The quasi-metrizability of hyperspaces

Abstract

For a space $X$, let $(CL(X), \tau_V)$, $(CL(X), \tau_{locfin})$ and $(CL(X), \tau_F)$ be the set $CL(X)$ of all nonempty closed subsets of $X$ which are endowed with Vietoris topology, locally finite topology and Fell topology respectively. We prove that $(CL(X), \tau_V)$ is quasi-metrizable if and only if $X$ is a separable metrizable space and the set of all non-isolated points of $X$ is compact, $(CL(X), \tau_{locfin})$ is quasi-metrizable or symmetrizable if and only if $X$ is metrizable and the set of all non-isolated points of $X$ is compact, and $(CL(X), \tau_F)$ is quasi-metrizable if and only if $X$ is hemicompact and metrizable. As an application, we give a negative answer to a Conjecture in \cite{LL2022}.