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

Let X   be a metrizable space and Comp(X)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 Comp(X)Comp(X) being homeomorphic to a non-separable Hilbert space. Moreover, we consider the topological structure of pair (Comp(X‾),Fin(X)) of hyperspaces of X   and its completion X‾, where Fin(X)Fin(X) is the hyperspace of non-empty finite sets in X.