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

TL;DR

This paper characterizes when the hyperspace of finite subsets of a metrizable space is topologically equivalent to a specific subspace of a non-separable Hilbert space, revealing deep connections between topology and functional analysis.

Contribution

It provides a characterization of spaces whose finite-set hyperspaces are homeomorphic to a linear subspace of a non-separable Hilbert space.

Findings

Identifies conditions for ${ m Fin}(X)$ to be homeomorphic to a Hilbert space subspace

Connects hyperspace topology with non-separable Hilbert space structures

Advances understanding of hyperspaces in non-separable metrizable spaces

Abstract

Let ${\rm Fin}(X)$ 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 ${\rm Fin}(X)$ is homeomorphic to the linear subspace spanned by the canonical orthonormal basis of a non-separable Hilbert space.