Hyperspaces of finite subsets of non-separable Hilbert spaces

Abstract

Let $\ell_{2}(\tau)$ be the Hilbert space with weight $\tau$ and $\ell_{2}^{f}$ be the linear span of the canonical orthonormal basis of the separable Hilbert space $\ell_{2}$. In this paper, we prove that if a metric space $X$ ishomeomorphic to $\ell_{2}(\tau)$ or $\ell_{2](\tau) \times \ell_{2}^{f}$ then the hyperspace $Fin_{H}(X)$ of non-empty finite subsets of $X$ with the Hausdorff metric is homeomorphic to $\ell_{2}(\tau) \times \ell_{2}^{f}$.