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Infima of Hyperspace Topologies INFIMA OF HYPERSPACE TOPOLOGIES C. COSTANTINI, S. LEVI AND J. PELANT Abstract. We study infima of families of topologies on the hyperspace of a metrizable space. We prove that Kuratowski convergence is the infimum, in the lattice of convergences, of all Wijsman topologies and that the cocompact topology on a metric space which is complete for a metric d is the infimum of the upper Wijsman topologies arising from metrics that are uniformly equivalent iod. §1. Introduction. The main problem we address in this article is the follow- ing: characterize the infimum of the families {wp} and {wp}, where p varies over all admissible metrics on X and wp is the Wijsman (or wp~ the upper Wijsman) topology associated to p (see Section 2 for the definitions). This is one in a series of problems which arise in a natural way: the metriz- able structure of X allows us to define a hyperspace topology for each admissible metric on X; suprema or infima of these topologies depend only on the topology of the base space and we look for intrinsic characterization of these topologies. The central role played by Wijsman topologies among hyperspace topolog- ies is illustrated by the following results concerning suprema. The slice topology on the hyperspace of closed convex subsets of a normed linear space (a useful topology for which polarity is continuous) is the supremum of the Wijsman topologies arising from equivalent norms [Be 2]. The upper Hausdorff topology corresponding to a metric is the supremum of the upper Wijsman topologies over all uniformly equivalent metrics [BLLN]. The Vietoris topology is the supremum of all Wijsman topologies [BLLN]. In this paper we will prove the following "dual" result. Upper Kuratowski convergence is the infimum of all wp~'s (Theorem 3.1) and, consequently, Kuratowski convergence is the infimum of all Wijsman topologies. To our knowledge, the only results concerning infima of hyperspace topol- ogies can be found in [LLP] and the forthcoming paper [CV]. We note that the study of infima is more complex than that of suprema, since infima depend on the lattice of reference (see Section 2). In Section 2 we give the definitions that we will need and describe the lattices of convergences and of topologies on the hyperspace. In Section 3 we prove, through the construction of an appropriate admis- sible metric, the above-mentioned result which describes Kuratowski conver- gence in terms of the Wijsman topologies. We deduce some corollaries, among which the fact that the topological infimum of the Wijsman topologies is the topologization of Kuratowski convergence. [MATHEMATIKA, 42 (1995), 67-86] Downloaded from https://www.cambridge.org/core. University of Athens, on 05 Oct 2021 at 11:47:10, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0025579300011360 68 C. COSTANTINI, S. LEVI AND J. PELANT In Section 4 we investigate in more details topological infima. Our main result here is that in a complete metric space (X, d), the cocompact topology can be obtained as the infimum of the upper Wijsman topologies relative to the metrics that are uniformly equivalent to d. In Section 5 we show that the results obtained in the two previous sections still hold if we replace the families of upper Wijsman topologies with the families of corresponding "ball" and "ball proximal" topologies (note that these topol- ogies are finer than the corresponding Wijsman's). Finally, Section 6 deals with the two possible definitions of distance of points to the empty set and studies the differences that may arise in the hyperspace. The third named author was partially supported by grants from GACR and the Czech. Acad. §2. Convergences and topologies. A convergence on a set X is a function which associates to each net inla subset, possibly empty, of X. We will denote by # the collections of all convergences on X; if ceW and {Xj)jeJ is a net in X, we will say that (x,) c-converge to xeX if x belongs to the set that c associates to (xj). We will also write Xj *x. Two convergences are equal if they have the same converging nets. If c and c' are elements of #, we will write if every net that is c'-convergent to a point is also c-convergent to the same point. Thus (<&, <) is an ordered space which is a complete lattice under the operations of sup and inf. Given a family (c,),e/ of convergences, its supremum V/e/ Ci has the following description: the net (Xj)JeJ (V;e/c,)-converges to x if (XJ) c,-converges to x for every iel; while its infimum has the following description: (xj)jej (Aie/c<)-converges to x if (xj) c,-converges to x for some iel. Within the lattice #, we can consider the subfamily of topological conver- gences and we will identify each of these with the corresponding topology; thus &~, the family of all topologies on X, is a subset of # which inherits the same order and the same supremum operation but not the same infimum: the infi- mum of a family of topologies on X is the intersection-topology. Hence the symbols < and v (or sup) are unambiguously defined, while we will need to specify for the A (or inf) operation the lattice of reference. Given a convergence c onX,a "closest" topology T(c) is defined in the follow- ing manner: Note that the operator T is monotone and is the identity on 2T. We will say that a subset A of X is c-closed if the conditions (aj)Jej^A and (a,) c-converges to a imply that aeA. We will, from now on, consider only convergences on X which have the additional property that if a net (a,-) c-converges to aeX, each subnet of (a,-) converges to a. Downloaded from https://www.cambridge.org/core. University of Athens, on 05 Oct 2021 at 11:47:10, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0025579300011360 INFIMA OF HYPERSPACE TOPOLOGIES 69 We recall the following two results; for proofs and more information see [DG]. LEMMA 2.1. Given ce#, let T'(c) be the family of all c-closed subsets of X. Then (i) T'(c) is a topology; (ii) OT'(c); and (iii) T'(C)^T for each topology r<c. The three points above combine to prove that T'(c) = T(c); we obtain in this way a different description of the topology T(c). PROPOSITION 2.2 Let <& = (c,),e/ be a family of convergences on X. Then (i) Inf« c,^Infr T(ct); and (ii) By (ii) above, the operator T distributes over the infima, but, as Example 4.4 shows (or as we can deduce by a general lattice-theoretic argument), T does not even distribute over finite suprema. Let us now define the convergences we will be working with. X will denote a metrizable space and c(X) the hyperspace of X, that is, the collection of closed subsets of A'; we define co(X) = c(X)\{0}. Let M be the set of all admissible metrics on X; for d&Jl we say that the net (Aj)jsJcc(X) Wijsman-converges to Aec(X), according to the metric d, and write Aj-^+A, if VxeX: d(x, A) = lim d(x, AJ), where d(x, A) = M{d(x, a)\aeA} if A*0 and d{x, 0) = sup {d{x,y)\yeX}. Recall that wd= wj v wj , where the upper and lower parts of the Wijsman convergence are defined in such a way that the net {Aj)jsJ converges to A in w^ [respectively, wj] if d(x, A)^\im\n{Jsjd(x, Aj) [respectively, d(x, A)^ lim supjej d{x, Aj)] for each xeX. The convergence wj coincides with the lower Vietoris topology V~, whereas the (topological) convergence Wd does depend on the choice of the metric. Given the net (Aj)JeJzc(X), we define LsAj={yeX\VVnbd ofy: V/e/: Sj'^j: VnAr^0}, and UAj={yeX\VVTibdofy: 3jeJ:Vj'^j: VnAr*0), + and say that the net {Aj)jeJ fc -converges to A if LsAj^A; the condition A^LiAj is again equivalent to F~-convergence of (Aj)JeJ to A. It is well known that A:+-covergence is topological if, and only if, X is locally compact, ^-convergence (after Kuratowski) is k = k+ v V~. In [DGL], the topologization T(k+) was studied and the problem of establishing for which spaces T(k+) is the cocompact convergence was raised; we will address this problem briefly in Section 4. Downloaded from https://www.cambridge.org/core. University of Athens, on 05 Oct 2021 at 11:47:10, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0025579300011360 70 C. COSTANTINI, S. LEVI AND J. PELANT Cocompact convergence (or c-convergence) of (Aj)JeJ to A is given by the condition: for every compact subset K^X such that Kn A = 0, Kr\Aj=0 eventually. This c is a topological convergence. Fell convergence is F= c v V~. c + Finally, given a metric deJt, we say that (Aj)jej (B d) -converges to A if for every xeX and r>0 the relation B{{x)nA = 0 implies that Bd(x) r\Aj=0 eventually (where Bd(x) denotes the open rf-ball of centre x and radius r and Bd(x) the closed ball).
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