) the Hausdorff metric topology which is available only in a metric space. After the discovery of uniform spaces by Weil, the second one was generalized to Hausdorff-Bourbaki uniform topology. In a seminal paper, Michael [18] made a detailed study of above topologies in the middle of the last century. Then in 1966 Wijsman [25], in studying Convex Analysis, introduced a con- vergence for nets of closed subsets of a metric space (X,d) viz: An → A iff for each x ∈ X, d(x,An) → d(x,A). The lower part of this convergence is equivalent to convergence in the lower Vietoris topology. Attempts to topologize the upper part of the convergence were only partially successful until recently. The first attempt resulted in the ball topology ([1]) wherein a typical neighbourhood of a closed set consists of closed subsets that do not intersect a proper closed ball in the metric space. The next attempt led to proximal ball topology ([10]) wherein a typical neigh- bourhood of a closed set consists of closed sets that are far (w.r.t. the metric proximity) from a proper closed ball. This was more successful since it equalled the Wijsman topology in nice metric spaces, which included the normed linear spaces. But the topologization for all metric spaces defied attempts for the past 36 years. A major part of the literature on hyperspaces consists of comparisons of various hypertopologies with one another. In this, the Wijsman topologies play an important role as the building blocks of many other topologies ([2]). But they are not easy to handle as the Wijsman topology depends strongly on the metric d. For example, even two uniformly equivalent metrics can give rise to unequal Wijsman topologies and non-uniformly equivalent metrics can induce equal Wijman topologies! Moreover, it is not easy to compare objects of different kinds and the absence of topologization of Wijsman topology was a major hurdle. So the literature contains long and complicated proofs involving epsilonetics. Recently, one of us [20] discovered a simple way of topologizing the upper Wijsman convergence. In this approach, a typical neighbourhood of a closed set A in a metric space (X,d) consists of closed sets that do not intersect a proper closed ball B that is far from A in the metric proximity. With this new approach it is possible to give simple conceptual proofs of results involving comparisons of the Wijsman topologies among themselves and with others. This new way Bombay hypertopologies 423 of looking at upper Wijsman topology interprets those results in a simple and direct way. Moreover, it leads to a generalization of the Wijsman topology to the hyperspace of any topological space satisfying some simple conditions. When it was further scrutinized, it was found that the above generalization of the upper Wijsman topology can be further generalized to represent all known upper hypertopologies. In this way we arrived at the upper Bombay topology which can be defined in any topological space equipped with two compatible proximities and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Joining the upper Bombay topology to the lower one generated by a collection IL of locally finite families of open sets give rise to the IL-locally finite Bombay topology which includes all known hypertopologies as special cases. We thus achieved our goal of unification stated at the beginning. For references on hyperspaces up to 1993, we generally refer to [1], except when a specific reference is needed. For proximities see [13], [22] and [19]. 2. Preliminaries. Let (X,τ) denote a T1 space. For any E ⊂ X, clXE, intE and Ec stand for the closure, interior and complement of E in X, respectively. Let γ, γ1 and γ2 be compatible LO-proximities on X. We always assume γ1 ≤ γ2 (i.e. A6γ1B implies A6γ2B (see [22]) where 6γ stands for the negation of γ). We use the symbol γ0 to denote the fine LO-proximity on X given by Aγ0B iff clXA∩ clXB 6= ∅. (γ0 is called the Wallman proximity). In the case (X,τ) is Tychonoff, then: γ] denotes the fine EF-proximity on X given by A6γ]B iff they can be separated by a continuous function f : X → [0, 1]. (γ] is called the functionally indistinguishable proximity on X). If U is a compatible separated uniformity on X, then γ(U) denotes the EF-proximity on X given by Aγ(U)B iff U(A) ∩B 6= ∅ for each U ∈U. (γ(U) is called the uniform proximity induced by U). If (X,τ) is a metrizable space with metric d, then γ(d) is the EF-proxi- mity on X given by Aγ(d)B iff Dd(A,B) = inf{d(a,b) : a ∈ A,b ∈ B} = 0. (γ(d) is called the metric proximity induced by d). CL(X) (resp. K(X)) is the family of all nonempty closed subsets of X (resp. the family of all nonempty closed and compact subsets of X). ∆ is a nonempty subfamily of CL(X) which is closed under finite unions and which contains all singletons. We call ∆ a cobase ([23]). Let us note that in the literature ∆ is usually assumed to contain merely all singletons. 424 G. Di Maio, E. Meccariello and S. Naimpally In our view the above assumption simplifies the results. In fact, it allows to display transparent statements and makes theory much simpler. For any set E ⊂ X and a subfamily E ⊂ τ we use the following standard notation: E− = {A ∈ CL(X) : A∩E 6= ∅}; E− = {A ∈ CL(X) : A∩E 6= ∅ for each E ∈E}. Furthermore, if γ is a compatible proximity on X, we set E++γ = {A ∈ CL(X) : A �γ E, i.e. A6γEc}. Note that γ1 ≤ γ2 is equivalent to U++γ1 ⊂ U ++ γ2 for every U ∈ τ. We omit γ if it is clear from the context and write E++γ simply as E ++. Moreover E+ = {A ∈ CL(X) : A ⊂ E, i.e. A �γ0 E}. Now we describe some hypertopologies on CL(X). The upper proximal ∆-topology (w.r.t. γ) σ(γ; ∆)+ is generated by the basis {E++ : Ec ∈ ∆}. When γ = γ0 we have the upper ∆-topology τ(∆)+ = σ(γ0; ∆)+. The lower Vietoris (or finite) topology τ(V −) has a basis {E− : E ⊂ τ is finite}. The lower locally finite topology τ(LF−) has a basis {E− : E ⊂ τ is locally finite}. The IL-lower locally finite topology τ(IL−) has a basis {E− : E ⊂ IL} provided IL ⊂ {E ⊂ τ : E is locally finite} satisfies the following filter condition: (∗) whenever E, F ∈ IL, then there exists G ∈ IL such that G− ⊂E− ∩F− (see [17]). The proximal (finite) ∆-topology (w.r.t. γ) σ(γ; ∆) = σ(γ; ∆)+∨τ(V −). We omit γ if it is obvious from the context and write σ(∆) for σ(γ; ∆). The ∆-topology τ(∆) = τ(∆)+∨τ(V −) = σ(γ0; ∆)+∨τ(V −) = σ(γ0; ∆). The proximal locally finite ∆-topology (w.r.t. γ) σ(γ; LF, ∆) = σ(γ; ∆)+ ∨ τ(LF−). Bombay hypertopologies 425 The proximal IL-locally finite ∆-topology (w.r.t. γ) σ(γ; IL, ∆) = σ(γ; ∆)+ ∨ τ(IL−). We omit the prefix ”proximal” and replace σ by τ if γ = γ0. Well known special cases are: (a) when ∆ = CL(X), τ(∆) = τ(V ) the Vietoris or finite topology; σ(γ; ∆) = σ(γ) the proximal topology (w.r.t. γ); τ(LF∆) = τ(LF) the locally finite topology; σ(γ; LF, ∆) = σ(γ; LF) the proximal locally finite topology. (b) When ∆ = K(X), τ(∆) = τ(F) the Fell topology. (c) Let (X,d) be a metric space, γ(d) the metric proximity induced by d and ∆ denote the cobase B generated by all finite unions of closed balls of nonnegative radii. Then τ(∆) = τ(B) the ball topology; σ(γ(d); ∆) = σ(γ(d);B) = σ(B) the proximal ball topology. It was shown in [20] that a typical neighbourhood at A ∈ CL(X) in the upper Wijsman topology τ(Wd)+ is of the form: U+ where Uc ∈B and A6γ(d)Uc. Thus three parameters are involved: (i) γ(d) the metric proximity in A6γ(d)Uc, (ii) γ0 the fine LO-proximity in U+, and (iii) the cobase B which contains Uc. We note that there are two proximities, namely γ(d), γ0 with γ(d) ≤ γ0 and a cobase. By replacing the two proximal parameters γ(d), γ0 by two LO- proximities γ1, γ2 with γ1 ≤ γ2 and the cobase B by ∆ we have the following definition: Definition 2.1. Let (X,τ) be a T1 space with compatible proximities γ1, γ2 with γ1 ≤ γ2 and ∆ a cobase. Then a typical neighbourhood of A ∈ CL(X) in the upper Bombay topology σ(γ1,γ2; ∆)+ is: U++γ2 where U c ∈ ∆ and A6γ1Uc (or equivalently A ∈ U++γ1 ). (Note that since γ1 ≤ γ2 A6γ1Uc implies A6γ2Uc which in turn is equivalent to A ∈ U++γ2 ). γ1, γ2 and ∆ are the three parameters of the upper Bombay hypertopology: γ1, γ2 are the proximal parameters and ∆ is the cobase. 426 G. Di Maio, E. Meccariello and S. Naimpally Furthermore, σ(γ1; ∆)+ (respectively, σ(γ2; ∆)+) represents the first coordinate upper topology (the second coordinate upper topology). (i) The Bombay topology σ(γ1,γ2; ∆) is the join of the upper Bombay topo- logy σ(γ1,γ2; ∆)+ and the lower Vietoris topology τ(V −) i.e. σ(γ1,γ2; ∆) = σ(γ1,γ2; ∆)+ ∨ τ(V −). (ii) The locally finite Bombay topology σ(γ1,γ2; LF, ∆) = σ(γ1,γ2; ∆)+ ∨ τ(LF−). Let IL ⊂{E : E ∈ τ is locally finite} satisfy the following filter condition: (∗) whenever E, F ∈ IL, then there is G ∈ IL such that G− ⊂E− ∩F−. (iii) The IL-locally finite Bombay topology σ(γ1,γ2; IL, ∆) = σ(γ1,γ2; ∆)+ ∨ τ(IL−). Remark 2.2. (a) If in (iii) of above Definition each member of IL is finite, then σ(γ1,γ2; IL, ∆) equals σ(γ1,γ2; ∆), since τ(IL −) = τ(V −) (see also Lemma 3.2 below). (b) Again if in (iii) of Definition 2.1 we choose γ1 = γ2 = γ, then we have the proximal IL-locally finite ∆ topology σ(γ; IL, ∆) from which we obtain all classical hypertopologies defined above. (c) Let (X,d) be a metric space, ∆ = B the cobase generated by all closed balls, and γ = γ(d) the metric proximity induced by d. Then σ(γ,γ0; ∆) = σ(γ,γ0;B) = τ(Wd) is the Wijsman topology, i.e. the Bombay topology is a generalization of the Wijsman topology. In ad- dition, suppose (X,τ) is a T1 space, γ a compatible LO-proximity coarser than γ0 and ∆ a cobase, then we refer the upper-Bombay topology σ(γ,γ0; ∆)+ (the Bombay topology σ(γ,γ0; ∆)) as the up- per g-Wijsman topology (the g- Wijsman topology). Here g stands for generalized (w.r.t. ∆ and γ since γ0 is kept fixed). (d) Let (X,U) be a separated uniform space and γ = γ(U) the compati- ble EF-proximity induced by U. Let IL be the collection of all families of open sets of the form {U(x) : x ∈ Q ⊂ A}, where A ∈ CL(X), U ∈U and Q is U-discrete, i.e. x, y ∈ Q, x 6= y implies x 6∈ U(y). Then the Hausdorff Bourbaki or H-B uniform topology associated to U on CL(X) is τ(UH) = τ(UH)+∨τ(U−H) = σ(γ) +∨τ(IL−) (see [20]). More- over, if the uniformity U comes from a metric d we get the Hausdorff metric topology τ(Hd). Thus all Hausdorff-Bourbaki topologies are proximal locally finite. Bombay hypertopologies 427 (e) Many new hypertopologies can be defined by replacing the lower Vi- etoris topology τ(V −) by the lower IL-locally finite topology. Thus we have the IL-locally finite Fell topology, the IL-locally finite Wijsman topology, the IL-locally finite ball topology, the proximal IL-locally finite ∆-topology, etc. In addition, by a proper choice of the two proximal coordinates in a Bombay topology, one can get infinitely many new hypertopologies. 3. Principal results. In this section we plan to find necessary and sufficient conditions for one IL-locally finite Bombay topology to be coarser than another. It is well known that in the comparison of hit-and-miss hypertopologies, the lower and the upper parts play their roles separately. Hence, following Lemmas hold. Lemma 3.1. Let (X,τ) be a T1 space with compatible LO-proximities γ1, γ2, η1, η2 satisfying γ1 ≤ γ2 and η1 ≤ η2. Let ∆, Λ be cobases and IL1, IL2 two collections of locally finite families of open sets satisfying condition (∗). The following are equivalent: (a) σ(γ1,γ2; IL1, ∆) ≤ σ(η1,η2; IL2, Λ); (b) τ(IL−1 ) ≤ τ(IL − 2 ) and σ(γ1,γ2; ∆) + ≤ σ(η1,η2; Λ)+. Lemma 3.2. Let (X,τ) be a T1 space and IL, IL1, IL2 collections of locally finite families of open sets satisfying condition (∗). Then: τ(IL−1 ) ≤ τ(IL − 2 ) if and only if for each E ∈ IL1, there exists F ∈ IL2 such that F− ⊂E−. Thus if all members of IL2 are finite, then the same is true of the members of IL1 and hence τ(IL −) ≤ τ(V −) if and only if all members of IL are finite. Now, we turn our attention to the upper Bombay topologies and consider the principal result of this paper showing that it includes most of the results in the literature involving comparison of various hypertopologies. Next Definition and Lemma play a key role. Definition 3.3. Let (X,τ) be a T1 space and γ a compatible LO-proximity on X. If B ⊂ X, set γ(B) = {F ⊂ X : FγB} ([24]), i.e. γ(B) is the collection of all subsets F of X near to B w.r.t. γ. Lemma 3.4. Let (X,τ) be a T1 space, γ and η compatible LO-proximities on X and C and D nonempty closed subsets of X. The following are equivalent: (a) (Dc)++η ⊂ (Cc)++γ ; (b) C ⊂ D and γ(C) ⊂ η(D). Proof. (a) ⇒ (b). Assume not, then either i) C 6⊂ D or ii) γ(C) 6⊂ η(D). If i) occurs, then there exists c ∈ C \ D. Choose F ∈ (Dc)++η and consider 428 G. Di Maio, E. Meccariello and S. Naimpally F ′ = F ∪{c}. Then F ′ ∈ (Dc)++η but F ′ 6∈ (Cc)++γ ; a contradiction. If ii) occurs, then there exists an F ⊂ X such that FγC but F 6ηD. Since FγC, then F 6= ∅. Let E = clXF. Then E ∈ CL(X), EγC but E 6ηD. Therefore E ∈ (Dc)++η but E 6∈ (Cc)++γ ; a contradiction. (b) ⇒ (a). Assume not, then (Dc)++η 6⊂ (Cc)++γ . Hence there exists F ∈ CL(X) such that F 6ηD but FγC, i.e. F ∈ γ(C) but F 6∈ η(D); a contradiction. � Theorem 3.5. (Main Theorem) Let (X,τ) be a T1 space with compatible LO-proximities γ1, γ2, η1, η2 satisfying γ1 ≤ γ2 and η1 ≤ η2 and ∆ and Λ cobases. The following are equivalent: (a) σ(γ1,γ2; ∆)+ ≤ σ(η1,η2; Λ)+; (b) for each B ∈ ∆ and W ∈ τ, W 6= X, with B �γ1 W , there exists a B′ ∈ Λ such that: (i) B ⊂ B′ �η1 W , and (ii) γ2(B) ⊂ η2(B′). Proof. σ(γ1,γ2; ∆)+ ≤ σ(η1,η2; Λ)+ if and only if for each A ∈ CL(X), A 6= X, A ∈ U++γ2 ∈ σ(γ1,γ2; ∆) + - where Uc ∈ ∆ and A ∈ U++γ1 - there exists a V ∈ τ such that V c ∈ Λ, A ∈ V ++η1 and A ∈ V ++ η2 ⊂ U++γ2 . Noting that A ∈ V ++ η1 is equivalent to V c �η1 Ac and using Lemma 3.4 we have (i) and (ii) where W = Ac, B = Uc and B′ = V c. � Corollary 3.6. Let (X,τ) be a T1 topological space with compatible LO-proxi- mities γ1, γ2, η1, η2 satisfying γ1 ≤ γ2 and η1 ≤ η2, ∆ and Λ cobases and IL1 and IL2 collections of locally finite families of open sets satisfying condition (∗). The following are equivalent: (a) σ(γ1,γ2; IL1, ∆) ≤ σ(η1,η2; IL2, Λ); (b) IL2 refines IL1 and whenever B ∈ ∆, W ∈ τ, W 6= X, with B �γ1 W , then there exists a B′ ∈ Λ such that: (i) B ⊂ B′ �η1 W , and (ii) γ2(B) ⊂ η2(B′). Remark 3.7. (a) For future reference we note that FγB implies that there is a net in X whose range C ⊂ F and CγB (cf. Lemma 3.2 in [5]). (b) We note that if η2 ≤ γ2, then (ii) at (b) of the Main Theorem is auto- matically satisfied. (c) If γ1 is an EF-proximity, then A ∈ U++γ1 implies the existence of E ∈ CL(X) with E ∈ U++γ1 and A ∈ (intE) ++ γ1 . So setting W ′ = intE, (i) at (b) of the Main Theorem can be written as B ⊂ B′ �η1 W ′ �γ1 W . (d) The proximal topologies σ(γ1; ∆) and σ(γ2; ∆) are the proximal coordi- nate topologies of the given (finite) Bombay topology σ(γ1,γ2; ∆). Bombay hypertopologies 429 Let γ and η, with γ ≤ η, be compatible LO-proximities on a given topological space X and ∆ a cobase. In the motivation we saw that the (finite) Bombay topology σ(γ,η; ∆) is a generalization of the g-Wijsman topology σ(γ,γ0; ∆), which in turn, is a generalization of the Wijsman topology σ(γ(d),γ0;B). We conclude this section by deriving some results in the general case and we give appropriate references. We reserve the special cases: 1) (X,τ) is a metrizable space with metric d and the first proximal pa- rameter γ = γ(d) is the EF-proximity associated to d, and 2) (X,τ) is a Tychonoff space and the first proximal parameter γ is EF or (X,τ) is a uniformizable space with separated uniformity U and the first proximal parameter γ = γ(U) is the proximity induced by U, to next sections. The results given below are new and follow easily from the Main Theorem or from the definitions and so we omit the proofs. We start to compare a (finite) Bombay topology σ(γ1,γ2; ∆) with its proxi- mal coordinate topologies. Theorem 3.8. (cf. [1] Pages 45, 53) Let (X,τ) be a T1 space with compatible LO-proximities γ1, γ2, satisfying γ1 ≤ γ2 and ∆ a cobase. Then: (a) σ(γ1,γ2; ∆) ≤ σ(γ1; ∆); (b) σ(γ1,γ2; ∆) ≤ σ(γ2; ∆). Corollary 3.9. Let (X,τ) be a T1 space, γ a compatible LO-proximity and ∆ a cobase. Then: (a) σ(γ,γ0; ∆) ≤ σ(γ; ∆) (i.e. the g-Wijsman topology is coarser than its first proximal coordinate topology); (b) σ(γ,γ0; ∆) ≤ τ(∆) (i.e. the g-Wijsman topology is coarser than its second coordinate topology). Next Theorem and Corollary show when a Bombay topology σ(γ1,γ2; ∆) and a g-Wijsman topology σ(γ,γ0; ∆) are finer than their proximal coordinate topologies. Theorem 3.10. (cf. [1] Pages 45, 52, 53) Let (X,τ) be a T1 space with com- patible LO-proximities γ1, γ2 satisfying γ1 ≤ γ2 and ∆ a cobase. The following are equivalent: (a) σ(γ2; ∆) ≤ σ(γ1,γ2; ∆); (b) σ(γ1,γ2; ∆) = σ(γ1; ∆) = σ(γ2; ∆); 430 G. Di Maio, E. Meccariello and S. Naimpally (c) for each B ∈ ∆ and W ∈ τ, W 6= X, with B �γ2 W , there exists a B′ ∈ ∆ such that B ⊂ B′ �γ1 W ; (d) whenever A ∈ CL(X), B ∈ ∆ and A6γ2B, then A6γ1B. Corollary 3.11. Let (X,τ) be a T1 space, γ a compatible LO-proximity on X and ∆ a cobase. The following are equivalent: (a) τ(∆) ≤ σ(γ,γ0; ∆); (b) σ(γ,γ0; ∆) = σ(γ; ∆) = τ(∆); (c) for each B ∈ ∆ and W ∈ τ, W 6= X, with B ⊂ W , there exists a B′ ∈ ∆ such that B ⊂ B′ �γ W ; (d) whenever A ∈ CL(X), B ∈ ∆ and A∩B = ∅, then A6γB. Now, we compare two proximal hit-and-miss hypertopologies associated to the same cobase ∆. Theorem 3.12. (cf. [1] Page 53 and Theorem 3.3 in [5]) Let (X,τ) be a T1 space with compatible LO-proximities γ1, γ2 satisfying γ1 ≤ γ2 and ∆ a cobase. The following are equivalent: (a) σ(γ1; ∆) ≤ σ(γ2; ∆); (b) for each B ∈ ∆ and W ∈ τ, W 6= X, with B �γ1 W , there exists a B′ ∈ ∆ such that B ⊂ B′ �γ2 W and γ1(B) ⊂ γ2(B′); (c) whenever A ∈ CL(X), B ∈ ∆ and A6γ1B, then there exists a B′ ∈ ∆ such that i) B ⊂ B′, A6γ2B′ and ii) whenever C is the range of a net satisfying C ⊂ (B′)c and Cγ1B then Cγ2B′. Corollary 3.13. Let (X,τ) be a T1 space, γ a compatible LO-proximity on X and ∆ a cobase. The following are equivalent: (a) σ(γ; ∆) ≤ τ(∆); (b) for each B ∈ ∆ and W ∈ τ, W 6= X, with B �γ W , there exists a B′ ∈ ∆ such that B ⊂ B′ �γ0 W and γ(B) ⊂ γ0(B′); (c) whenever A ∈ CL(X), B ∈ ∆ and A6γB, then there exists a B′ ∈ ∆ such that i) B ⊂ B′, A∩B′ = ∅ and ii) whenever {xλ : λ ∈ Σ} is a net whose range C is contained in (B′)c and CγB, then {xλ : λ ∈ Σ} has a cluster point. Theorem 3.14. (cf. [1] Page 53 and Theorem 3.11 in [5]) Let (X,τ) be a T1 space with compatible LO-proximities γ1, γ2 satisfying γ1 ≤ γ2 and ∆ a cobase. The following are equivalent: (a) σ(γ2; ∆) ≤ σ(γ1; ∆); (b) σ(γ2; ∆) = σ(γ1; ∆); (c) for each B ∈ ∆ and W ∈ τ, W 6= X, with B �γ2 W , there exists a B′ ∈ ∆ such that B ⊂ B′ �γ1 W ; (d) whenever A ∈ CL(X), B ∈ ∆ and A6γ2B, then A6γ1B. Corollary 3.15. Let (X,τ) be a T1 space, γ a compatible LO-proximity on X and ∆ a cobase. The following are equivalent: Bombay hypertopologies 431 (a) τ(∆) ≤ σ(γ; ∆); (b) τ(∆) = σ(γ; ∆); (c) for each B ∈ ∆ and W ∈ τ, W 6= X, with B ⊂ W , there exists a B′ ∈ ∆ such that B ⊂ B′ �γ W ; (d) whenever A ∈ CL(X), B ∈ ∆ and A∩B = ∅, then A6γB. Next Theorem compares two Bombay topologies which have the same second proximity parameter but different cobases. Theorem 3.16. (cf. [1] Page 39) Let (X,τ) be a T1 space and γ, δ and η compatible LO-proximities on X such that γ ≤ η as well as δ ≤ η. Let ∆ and Λ be cobases. The following are equivalent: (a) σ(γ,η; ∆) ≤ σ(δ,η; Λ); (b) for each B ∈ ∆ and W ∈ τ, W 6= X, with B �γ W , there exists a B′ ∈ Λ such that B ⊂ B′ �δ W . Corollary 3.17. Let (X,τ) be a T1 space, γ and δ compatible LO-proximities on X and ∆ and Λ cobases. The following are equivalent: (a) σ(γ,γ0; ∆) ≤ σ(δ,γ0; Λ); (b) for each B ∈ ∆ and W ∈ τ, W 6= X, with B �γ W , there exists a B′ ∈ Λ such that B ⊂ B′ �δ W . Corollary 3.18. Let (X,τ) be a T1 space, ∆ and Λ cobases. The following are equivalent: (a) τ(∆) ≤ τ(Λ); (b) for each B ∈ ∆ and W ∈ τ, W 6= X, with B ⊂ W , there exists a B′ ∈ Λ such that B ⊂ B′ ⊂ W . 4. The metric case. A large part of the literature is in the setting of a metric space. Let (X,τ) be a metrizable space with metric d, γ = γ(d) the d-metric proximity, B(d) the family of finite unions of all d-closed balls and TB(d) denote the family of all closed d-totally bounded subsets of X (we omit d if it is obvious from the context and write respectively B and TB for B(d) and TB(d)). Remark 4.1. Let (X,τ) be a metrizable space with metric d and γ = γ(d) the d-metric proximity. Let α, δ and ε be positive reals with ε < δ < α and Sd(x,ε) and Bd(x,ε) be the open and the closed d-balls centered at x of radius ε. We omit the subscript d if it is clear from the context. Then: (a) If X is a metrizable space with metric d, then TB = TB(d) is a cobase and it is even hereditarily closed. (b) Whenever α, δ and ε are positive reals with ε < δ < α, then S(x,ε) ⊂ B(x,ε) �γ S(x,δ) ⊂ B(x,δ) �γ S(x,α) ⊂ B(x,α). 432 G. Di Maio, E. Meccariello and S. Naimpally (c) A set D is said to be strictly d-included in E (D ⊂⊂d E) iff there is a finite set of points {x1, · · · ,xn} of E and positive reals εk < αk, k = 1, · · · ,n, such that (SDI) D ⊂ n⋃ k=1 S(xk,εk) ⊂ n⋃ k=1 S(xk,αk) ⊂E ([1] Page 38). So, using (b) it can be shown that open balls can be replaced by closed balls in (SDI) as well as (SDI) is in turn equivalent to D ⊂ n⋃ k=1 S(xk,εk) �γ n⋃ k=1 S(xk,αk) ⊂E. (d) If A is a nonempty subset of X, B′ is a finite unions of balls, W ∈ τ and A ⊂ B′ �γ W , then there exists a finite union of balls B′′ such that A ⊂ B′ ⊂⊂d B′′ �γ W and thus by above A �γ B′′ �γ W . Indeed, if B′ �γ W , then D(B′,Wc) = inf{d(b,y) : b ∈ B′,y ∈ Wc} = ε > 0. Since B′ = n⋃ k=1 S(xk,εk), select t = ε 2 and set B′′ = n⋃ k=1 S(xk,εk + t). Thus whenever A ∈ CL(X) and W ∈ τ with W 6= X, the following are equivalent: (1) there exists a B′ ∈B such that A ⊂ B′ �γ W ; (2) A ⊂⊂d W ; (3) there exists a B′′ ∈B such that A �γ B′′ �γ W . (e) We note that from Remark 3.7 (a) if C and C′ are nonempty subsets of X with C ⊂ C′, then the inclusion γ(C) ⊂ γ0(C′) occurs if and only if whenever there is a sequence of points {xn : n ∈ IN} in (C′)c with limn→∞d(xn,C) = 0, then the sequence {xn : n ∈ IN} has a cluster point (see [1] Page 52). (f) Let X be a metrizable space with compatible metrics d and e inducing the metric proximities γ = γ(d) and η = η(e), respectively. Let ∆ = B = B(d) and Λ = B′ = B(e). The following are equivalent: (1) τ(Wd) ≤ τ(We); (2) for each B ∈ B and W ∈ τ with B �γ W there exists a B′ ∈ B′ such that B ⊂ B′ �η W ; (3) for each B ∈ B and W ∈ τ with B �γ W there exists a B′′ ∈ B′ such that B ⊂⊂e B′′ ⊂⊂e W (cf. [3], [15]); (4) each proper open d-sphere is strictly e-included in every its open enlargement (cf. Theorem 2.1.10 in [1]). Bombay hypertopologies 433 (g) From (d) we have that strict d-inclusion is equivalent to weak total boun- dedness. We recall that a closed subset E of X is said to be weakly totally bounded or w-TB in a open set W iff there exists a B ∈B such that E �γ B �γ W (see (16.1) in [10]). Moreover: (g1) In any infinite dimensional Banach space, the closed unit ball is not totally bounded but it is weakly totally bounded in any open ball centered at 0 and radius greater than 1; (g2) let l2 be the Hilbert space of square summable sequences, θ the origin and {en : n ∈ IN} the standard orthonormal base for l2. Let (X,d) be the metric subspace of l2 where X = {θ}∪{en : n ∈ IN}. Then {e2n+1 : n ∈ IN} is not w-TB in X \{e2n : n ∈ IN}. Theorem 4.2. (cf. 3.9) Let (X,d) be a metric space, γ = γ(d) the metric proximity induced by d and B the cobase generated by all closed d-balls. Then: (a) τ(Wd) ≤ σ(B); (b) τ(Wd) ≤ τ(B). Theorem 4.3. (cf. 3.11) Let (X,d) be a metric space, γ = γ(d) the metric proximity induced by d and B the cobase generated by all closed d-balls. In the following (a), (b), (c) and (d) are equivalent and each implies (e) which is equivalent to (f). (a) τ(B) ≤ τ(Wd); (b) τ(Wd) = σ(B) = τ(B); (c) for each B ∈B and W ∈ τ, W 6= X, with B ⊂ W implies B ⊂⊂d W ; (d) X is ball Atsuji (i.e. disjoint closed sets, one of which is a ball, are far). (e) σ(B) ≤ τ(Wd); (f) for each positive real ε and each B ∈ B, B is strictly d-included in its ε-enlargement, i.e. B ⊂⊂d S(B,ε). Theorem 4.4. (cf. 3.13) Let (X,d) be a metric space, γ = γ(d) the metric proximity induced by d and B the cobase generated by all closed d-balls. The followings are equivalent: (a) σ(γ;B) ≤ τ(B); (b) for each B ∈ B and W ∈ τ, W 6= X, with B �γ W , there exists a B′ ∈B such that B ⊂ B′ ⊂ W and γ(B) ⊂ γ0(B′); (c) for each B ∈ B and W ∈ τ, W 6= X, with B �γ W , there exists a B′ ∈B such that B ⊂ B′ ⊂ W and for each sequence {xn : n ∈ IN} in (B′)c with lim n→∞ d(xn,B) = 0 has a cluster point. Theorem 4.5. (cf. Theorem 3.1 in [8]) Let X be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d and ∆ a cobase. The following are equivalent: (a) τ(Wd) ≤ τ(∆); 434 G. Di Maio, E. Meccariello and S. Naimpally (b) for every x ∈ X and every reals α, β with 0 < α < β and S(x,β) 6= X, there exists a B′ ∈ ∆ such that S(x,α) ⊂ B′ �γ0 S(x,β); (c) τ(Wd) ≤ σ(γ; ∆). Lemma 4.6. Let X be a metrizable space with metric d and A ∈ CL(X). The following are equivalent: (a) A is totally bounded; (b) for each positive real ε and each closed subset E of A, E ⊂⊂d S(E,ε); (c) for each positive real ε and each closed subspace E of A, E is w-TB in S(E,ε) (i.e. there exists B ∈B such that E � B � S(E,ε)). Proof. It suffices to show that (c) ⇒ (a), since (a) ⇒ (b) and (b) ⇒ (c) follow from the definition of total boundedness and Remark 4.1 (d) respectively. Suppose (c) holds but not (a). Then, without any loss of generality, we may suppose that A = {an : n ∈ IN} is bounded and there is an ε > 0 such that d(ap,aq) > ε for all p 6= q. Now we use the ”milky way ” technique. Set a1,n = an for all n ∈ IN. Since {a1,n : n ∈ IN} is bounded, it has a subsequence {a2,n : n ∈ IN} such that d(a1,1,a2,n) → α1 ≥ ε. Inductively, for each r ∈ IN, {ar,n : n ∈ IN} has a subsequence {ar+1,n : n ∈ IN} such that d(ar,r,ar+1,n) → αr ≥ ε. We note that for all infinite subsets D ⊂{as,n : s > r,n ∈ IN} d(ar,r,D) = αr. Set E = {a2r,2r : r ∈ IN} and η = inf{αr : r ∈ IN}. Clearly, (∗) 0 < ε ≤ η < +∞. We claim that E is not w-TB in S(E,η). For if not, there is a p ∈ IN and a t > 0 such that S(a2p,2p; t) � S(E,η) and S(a2p,2p; t) contains an infinite subset F ⊂{a2n,2n : n ∈ IN}. From what we have done so far, it follows that t < η and d(a2p,2p,F) = α2p ≤ t. So, α2p ≤ t < η ≤ inf{αr : r ∈ IN} which contradicts (∗). � Corollary 4.7. Let X a metrizable space with metric d and B ∈ CL(X). The following are equivalent: (a) B is compact; (b) for each closed subset E of B and each W ∈ τ with E ⊂ W , E ⊂⊂d W ; (c) for each closed subset E of B and each W ∈ τ with E ⊂ W , E is w-TB in W. Proof. (a) ⇒ (b) is clear because since γ is an EF-proximity and B is compact, then B ⊂ W is equivalent to B �γ W . Hence apply (d) in Remark 4.1. Again, (b) ⇔ (c) follows from (d) in Remark 4.1. (c) ⇒ (a) It suffices to prove that if every closed subset of B is w-TB in a larger open set, then every sequence {xn : n ∈ IN} in B has a convergent subsequence. By above Lemma {xn : n ∈ IN} has a Cauchy subsequence {yk : k ∈ IN}. If {yk : k ∈ IN} does not converge, then the closed set {y2k+1 : k ∈ IN} is not w-TB in the open set X \{y2k : k ∈ IN}. � From (d) in Remark 4.1, Main Theorem, Lemma 4.6 and Corollary 4.7 we have: Bombay hypertopologies 435 Theorem 4.8. (cf. Proposition 4.1 in [8]) Let X be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d, B = B(d) the family of finite unions of all closed d-balls and ∆ a cobase. The following are equivalent: (a) σ(γ; ∆) ≤ τ(Wd); (b) for every B ∈ ∆ \{X} and every positive real ε there exists a B′ ∈ B such that B ⊂ B′ �γ S(B,ε); (c) for every B ∈ ∆ \{X} and every positive real ε, B ⊂⊂d S(B,ε). Thus if ∆ is hereditarily closed, then σ(γ; ∆) ≤ τ(Wd) if and only if every B ∈ ∆ is totally bounded. Corollary 4.9. (cf. Theorem (5.5) in [2]) Let (X,d) be a metric space and γ the metric proximity induced by d. The following are equivalent: (a) τ(Wd) = σ(γ); (b) for each positive real ε and each proper closed set E, E ⊂⊂d S(E,ε); (c) for each positive real ε and each closed subspace B of X, B is w-TB in S(B,ε); (d) X is totally bounded. Theorem 4.10. (cf. Proposition 4.2 in [8]) Let X be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d, B = B(d) the family of finite unions of all closed d-balls and ∆ a cobase. The following are equivalent: (a) τ(∆) ≤ τ(Wd); (b) for every B ∈ ∆ and every W ∈ τ, W 6= X, with B ⊂ W , there exists a B′ ∈B such that B ⊂ B′ �γ W ; (c) whenever B ∈ ∆ and W ∈ τ, W 6= X, with B ⊂ W , then B ⊂⊂d W . Thus if ∆ is hereditarily closed, then τ(∆) ≤ τ(Wd) if and only if every B ∈ ∆ is compact. Corollary 4.11. (cf. Corollary (5.6) in [2]) Let (X,d) be a metric space and γ the metric proximity induced by d. The following are equivalent: (a) τ(Wd) = τ(V ); (b) whenever E is a proper closed subset of X and W ∈ τ is such that E ⊂ W , then E ⊂⊂d W ; (c) for each closed subspace B of X and each W ∈ τ with B ⊂ W , B is w-TB in W ; (d) X is compact. Theorem 4.12. (cf. Proposition 3.1 in [7]) Let X be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d, TB = TB(d) the cobase of all closed d-totally bounded set of X. The following are equivalent: (a) τ(TB) ≤ τ(Wd); (b) X is complete. Proof. The Main Theorem shows that the range of any subsequence of a Cauchy sequence (which is totally bounded) is far from every disjoint closed set. � 436 G. Di Maio, E. Meccariello and S. Naimpally Theorem 4.13. Let X be a metrizable space and γ = γ(d) the metric proxi- mity induced by d. The following are equivalent: (a) τ(V ) = σ(γ); (b) γ = γ0; (c) X is Atsuji. Proof. Use the Main Theorem noting that τ(V ) = σ(γ0) = σ(γ) if and only if τ(V )+ = σ(γ0)+ = σ(γ)+. � 5. The Tychonoff case. As we noted above, quite a bit of the literature on hyperspaces, especially that concerning the Wijsman topology, is based on metric spaces. Here we show that many results are valid in (generalized) uniform spaces (see [6], [10], [11] and [5]). In this section (X,τ) denotes a Tychonoff space, γ a compatible EF-proximity, whereas γ0 and γ] denote respectively the Wallman (the fine LO-) proximity and the functionally indistinguishable (the fine EF-) proximity on X (see Pre- liminaries). It is a well known fact that γ] = γ0 if and only if X is normal (Urysohn’s Lemma). Π(γ) denotes the family of all uniformities U compatible with γ, U? is the coarsest totally bounded member of Π(γ) and U] the fine uniformity. Π(τ) denotes the family of all uniformities U compatible with τ. It is a well kown fact that Π(γ) ⊂ Π(τ). Without loss of generality we may assume that each U ∈ Π(τ) consists of members which are symmetric and closed. For each U ∈ Π(τ), γ(U) is the uniform proximity induced by U, whereas B(U) denotes the collection of finite unions of (generalized) closed balls w.r.t. U, where for each x ∈ X and U ∈ U, U[x] is the (generalized) U ball in X centered at x. TB(U) denotes the collection of all closed U-totally bounded subsets of X. The result below cannot be extended to LO-proximity spaces (see Lemma 6.1 in [5]). Theorem 5.1. (cf. Theorem 2.7 in [11], [1] Page 50) Let (X,τ) be a Tychonoff space and γ a compatible EF -proximity on X. Then sup{σ(γ,γ0;B(U)) : U ∈ Π(γ)} = σ(γ). Proof. It suffices to prove σ(γ) ≤ sup{σ(γ,γ0;B(U)) : U ∈ Π(γ)}. Suppose A ∈ W ++γ ∈ σ(γ), W ∈ τ, W 6= X and A ∈ CL(X). Then there are a closed entourage U ∈ U? and a finite subset F of X such that A ⊂ U[F] ⊂ U2[F] �γ W . Thus σ(γ) ≤ σ(γ,γ0;B(U?)) and hence the claim holds. � Corollary 5.2. (cf. [11]) Let (X,τ) be a Tychonoff space. Then sup{σ(γ,γ0;B(U)) : U ∈ Π(τ)} = σ(γ]). Bombay hypertopologies 437 Corollary 5.3. (cf. [21]) Let (X,τ) be a Tychonoff space. The following are equivalent: (a) sup{σ(γ,γ0;B(U)) : U ∈ Π(τ)} = τ(V ); (b) X is normal; (c) γ] = γ0. Corollary 5.4. (cf. [2]) Let X be a metrizable space with metric d and γ = γ(d) the metric proximity induced by d. Then: (a) sup{τ(We): e varies in the set of all metrics uniformly equivalent to d} = σ(γ); (b) sup{τ(We): e varies in the set of all metrics topologically equivalent to d} = τ(V ). Theorem 5.5. (cf. [7] and Lemma 6.12 in [5]) Let (X,τ) be a Tychonoff space, γ a compatible EF -proximity on X, U ∈ Π(γ) and TB(U) the family of all totally bounded closed subsets of X w.r.t. U. Then σ(γ; TB(U)) ≤ σ(γ,γ0;B(U)). Proof. Let B ∈ TB(U) and W ∈ τ with B �γ W 6= X. Then there are an entourage U ∈ U and a finite set F ⊂ X such that B ⊂ U[F] ⊂ U2[F] ⊂ W . Observe that U[F] ∈ B(U) and U[F] ⊂ U2[B] ⊂ W implies U[F] �γ W . The result follows from the Main Theorem. � Let U be a separated uniformity on X and U ∈U. We recall that that U′ ∈U is composably contained in U iff there is a U′′ ∈U such that U′ ◦U′′ ⊂ U (see Definition (3.3) in [10]). Next result generalizes Theorem 3.1 in [8] to Tychonoff spaces. Theorem 5.6. (cf. Theorem 3.1 in [8]) Let (X,τ) be a Tychonoff space, U ∈ Π(τ), the compatible EF -proximity γ = γ(U) and ∆ a cobase. The following are equivalent: (a) σ(γ,γ0;B(U)) ≤ τ(∆); (b) for each x ∈ X, for each U ∈ U with U[x] 6= X and for each U′ ∈ U composably contained in U there is a B′ ∈ ∆ such that U′[x] ⊂ B′ �γ U[x]; (c) σ(γ,γ0;B(U)) ≤ σ(γ; ∆). Proof. It suffices to use the Main Theorem observing that if U′ ∈U is compo- sably contained in U ∈U, then U′[x] �γ U[x] for each x ∈ X. � Using the notion of composably contained we have the next definition which extends to uniform setting the concept of strict inclusion. Definition 5.7. (cf. [3], [15] and [1] Page 38) Let (X,τ) a Tychonoff space, U ∈ Π(τ) and D, E ⊂ X. We say that D is strictly U-included in E (D ⊂⊂U E) iff there exist a finite set F ⊂ D and entourages U, U′ ∈U with U′ composably contained in U such that 438 G. Di Maio, E. Meccariello and S. Naimpally (?) D ⊂ U′[F] ⊂ U[F] ⊂ E. Remark 5.8. If γ = γ(U), then in above definition condition (?) is equivalent to the following one (?′) D ⊂ U′[F] �γ U[F] ⊂ E. Thus whenever γ is a compatible EF-proximity on X and U ∈ Π(γ), a set D is stricly U-included in E iff there exist a finite subset F and entourages U, U′ ∈U such that D ⊂ U′[F] �γ U[F] ⊂ E. We also note that if U is the metric uniformity induced by d, then for each U ∈U there exists some positive ε such that U = {(x,y) ∈ X×X : d(x,y) ≤ ε}. Thus U[x] = B(x,ε) for each x ∈ X and D ⊂⊂U E is equivalent to D ⊂⊂d E (cf. (c) in Remark 4.1). The next result generalizes Theorems 4.1 and 4.2 in [8] to the EF-proximities setting. We omit the proof since it is easily derived from definitions, the Main Theorem and the above Remark. Theorem 5.9. Let (X,τ) be a Tychonoff space, γ a compatible EF -proximity on X, U ∈ Π(γ) and ∆ a cobase. In the following (a) and (b) are equivalent and each implies (c) which is equivalent to (d). (a) τ(∆) ≤ σ(γ,γ0;B(U)); (b) whenever B ∈ ∆ \{X} and W ∈ τ with B ⊂ W , then B ⊂⊂U W . (c) for every B ∈ ∆ \{X} and U ∈U, B ⊂⊂U U[B]; (d) σ(γ; ∆) ≤ σ(γ,γ0;B(U)). Corollary 5.10. (cf. 3.11) Let (X,τ) be a Tychonoff space, γ a compatible EF -proximity on X and U ∈ Π(γ). In the following (a), (b), (c), (d) and (e) are equivalent and each implies (f) which is equivalent to (g). (a) τ(B(U)) ≤ σ(γ,γ0;B(U)); (b) σ(γ,γ0;B(U)) = τ(B(U)) = σ(γ;B(U)); (c) whenever B ∈B(U) and W ∈ τ with B ⊂ W , then B ⊂⊂U W ; (d) for every B ∈ B(U) and W ∈ τ, W 6= X, with B ⊂ W , there exists a B′ ∈B(U) such that B ⊂ B′ �γ W ; (e) X is B(U)-Atsuji space w.r.t. γ (i.e. disjoint closed sets, one of which is a member of B(U), are far w.r.t. γ). (f) σ(γ;B(U)) ≤ σ(γ,γ0;B(U)); (g) for every B ∈B(U) and U ∈U, B ⊂⊂U U[B]. We end this section with the following results derived from (d) in Remark 2.2 and Lemma 3.2. Theorem 5.11. (cf. [6]) Let (X,τ) be a Tychonoff space, γ a compatible EF -proximity on X and U ∈ Π(γ). Then the proximal topology σ(γ) and the Hausdorff-Bourbaki topology τ(UH) are equal if and only if the uniformity U equals the coarsest member U? ∈ Π(γ), and thus U is totally bounded. Bombay hypertopologies 439 Proof. The result follows from the fact that U is totally bounded iff maximal U-discrete sets are finite for all U ∈U. � Corollary 5.12. (cf. [2]) Let (X,τ) be a Tychonoff space, γ a compatible EF -proximity on X and U ∈ Π(γ). The following are equivalent: (a) σ(γ,γ0;B(U)) = τ(UH); (b) U is totally bounded. 6. Bounded Hypertopologies. In the literature on hyperspaces of a metric space (X,d), there are sev- eral bounded hypertopologies such as (i) the bounded Vietoris τ(bV ), (ii) the bounded proximal σ(bγ) w.r.t. γ, (iii) the bounded Hausdorff metric topology (or the Attouch-Wets topology) τ(AWd), etc. In this section, with the help of a slight generalization of the concept of abstract boundedness due to Hu ([16], also see [12] and [22]) we show that all bounded hypertopologies can be subsumed under the locally finite Bombay topologies. Definition 6.1. A nonempty collection ∆ ⊂ CL(X) is called a boundedness in a topological space X iff ∆ is closed under finite unions and is closed hereditary. We also assume that ∆ contains the singletons. Remark 6.2. (a) The followings are examples of boundedness: (i) B(X) the family of all closed d-bounded subsets of a metric space (X,d). This can be extended easily to a uniform space (X,U) by declaring a closed set A is U-bounded iff there is a finite set {x1, · · · ,xn}⊂ X and entourages Uk ∈U\{X×X}, k = 1, · · · ,n, such that A ⊂ n⋃ k=1 Uk[xk] or using notation of Section 4, there is a B ∈B(U) such that A ⊂ B. Let ∆ = B(U) denote the family of all U- bounded subsets of X and γ = γ(U) the EF-proximity induced by U. (ii) The family K(X) of all nonempty compact subsets of a Hausdorff topological space. (b) Let (X,U) be a Hausdorff uniform space, ∆ a boundedness and γ = γ(U). Then the ∆-bounded Hausdorff Bourbaki filter (or ∆-Attouch- Wets filter) U∆ is generated by sets of the form: [B,U] = {(A1,A2) ∈ CL(X) ×CL(X) : A1 ∩B ⊂ U[A2] and A2 ∩B ⊂ U[A2]}, where B ∈ ∆ and U ∈U. 440 G. Di Maio, E. Meccariello and S. Naimpally The associated topology τ(U∆) is the ∆-bounded Hausdorff topology (or the ∆-Attouch-Wets topology). We note that if X ∈ ∆ we get the Hausdorff Bourbaki-uniformity. If U is induced by a metric d and ∆ = TB(d), then the τ(U∆) is the bounded Hausdorff topology (i.e. the Attouch-Wets topology) τ(AWd). It was shown in [20] that if IL = ILU,∆, then the family of sets {U[x] : x ∈Q} where Q is a maximal discrete subset of B ∈ ∆ for U ∈U describes τ(ILU,∆). Thus τ(U∆) = τ(IL−U,∆) ∨σ(δ; ∆) +. It is now obvious that ∆-bounded hypertopologies are a part and parcel of ∆-Bombay topologies and the usual metric d-boundedness is also included in our definition of ∆, and by above it follows that: Theorem 6.3. The ∆-bounded Hausdorff topology (i.e. the ∆-Attouch-Wets topology) τ(U∆) and the bounded Hausdorff topology (i.e. the Attouch-Wets topology) τ(AWd) are proximal IL-locally finite. Moreover, using the Main Theorem and the decomposition property of the locally finite Bombay topology we have: Theorem 6.4. Let (X,τ) be a T1 topological space with compatible LO-proximities γ1, γ2, η1, η2 satisfyng γ1 ≤ γ2 and η1 ≤ η2, ∆, Λ two boundedness and IL1 and IL2 locally finite collections of open subsets of X. The following are equivalent: (a) σ(γ1,γ2; IL1, ∆) ≤ σ(η1,η2; IL2, Λ); (b) IL2 refines IL1 and whenever B ∈ ∆, W ∈ τ, W 6= X, with B �γ1 W , then there exists a B′ ∈ Λ such that: (i) B ⊂ B′ �η1 W , and (ii) γ2(B) ⊂ η2(B′). Let (X,d) be a metric space and as usual γ = γ(d) the metric proximity, U the separated uniformity associated to d, B the cobase generated by all closed d-balls and B(X) the family of all closed d-bounded subsets of X. We have: (see [1] Page 115) (1) the bounded proximal topology σ(δ; B(X)) = τ(V −) ∨σ(δ; B(X))+; (2) the dual bounded proximal topology σ(γ; IL,B) = τ(IL−U,B)∨σ(γ,γ0;B) +, which is clearly a IL-locally finite bounded Wijsman topology. We point out that from above Remark (b) if in (2) if we replace B with B(X), then σ(γ; IL,B(X)) = τ(IL−U,B(X)) ∨σ(γ,γ0; B(X)) +. Bombay hypertopologies 441 Thus from Theorem 6.4 and above Remark the following results are obvious and certainly they have generalizations: Theorem 6.5. Let X be a metrizable space with metric d, γ = γ(d) the metric proximity induced by d, B the cobase generated by all proper closed d balls and B(X) the family of all closed d bounded subsets of X. Then: (a) σ(γ,γ0;B) ≤ σ(γ;B) ≤ σ(γ; B(X)) ≤ σ(γ); (b) σ(γ,γ0;B) ≤ σ(γ;B) ≤ σ(γ; B(X)) ≤ τ(AWd). Theorem 6.6. (cf. [1], Page 93) Let X be a metrizable space, d and e compa- tible metrics, γ and η the metric proximities induced by d and e respectively, B(X) the family of all closed d-bounded sets and B′(X) the family of all closed e-bounded sets. The following are equivalent: (a) τ(AWd) = τ(AWe); (b) δ = δ′, B(X) = B′(X) (i.e. every closed d-bounded subset is e- bounded and vice versa) and they have the same uniformly continuous functions on bounded sets. Theorem 6.7. Let X be a metrizable space with metric d, γ = γ(d). As usual, let B(X) and TB = TB(d) denote respectively the family of all closed d-bounded sets and the family of all closed d-totally bounded sets. The following are equivalent: (a) B(X) ⊂ TB; (b) τ(AWd) = τ(Wd); (c) τ(AWd) = σ(γ; B(X)). 7. Uniformizable Bombay hypertopologies. Let (X,τ) be a T1 space. This section is devoted to the characterization of the uniformizable Bombay topologies associated with a cobase ∆ and compa- tible LO-proximities γ, η satysfying γ < η. We do not consider the case γ = η because from (a) and (b) in Remark 2.2 we get the proximal ∆ topologies σ(γ; ∆) (w.r.t. γ) and the uniformizable proximal ∆ topologies σ(γ; ∆) have been treated in [1], [6] and [9] for example. Furthermore, we omit the proofs as they are similar to those in [9]. First we need the following definitions. Definition 7.1. Let (X,τ) be a T1 space, γ a compatible LO-proximity and ∆ ⊂ CL(X). (a) ∆ is called γ-Urysohn iff whenever D ∈ ∆ and A ∈ CL(X) are far w.r.t. γ, there exists an S ∈ ∆ such that D �γ S �γ Ac (see also [4]) (b) ∆ is called Urysohn iff (a) above is true w.r.t. the LO-proximity γ0, i.e. whenever D ∈ ∆ and A ∈ CL(X) are disjoint, there exists an S ∈ ∆ such that D ⊂ intS ⊂ S ⊂ Ac. 442 G. Di Maio, E. Meccariello and S. Naimpally Lemma 7.2. (cf. Theorem 1.6 in [9]) Let X be a T1 space with compatible LO-proximities γ, η satysfying γ < η and ∆ a cobase. If ∆ is γ-Urysohn, then the relation π defined on the power set of X by: (?) A6πB iff clA6γclB and either clA ∈ ∆ and clB �η (clA)c or clB ∈ ∆ and clA �η (clB)c is a compatible EF -proximity on X. Moreover, π ≤ γ and ∆ is γ-Urysohn iff ∆ is π-Urysohn. Remark 7.3. (a) Observe that even if the starting proximities γ and η are just LO, the new compatible proximity π is always EF. Thus the base space X is automatically completely regular. Note that we have a procedure that allows us to construct an EF-proximity on a Tychonoff space X by using as seeds LO-proximities γ, η with γ < η and a cobase ∆ which is γ-Urysohn. (b) Let X be an infinite set, τ the cofinite topology, ∆ = CL(X) and γl the coarsest compatible LO-proximity on X defined by A6γlB iff A6γ0B and either A or B is finite. Then X is T1 and it is easy to check that ∆ is not γ-Urysohn for each compatible LO-proximity γ with γl ≤ γ ≤ γ0. Lemma 7.4. (cf. Theorem 2.2 in [9]) Let X be a T1 space with compa- tible LO-proximities γ and η satisfying γ < η and ∆ a cobase. If ∆ is γ-Ury- sohn, then the upper Bombay topology σ(δ,η; ∆)+ equals the upper proximal ∆-topology σ(π; ∆)+ w.r.t. π, where π is the EF -proximity on X constructed in above Lemma 7.2. Theorem 7.5. (cf. Theorem 2.1 in [9]) Let X be a T1 space with compa- tible LO-proximities γ, η satisfying γ < η and ∆ a cobase hereditarily closed. The following are equivalent: (a) ∆ is γ-Urysohn; (b) the Bombay topology σ(γ,η; ∆) is Tychonoff. Let X be a Tychonoff space with a compatible uniformity U (as usual we assume that all elements U ∈U are symmetric and closed), B(U) the cobase generated by all (generalized) closed balls w.r.t. U (see preliminaries in Section 4) and γ = γ(U) denote the uniform proximity induced by U. Then it is easy to show that B(U) is γ-Urysohn. It therefore follows that: Corollary 7.6. (cf. [14]) Let U be a compatible uniformity on X whose ele- ments are symmetric and closed, B(U) the cobase generated by all (gene- ralized) U-balls and γ the uniform proximity induced by U. Then the Wijsman topology τ(W(U)) is Tychonoff. Problem 7.7. Is it true that in a Tychonoff space every EF -proximity can be constructed using two LO-proximities as in Lemma 7.2 ? Bombay hypertopologies 443 Acknowledgements. We thank the referee for a careful reading of the manuscript and valuable suggestions. References [1] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, (Kluwer Academic Publishers, 1993). [2] G. Beer, A. Lechicki, S. Levi and S. Naimpally, Distance functionals and suprema of hyper- space topologies, Annali di Matematica pura ed applicata 162 (1992), 367–381. [3] C. Costantini, S. Levi and J. Zieminska, Metrics that generate the same hyperspace conver- gence, Set-Valued Analysis 1 (1993), 141–157. [4] D. Di Caprio and E. Meccariello, Notes on Separation Axioms in Hyperspaces, Q. & A. in General Topology 18 (2000), 65–86. [5] D. Di Caprio and E. Meccariello, G-uniformities, LR-proximities and hypertopologies, Acta Math. Hungarica 88 (1-2) (2000), 73–93. [6] A. Di Concilio, S. Naimpally and P.L. Sharma, Proximal hypertopologies, Proceedings of the VI Brasilian Topological Meeting, Campinas, Brazil (1988) [unpublished]. [7] G. Di Maio and Ľ. Holá, A hypertopology determined by the family of totally bounded sets is the infimum of upper Wijsman topologies, Q. & A. in General Topology, 15 (1997), 51–66. [8] G. Di Maio and Ď. Holý, Comparison among Wijsman topology and other hypertopologies, Atti Sem. Mat. Fis. Univ. di Modena 48 (2000), 121–133. [9] G. Di Maio, E. Meccariello and S. Naimpally, Uniformizing (proximal) ∆-topologies, Topo- logy and its Applications, (to appear). [10] G. Di Maio and S. Naimpally, Comparison of hypertopologies, Rendiconti di Trieste 22 (1990), 140–161. [11] G. Di Maio and S. Naimpally, Abstract measure of farness and Wijsmann convergence, Zbornik Radova 5 (1991), 109–112. [12] G. Di Maio and S. Naimpally, Some notes on hyperspace topologies, Ricerche di Matematica, (to appear). [13] R. Engelking, General topology, Revised and completed version, Helderman Verlag, (Helder- man, Berlin, 1989.) [14] S. Francaviglia, A. Lechicki and S. Levi, Quasi-uniformization of hyperspaces and conver- gence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347–370. [15] Ľ. Holá and R. Lucchetti, Equivalence among hypertopologies, Set-Valued Analysis 3 (1995), 339–350. [16] S.T. Hu, Boundedness in topological space, J. Math. Pures. Appl. 28 (1949), 287–340. [17] M. Marjanovic, Topologies on collections of closed subsets, Publ. Inst. Math. (Beograd) 20 (1966), 196–130. [18] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. [19] C.J. Mozzochi, M. Gagrat and S. Naimpally, Symmetric generalized topological structures, Exposition Press, (Hicksville, New York, 1976.) [20] S. Naimpally, All Hypertopologies are hit-and-miss, Applied General Topology 3 (2002), 45–53. [21] S. Naimpally and P. Sharma, Fine uniformity and locally finite hyperspace topology on 2X, Proc. Amer. Math. Soc. 103 (1988), 641–646. [22] S. Naimpally, B. Warrack, Proximity spaces, Cambridge Tracts in Mathematics 59, (Cam- bridge University Press, 1970.) [23] H. Poppe, Eine Bemerkung über Trennungsaxiome in Raum der abgeschlossenen Teilmen- gen eines topologischen Raumes, Arch. Math. 16 (1965), 197–199. [24] W.J. Thron, Proximity Structures and Grills, Math. Ann. 206 (1973), 35–62. 444 G. Di Maio, E. Meccariello and S. Naimpally [25] R. Wijsman, Convergence of sequences of convex sets, cones, and functions, II, Trans. Amer. Math. Soc. 123 (1966), 32–45. Received February 2002 Revised February 2003 Giuseppe Di Maio Seconda Università degli Studi di Napoli, Facoltà di Scienze, Dipartimento di Matematica, Via Vivaldi 43, 81100 Caserta, Italia E-mail address : giuseppe.dimaio@unina2.it Enrico Meccariello Università del Sannio, Facoltà di Ingegneria, Piazza Roma, Palazzo B. Lu- carelli, 82100 Benevento, Italia E-mail address : meccariello@unisannio.it Somashekhar Naimpally 96 Dewson Street, Toronto, Ontario, M6H 1H3, Canada E-mail address : sudha@accglobal.net