Fine uniformity and the locally finite hyperspace topology

Abstract

ABSTRACT. It is shown that a uniformizable space X is normal iff the locally finite topology eT on the hyperspace 2X coincides with the topology transmitted by the fine uniformity of X. We also prove that, for X normal, the topology eT is first countable only if the set of limit points X' of X is countably compact. Applications of these results to pseudocompactness and Atsuji spaces are given. Introduction.In a recent paper [4], Beer et al. proved that the locally finite topology eT on the hyperspace 2X of a metrizable space X is the supremum of all the Hausdorff metric topologies corresponding to equivalent metrics on X. In view of this result, and because a Hausdorff metric topology generally lacks topological invariance, the authors of [4] justifiably contended that the locally finite topology on a hyperspace would be a more useful structure in most applications.In this paper, we determine the appropriate context for the locally finite topology, namely that this topology is a uniform notion. Specifically, we prove that, in the class of uniformizable spaces, it is precisely the normal spaces for which the locally finite topology on a hyperspace coincides with the topology transmitted by the fine uniformity of the space. This result specialized to metrizable spaces yields the aforementioned result of Beer et al. We also show that if X is paracompact, then for eT to be first countable, X' must be compact. From this, several known characterizations of Atsuji spaces follow. All topological spaces considered in this paper are assumed to be Ti. Much of our terminology about hyperspaces is the same as in Michael's paper [11].Let (X, t) be a Ti space. The hyperspace 2X oí X is the set {E Ç X: E is closed, E ^ 0}. For a collection sf of subsets of X, we write

Somashekhar Naimpally, 1931–2014

“…With P.L. Sharma [148], Som proved that a Tychonoff space is normal if and only if the locally finite topology on its closed subsets coincides with the hyperspace topology determined by the fine uniformity on X. This shows that in the context of normal spaces, the locally finite topology is a uniform notion.…”

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The Baire property in hit-and-miss hypertopologies

“…It was shown in [2] that for a metrizable space X , τ L on 2 X is the supremum of all the Hausdorff metric topologies corresponding to equivalent metrics on X . Further, it was shown in [16] that for a normal space X , τ L on 2 X coincides with the Hausdorff-Bourbaki uniform topology induced by the fine uniformity on X .…”

On the Fell topology

“…Wijsman) topologies corresponding to topologically equivalent metrics is the locally finite (resp. Vietoris) topology ( [4], [25], [5]). …”