Bombay hypertopologies

@ Applied General Topology c Universidad Polit´ecnica 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 hypertopology the ILlocally 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 result includes, as special cases, many known results scattered throughout the literature. Moreover, our approach gives simple transparent proofs in comparison 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 convergence for nets of closed subsets of a metric space (X, d) viz: An→Aiff 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 neighbourhood 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 Ain a metric space (X, d) consists of closed sets that do not intersect a proper closed ball Bthat is far from Ain 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 T1space. For any E⊂X,clXE,intE and Ecstand for the closure, interior and complement of Ein X, respectively. Let γ,γ1and γ2 be compatible LO-proximities on X. We always assume γ1≤γ2(i.e. A6γ1B implies A6γ2B(see [22]) where 6γstands for the negation of γ). We use the symbol γ0to denote the fine LO-proximity on Xgiven by Aγ0Biff clXA∩clXB6=∅. (γ0is called the Wallman proximity). In the case (X, τ) is Tychonoff, then: γ]denotes the fine EF -proximity on Xgiven by A6γ]Biff they can be separated by a continuous function f:X→[0,1]. (γ]is called the functionally indistinguishable proximity on X). If Uis a compatible separated uniformity on X, then γ(U) denotes the EF -proximity on Xgiven by Aγ(U)Biff U(A)∩B6=∅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 -proximity on Xgiven by Aγ(d)Biff 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⊂Xand a subfamily E ⊂ τwe use the following standard notation: E−={A∈CL(X) : A∩E6=∅}; E−={A∈CL(X) : A∩E6=∅for each E∈ E}. Furthermore, if γis a compatible proximity on X, we set E++ γ={A∈CL(X) : AγE, i.e. A6γEc}. Note that γ1≤γ2is equivalent to U++ γ1⊂U++ γ2for every U∈τ. We omit γif it is clear from the context and write E++ γsimply as E++. Moreover E+={A∈CL(X) : A⊂E, i.e. Aγ0E}. 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=γ0we have the upper ∆-topology τ(∆)+=σ(γ0; ∆)+. The lower Vietoris (or finite)topology τ(V−) has a basis {E−:E ⊂ τis finite}. The lower locally finite topology τ(LF−) has a basis {E−:E ⊂ τis locally finite}. The IL-lower locally finite topology τ(IL−) has a basis {E−:E ⊂ IL} provided IL ⊂ {E ⊂ τ:Eis 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 γif 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 σby τif 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 Bgenerated 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) γ0the fine LO-proximity in U+, and (iii) the cobase Bwhich contains Uc. We note that there are two proximities, namely γ(d), γ0with γ(d)≤γ0 and a cobase. By replacing the two proximal parameters γ(d), γ0by two LOproximities γ1,γ2with γ1≤γ2and the cobase Bby ∆ we have the following definition: Definition 2.1. Let (X, τ) be a T1space with compatible proximities γ1,γ2 with γ1≤γ2and ∆ a cobase. Then a typical neighbourhood of A∈CL(X) in the upper Bombay topology σ(γ1, γ2; ∆)+is: U++ γ2where Uc∈∆ and A6γ1Uc(or equivalently A∈U++ γ1). (Note that since γ1≤γ2A6γ1Ucimplies A6γ2Ucwhich in turn is equivalent to A∈U++ γ2). γ1,γ2and ∆ are the three parameters of the upper Bombay hypertopology: γ1,γ2are 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 topology 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 ∈ τis 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, ∆ = Bthe 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 addition, suppose (X, τ) is a T1space, γa compatible LO-proximity coarser than γ0and ∆ a cobase, then we refer the upper-Bombay topology σ(γ, γ0; ∆)+(the Bombay topology σ(γ, γ0; ∆)) as the upper g-Wijsman topology (the gWijsman topology). Here gstands for generalized (w.r.t. ∆ and γsince γ0is kept fixed). (d) Let (X, U) be a separated uniform space and γ=γ(U) the compatible 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 Qis U-discrete, i.e. x,y∈Q,x6=yimplies x6∈ U(y). Then the Hausdorff Bourbaki or H-B uniform topology associated to Uon CL(X) is τ(UH) = τ(UH)+∨τ(U− H) = σ(γ)+∨τ(IL−) (see [20]). Moreover, if the uniformity Ucomes from a metric dwe 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 Vietoris 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 T1space with compatible LO-proximities γ1,γ2, η1,η2satisfying γ1≤γ2and η1≤η2. Let ∆,Λbe cobases and IL1,IL2two 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 T1space and IL,IL1,IL2collections 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 ∈ IL2such that F−⊂ E−. Thus if all members of IL2are finite, then the same is true of the members of IL1and 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 T1space and γa 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 Fof Xnear to Bw.r.t. γ. Lemma 3.4. Let (X, τ)be a T1space, γand ηcompatible LO-proximities on Xand Cand Dnonempty closed subsets of X. The following are equivalent: (a) (Dc)++ η⊂(Cc)++ γ; (b) C⊂Dand γ(C)⊂η(D). Proof. (a)⇒(b). Assume not, then either i) C6⊂ Dor ii) γ(C)6⊂ η(D). If i) occurs, then there exists c∈C\D. Choose F∈(Dc)++ ηand consider 428 G. Di Maio, E. Meccariello and S. Naimpally F0=F∪ {c}. Then F0∈(Dc)++ ηbut F06∈ (Cc)++ γ; a contradiction. If ii) occurs, then there exists an F⊂Xsuch that FγC but F6ηD. Since F γC, then F6=∅. Let E=clXF. Then E∈CL(X), EγC but E6ηD. Therefore E∈(Dc)++ ηbut E6∈ (Cc)++ γ; a contradiction. (b)⇒(a). Assume not, then (Dc)++ η6⊂ (Cc)++ γ. Hence there exists F∈CL(X) such that F6ηD but F γC, i.e. F∈γ(C) but F6∈ η(D); a contradiction.  Theorem 3.5. (Main Theorem) Let (X, τ)be a T1space with compatible LO-proximities γ1,γ2,η1,η2satisfying γ1≤γ2and η1≤η2and ∆and Λcobases. The following are equivalent: (a) σ(γ1, γ2; ∆)+≤σ(η1, η2; Λ)+; (b) for each B∈∆and W∈τ,W6=X, with Bγ1W, there exists a B0∈Λsuch that: (i) B⊂B0η1W, and (ii) γ2(B)⊂η2(B0). Proof. σ(γ1, γ2; ∆)+≤σ(η1, η2; Λ)+if and only if for each A∈CL(X), A6=X, A∈U++ γ2∈σ(γ1, γ2; ∆)+- where Uc∈∆ and A∈U++ γ1there exists a V∈τ such that Vc∈Λ, A∈V++ η1and A∈V++ η2⊂U++ γ2. Noting that A∈V++ η1 is equivalent to Vcη1Acand using Lemma 3.4 we have (i) and (ii) where W=Ac,B=Ucand B0=Vc. Corollary 3.6. Let (X, τ)be a T1topological space with compatible LO-proximities γ1,γ2,η1,η2satisfying γ1≤γ2and η1≤η2,∆and Λcobases and IL1 and IL2collections of locally finite families of open sets satisfying condition (∗). The following are equivalent: (a) σ(γ1, γ2;IL1,∆) ≤σ(η1, η2;IL2,Λ); (b) IL2refines IL1and whenever B∈∆,W∈τ,W6=X, with Bγ1W, then there exists a B0∈Λsuch that: (i) B⊂B0η1W, and (ii) γ2(B)⊂η2(B0). Remark 3.7. (a) For future reference we note that F γB implies that there is a net in Xwhose range C⊂Fand CγB (cf. Lemma 3.2 in [5]). (b) We note that if η2≤γ2, then (ii) at (b) of the Main Theorem is automatically satisfied. (c) If γ1is an EF-proximity, then A∈U++ γ1implies the existence of E∈ CL(X) with E∈U++ γ1and A∈(intE)++ γ1. So setting W0=intE, (i) at (b) of the Main Theorem can be written as B⊂B0η1W0γ1W. (d) The proximal topologies σ(γ1; ∆) and σ(γ2; ∆) are the proximal coordinate topologies of the given (finite) Bombay topology σ(γ1, γ2; ∆). Bombay hypertopologies 429 Let γand η, with γ≤η, be compatible LO-proximities on a given topological space Xand ∆ 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 dand the first proximal parameter γ=γ(d) is the EF -proximity associated to d, and 2) (X, τ) is a Tychonoff space and the first proximal parameter γis EF or (X, τ) is a uniformizable space with separated uniformity Uand 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 proximal coordinate topologies. Theorem 3.8. (cf. [1] Pages 45, 53) Let (X, τ)be a T1space with compatible LO-proximities γ1,γ2, satisfying γ1≤γ2and ∆a cobase. Then: (a) σ(γ1, γ2; ∆) ≤σ(γ1; ∆); (b) σ(γ1, γ2; ∆) ≤σ(γ2; ∆). Corollary 3.9. Let (X, τ)be a T1space, γa 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 T1space with compatible LO-proximities γ1,γ2satisfying γ1≤γ2and ∆a cobase. The following are equivalent: (a) σ(γ2; ∆) ≤σ(γ1, γ2; ∆); (b) σ(γ1, γ2; ∆) = σ(γ1; ∆) = σ(γ2; ∆); 436 G. Di Maio, E. Meccariello and S. Naimpally Theorem 4.13. Let Xbe a metrizable space and γ=γ(d)the metric proximity induced by d. The following are equivalent: (a) τ(V) = σ(γ); (b) γ=γ0; (c) Xis 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, γa compatible EF -proximity, whereas γ0and γ]denote respectively the Wallman (the fine LO-) proximity and the functionally indistinguishable (the fine EF -) proximity on X(see Preliminaries). It is a well known fact that γ]=γ0if and only if Xis normal (Urysohn’s Lemma). Π(γ) denotes the family of all uniformities Ucompatible 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 Ucompatible 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∈Xand U∈ U,U[x] is the (generalized) Uball 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 γa 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,W6=Xand A∈CL(X). Then there are a closed entourage U∈ U?and a finite subset Fof Xsuch that A⊂U[F]⊂U2[F]γW. 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) Xis normal; (c) γ]=γ0. Corollary 5.4. (cf. [2]) Let Xbe a metrizable space with metric dand γ=γ(d) the metric proximity induced by d. Then: (a) sup{τ(We):evaries in the set of all metrics uniformly equivalent to d}=σ(γ); (b) sup{τ(We):evaries 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, γa compatible EF -proximity on X,U ∈ Π(γ)and TB(U)the family of all totally bounded closed subsets of Xw.r.t. U. Then σ(γ;TB(U)) ≤σ(γ, γ0;B(U)). Proof. Let B∈T B(U) and W∈τwith BγW6=X. Then there are an entourage U∈ U and a finite set F⊂Xsuch that B⊂U[F]⊂U2[F]⊂W. Observe that U[F]∈ B(U) and U[F]⊂U2[B]⊂Wimplies U[F]γW. The result follows from the Main Theorem.  Let Ube a separated uniformity on Xand U∈ U. We recall that that U0∈ U is composably contained in Uiff there is a U00 ∈ U such that U0◦U00 ⊂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=Xand for each U0∈ U composably contained in Uthere is a B0∈∆such that U0[x]⊂B0γ U[x]; (c) σ(γ, γ0;B(U)) ≤σ(γ; ∆). Proof. It suffices to use the Main Theorem observing that if U0∈ U is composably contained in U∈ U, then U0[x]γU[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 Dis strictly U-included in E(D⊂⊂UE) iff there exist a finite set F⊂Dand entourages U,U0∈ U with U0composably contained in Usuch that 438 G. Di Maio, E. Meccariello and S. Naimpally (?)D⊂U0[F]⊂U[F]⊂E. Remark 5.8. If γ=γ(U), then in above definition condition (?) is equivalent to the following one (?0)D⊂U0[F]γU[F]⊂E. Thus whenever γis a compatible EF -proximity on Xand U ∈ Π(γ), a set D is stricly U-included in Eiff there exist a finite subset Fand entourages U, U0∈ U such that D⊂U0[F]γU[F]⊂E. We also note that if Uis the metric uniformity induced by d, then for each U∈ U there exists some positive εsuch that U={(x, y)∈X×X:d(x, y)≤ε}. Thus U[x] = B(x, ε) for each x∈Xand D⊂⊂UEis equivalent to D⊂⊂dE (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, γa 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∈τwith B⊂W, then B⊂⊂UW. (c) for every B∈∆\ {X}and U∈ U,B⊂⊂UU[B]; (d) σ(γ; ∆) ≤σ(γ, γ0;B(U)). Corollary 5.10. (cf. 3.11)Let (X, τ)be a Tychonoff space, γa compatible EF -proximity on Xand 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∈τwith B⊂W, then B⊂⊂UW; (d) for every B∈ B(U)and W∈τ,W6=X, with B⊂W, there exists a B0∈ B(U)such that B⊂B0γW; (e) Xis 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⊂⊂UU[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, γa compatible EF -proximity on Xand 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 Uis totally bounded. Bombay hypertopologies 439 Proof. The result follows from the fact that Uis 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, γa compatible EF -proximity on Xand U ∈ Π(γ). The following are equivalent: (a) σ(γ, γ0;B(U)) = τ(UH); (b) Uis totally bounded. 6. Bounded Hypertopologies. In the literature on hyperspaces of a metric space (X, d), there are several 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 Xiff ∆ 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 Ais U-bounded iff there is a finite set {x1,· · · , xn} ⊂ Xand 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 Ubounded subsets of Xand 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 ∆-AttouchWets 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 Uis induced by a metric dand ∆ = 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 Qis 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 T1topological space with compatible LO-proximities γ1,γ2,η1,η2satisfyng γ1≤γ2and η1≤η2,∆,Λtwo boundedness and IL1and IL2locally finite collections of open subsets of X. The following are equivalent: (a) σ(γ1, γ2;IL1,∆) ≤σ(η1, η2;IL2,Λ); (b) IL2refines IL1and whenever B∈∆,W∈τ,W6=X, with Bγ1W, then there exists a B0∈Λsuch that: (i) B⊂B0η1W, and (ii) γ2(B)⊂η2(B0). Let (X, d) be a metric space and as usual γ=γ(d) the metric proximity, U the separated uniformity associated to d,Bthe 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 Bwith 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 Xbe a metrizable space with metric d,γ=γ(d)the metric proximity induced by d,Bthe cobase generated by all proper closed dballs and B(X)the family of all closed dbounded 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 Xbe a metrizable space, dand ecompatible metrics, γand ηthe metric proximities induced by dand erespectively, B(X)the family of all closed d-bounded sets and B0(X)the family of all closed e-bounded sets. The following are equivalent: (a) τ(AWd) = τ(AWe); (b) δ=δ0,B(X) = B0(X) (i.e. every closed d-bounded subset is ebounded and vice versa) and they have the same uniformly continuous functions on bounded sets. Theorem 6.7. Let Xbe a metrizable space with metric d,γ=γ(d). As usual, let B(X)and T B =T B(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 T1space. This section is devoted to the characterization of the uniformizable Bombay topologies associated with a cobase ∆ and compatible LO-proximities γ,ηsatysfying 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 T1space, γa 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γSγAc(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 Xbe a T1space with compatible LO-proximities γ,ηsatysfying γ < η and ∆a cobase. If ∆is γ-Urysohn, then the relation πdefined on the power set of Xby: (?)A6πB iff clA6γclB and either clA ∈∆and clB η(clA)cor clB ∈∆ and clA η(clB)c is a compatible EF -proximity on X. Moreover, π≤γand ∆is γ-Urysohn iff ∆is π-Urysohn. Remark 7.3. (a) Observe that even if the starting proximities γand ηare just LO, the new compatible proximity πis always EF . Thus the base space Xis automatically completely regular. Note that we have a procedure that allows us to construct an EF -proximity on a Tychonoff space Xby using as seeds LO-proximities γ,ηwith γ < η and a cobase ∆ which is γ-Urysohn. (b) Let Xbe an infinite set, τthe cofinite topology, ∆ = CL(X) and γlthe coarsest compatible LO-proximity on Xdefined by A6γlBiff A6γ0Band either Aor Bis finite. Then Xis T1and it is easy to check that ∆ is not γ-Urysohn for each compatible LO-proximity γwith γl≤γ≤γ0. Lemma 7.4. (cf. Theorem 2.2 in [9]) Let Xbe a T1space with compatible LO-proximities γand ηsatisfying γ < η and ∆a cobase. If ∆is γ-Urysohn, then the upper Bombay topology σ(δ, η; ∆)+equals the upper proximal ∆-topology σ(π; ∆)+w.r.t. π, where πis the EF -proximity on Xconstructed in above Lemma 7.2. Theorem 7.5. (cf. Theorem 2.1 in [9]) Let Xbe a T1space with compatible LO-proximities γ,ηsatisfying 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 Xbe 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 Ube a compatible uniformity on Xwhose elements are symmetric and closed,B(U)the cobase generated by all (generalized) U-balls and γthe 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 hyperspace 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 convergence, 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 ˇ L. Hol´a, 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 ˇ D. Hol´y, 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, Topology 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, (Helderman, Berlin, 1989.) [14] S. Francaviglia, A. Lechicki and S. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347–370. [15] ˇ L. Hol´a 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, (Cambridge University Press, 1970.) [23] H. Poppe, Eine Bemerkung ¨uber Trennungsaxiome in Raum der abgeschlossenen Teilmengen 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 Universit`a degli Studi di Napoli, Facolt`a di Scienze, Dipartimento di Matematica, Via Vivaldi 43, 81100 Caserta, Italia E-mail address:[email protected] Enrico Meccariello Universit`a del Sannio, Facolt`a di Ingegneria, Piazza Roma, Palazzo B. Lucarelli, 82100 Benevento, Italia E-mail address:[email protected] Somashekhar Naimpally 96 Dewson Street, Toronto, Ontario, M6H 1H3, Canada E-mail address:[email protected]