Hyperspaces with a countable character of closed subsets

TL;DR

This paper characterizes when hyperspaces of closed subsets of a regular space have a countable character, linking these properties to compactness, metrizability, and separability of the underlying space.

Contribution

It provides necessary and sufficient conditions for hyperspaces with Fell and Vietoris topologies to have a countable character based on properties of the original space.

Findings

$(CL(X), \tau_F)$ has countable character iff $X$ is compact metrizable.

$(CL(X), \tau_F)$ has countable character on compact subsets iff $X$ is locally compact and separable metrizable.

$(\mathcal{K}(X), \tau_V)$ has the compact-$G_\delta$ property iff $X$ has it and all compact subsets are metrizable.

Abstract

For a regular space $X$, the hyperspace $(CL(X), \tau_{F})$ (resp., $(CL(X), \tau_{V})$) is the space of all nonempty closed subsets of $X$ with the Fell topology (resp., Vietoris topology). In this paper, we give the characterization of the space $X$ such that the hyperspace $(CL(X), \tau_{F})$ (resp., $(CL(X), \tau_{V})$) with a countable character of closed subsets. We mainly prove that $(CL(X), \tau_F)$ has a countable character on each closed subset if and only if $X$ is compact metrizable, and $(CL(X), \tau_F)$ has a countable character on each compact subset if and only if $X$ is locally compact and separable metrizable. Moreover, we prove that $(\mathcal{K}(X), \tau_V)$ have the compact-$G_\delta$ property if and only if $X$ have the compact-$G_\delta$ property and every compact subset of $X$ is metrizable.