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

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