On a hyperspace of compact subsets which is homeomorphic to a non-separable Hilbert space

Abstract

Let $X$ be a metrizable space and $\mathrm{Comp}(X)$ be the hyperspace consisting of non-empty compact subsets of $X$ endowed with the Vietoris topology. In this paper, we give a necessary and sufficient condition on $X$ for $\mathrm{Comp}(X)$ to be homeomorphic to a non-separable Hilbert space. Moreover, we consider the topological structure of pair $(\mathrm{Comp}(\overline{X}),\mathrm{Fin}(X))$ of hyperspaces of $X$ and its completion $\overline{X}$, where $\mathrm{Fin}(X)$ is the hyperspace of non-empty finite sets in $X$.

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