On separation axioms in hyperspaces

Abstract

It is the purpose of this paper to show how to use approach spaces to get a unified method of proving Baireness of various hyperspace topologies. This generalizes results spread in the literature including the general (proximal) hit-and-miss topologies, as well as various topologies generated by gap and excess functionals. It is also shown that the Vietoris hyperspace can be non-Baire even if the base space is a 2nd countable Hausdorff Baire space.2010 MSC: Primary 54B20; Secondary 54E52, 54A05

Baire spaces and hyperspace topologies revisited

“…It was observed that imposing separation axioms on the base space is frequently not necessary to obtain results on hypertopologies (see [6], [19], [24], [10]), which is the case throughout this paper as well.…”

“…15, no. 1 This property is satisfied in uniform and metric spaces, and a topological space X has property (P) iff X is weakly-R 0 [24], i.e. for all open U ⊆ X and x ∈ U there is a y ∈ {x} with {y} ⊂ U iff each nonempty difference of open sets contains a nonempty closed set.…”

Note on hit-and-miss topologies

“…This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topology for weakly-base spaces.…”

“…be a topological space and be the nonempty closed subsets of . Following [2], [3], [5], [16], [17], [19], [20], [21] we will continue to study hit-and-miss hyperspace topologies or -topologies on , where is a fixed subfamily of . Two of the most studied hit-and-miss topologies are the Vietoris topology ( [14], [13]) and the Fell topology ( [7], [13], [17]).…”

“…It is the purpose of this paper to show that quite similar characterizations hold with no preliminary conditions (or with much less restrictive conditions) on or , respectively. This is achieved by applying techniques and notions from [19] and a completely new approach is employed to characterize complete regularity of . Note that a characterization of normality of is not known except for some special cases, like the Vietoris topology ( [11], [18]) or the Fell topology ( [10]); for some more general results on normality see [6].…”