Hyperspaces with a countable character of closed subsets

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.