Annali di Matematica pura ed applicata (IV), Vol. CLXII (1992), pp. 367-381 Distance Functionals and Suprema of Hyperspace Topologies (*). GERALD BEER(**) - ALOJZY LECHICKI SANDRO LEVI(***) - SOMASHEKHAR NAIMPALLY Abstract. - Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the wea#est topology on CL(X) such that A ~ d(x,A) is continuous for each x e X and each admissible metric d. We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) --~ --*inf{d(a, b): a e A, b eB} is continuous on CL(X) x CL(X). 1. - Introduction. Let CL(X) denote the nonempty closed subsets of a metrizable topological space X. If d is an admissible metric for X, and A ~ CL(X), and x e X, then the distance from x to A is given by the familiar formula d(x, A) =- inf d(x, a). aeA Usually, one thinks of d as a function of the point variable, with the set A held fixed. Alternatively, we may hold x fixed to obtain a function d(x, .) on CL(X). The weakest topology on CL(X) such that A --. d(x, A) is continuous for each x E X is usually called the Wijsman topology in the literature ([16], [19]). We denote this topology by 7W(d) in the sequel. A basic question is this: what is the supremum of the Wijsman topologies, as d runs over the admissible metrics for X? Put differently, if ~ denotes the set of admissible metrics, what is the weakest topology on CL(X) such that for each x e X and each d E 0~, the functional d(x, .) is continuous on CL(X)? One main result of this (*) Entrata in Redazione il 9 novembre 1989; versione riveduta entrata il 20 aprile 1990. Indirizzo degli AA.: G. BEER: Department of Mathematics, California State University, Los Angeles, Los Angeles, California 90032, USA; A. LECHICKI: Hardstrasse 43, Furth D-8510, FRG; S. LEvi: Dipartimento di Matematica, Universit~ di Milano, Milano 20133, Italy; S. NAIM- PALLY: Department of Mathematics, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1. (**) Visiting the University of Minnesota. (***) Visiting California State University, Los Angeles. 368 G. BEER - A. LECHICKI - S. LEVI - S. NAIMPALLY: Distance functionals, etc. paper shows that this topology is none other than the familiar Vietoris topology. On the other hand, if we restrict our attention to metrics that determine the same unifor- mity, then the supremum is a properly smaller topology (unless with respect to this uniformity, each continuous real valued function on X is uniformly continuous). AI- though Wijsman topologies corresponding to uniformly equivalent metrics may dif- fer [19, p. 449], this supremum is actually determined by any one of the uniformly equivalent metrics, and is functionally characterized as follows: it is the weakest topology on CL(X) such that the gap or separation functional Dd: CL(X) • CL(X) --> --~ R, defined by Dd (A, B) = inf {d(a, b): a e A, b e B}, is continuous, where d is any one of the metrics. These results show that Wijsman topologies are building blocks in the lattice of hyperspace topologies. 2. - Preliminaries. In this section we provide basic notation, and introduce a few hyperspace topolo- gies on CL(X). In what follows, d will be a fixed metric on X. The open (resp. closed) ball of radius ~ about x e X will be denoted by S~ [x] (resp. B~ [x]). If A c X, we denote the union of all open d-balls of radius r whose centers run over A by S~ [A]. In terms of this notation, Hausdor(f distance on CL(X) is defined by the formula Hd (A, B) = inf {s: S~ [A] ~ B and S~ [B] ~ A}. Such a distance defines an infinite valued metric on CL(X), which is evidently finite valued when restricted to the closed and bounded subsets of X. We denote by ZH(d)the topology induced on CL(X) by Hd. Basic facts about Hausdorff distance can be found in [10] and in [18]. Most impor- tantly, lid (A, B) = sup Id(x, A) - d(x, B) I, so that the zH(d)-convergence of a net (A~) xeX to A is equivalent to the uniform convergence of (d(., A~)) to d(., A). As a result, rH(d) ~ ZW(d). By equicontinuity of distance functions, the topologies agree provided (X, d} is totally bounded; the converse is also true [7,16]. The Wijsman topology zW(d) is actually metrizable if and only if X is second countable [19]; in this case, Zw(d) is also second countable. That second countability of X implies second countability and metrizability of ~W(d) follows easily from the Urysohn metrization theorem [26, p. 166]: Zw(d) is second countable and Hausdorff because it has as a subbase all sets of the form d(x, .)- 1 (~, ~), where x ranges over a countable dense subset of X and ~ and ,6 are rational, and it is completely regular simply because it is a weak topology. There is a great deal of liter- ature on Wijsman convergence of sequences of sets, especially in a normed linear space. We refer the interested reader to [3, 7,23,25] and to [5]. To introduce the vietoris topology, the Fell topology, and the topology natural G. BEER - A. LECHICKI - S. LEVI - S. NAIMPALLY: Distance functionals, etc. 369 with respect to the gap functional Dg, we need some further notation. If A c X, the following collections of closed subsets of X are associated with A: A- = {FeCL(X): FNA ~ O}, A + = {Fe CL(X): FcA}, A ++ = {Fe CL(X): Dd(F, A c) > 0} = {Fe CL(X): for some z > 0, S~[F] cA}. Evidently, A ++ cA + and A c S~[A] ++. The Vietoris topology, also called the fi- nite topology or the exponential topology, is generated by all sets of the form V- and V +, where V runs over the open subsets of X. We denote the Vietoris topology by Zv in the sequel (it does not vary with d). For information on this topology (undoubted- ly, the most well-studied hyperspace topology), the reader may consult the funda- mental article of MICHAEL[20], the recent monograph of KLEIN and THOMPSON[18], or [15]. The Fell topology ZF [12], also called the topology of closed convergence [18], admits a similar presentation: it is generated by all sets of the form V- where V is an open subset of X, and W + where W has compact complement. Evidently, ZF C rV; more precisely, the results of [16] show that ~. c zm(d) c Zv for each d ~ ~. The Fell ~opology has been particularly well studied in relation to convergence and minimiza- tion problems for lower semicontinuous functions [2, 9]. A key construction in this article is the topology on CL(X) generated by all sets of the form V- and V ++, where V runs over the open subsets of X. We call this the d-proximal topology and denote it by ~o~). The motivation for this terminology and notation is as follows: sets of the form V ++ contain those closed sets that are far from V C with respect to the metric proximity associated with d (see, e.g., [21] or [26]). Since (VNW) ++= V ++ N W ++, a base for the topology ~-o~d/ consists of all sets of the form v++ n Vl n n ... n y:, where V, V1, ..., V~ are open. Evidently, a local base for the topology at A ~ CL(X) consists of all sets of the form St[A] ++ n S~[al]- n S~[az]- n ... n S~[a~]-, where {al, a2, ..., a~} cA and ~ > 0. From this presentation, it is clear that the d-proximal topology is compatible with Fisher convergence [13, 14] of sequences of sets, as considered by BARONTI and PAPINI [3]. It is also clear that the d-proximal topology is the supremum of the lower Vietoris topology and the upper Hausdorff metric topology induced by d, as discussed in [16]. Evidently, =o~d)=~o~/ if and only if d and p determine the same proximity (see, e.g., [21, Lemma 2.8]). In particular, if d and ~o determine the same uniformity, then Zo~)= zo~o) because a metric proximity is a special case of a proximity induced by a uniformity. In fact, the converse is true: for a metrizable space, there is a unique metric 370 G. BEER - A. LECHICKI - S. LEVI - S. NAIMPALLY: Distance functionals, etc. uniformity that gives rise to a given metric proximity [21, Theorem 12.18]. In the sequel, we write d ~ ~ provided d and ~ are uniformly equivalent. Finally we notice that z~(d) is coarser than either rv or rH(d); in the next section we show that it is finer than ~w(d). All of these facts can be shown to follow from results of [16]. 3. - Suprema of Wijsman topologies. Our f~'st result shows that the Vietoris topology is the weakest topology on CL(X) such that for each admissible metric d and each x e X, F---) d(x, F) is a continuous function on CL(X). THEOREM 3.1. - Let X be a metrizable space, and let ~ be the family of admissible metrics for its topology. Then ~v = sup {~w(d): d E O)}. PROOF. - Let z be the weak topology so described; Since zw(d) c rv for each d e 0~ ([16]), it is clear that z r To show r;,c z, it suffices to show that V- e ~ and V + E for each open subset V or X.