Topological structures of hyperspaces of finite sets in non-separable metrizable spaces

Let $\mathrm{F}\mathrm{i}\mathrm{n}(X)$ be the hyperspace consisting of non-empty finite subsets of a space $X$ endowed with the Vietoris topology. In this paper, we characterize a metrizable space $X$ whose hyperspace $\mathrm{F}\mathrm{i}\mathrm{n}(X)$ is homeomorphic to the linear subspace spanned by the canonical orthonormal basis of a non-separable Hilbert space.