Hyperspaces with a countable character of closed subsets

DOI10.1016/J.TOPOL.2023.108461arXiv2206.13026MaRDI QIDQ6403204

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Publication date: 26 June 2022

Abstract: For a regular space $X$, the hyperspace $(CL(X),a{u}_F)$ (resp., $(CL(X),a{u}_V)$) 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 $(CL(X),a{u}_F)$ (resp., $(CL(X),a{u}_V)$) with a countable character of closed subsets. We mainly prove that $(CL(X),a{u}_F)$ has a countable character on each closed subset if and only if $X$ is compact metrizable, and $(CL(X),a{u}_F)$ has a countable character on each compact subset if and only if $X$ is locally compact and separable metrizable. Moreover, we prove that $(mathcalK(X),a{u}_V)$ have the compact-$G_{d}elta$ property if and only if $X$ have the compact-$G_{d}elta$ property and every compact subset of $X$ is metrizable.

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