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

TL;DR

This paper characterizes when the hyperspace of non-empty compact subsets of a metrizable space is homeomorphic to a non-separable Hilbert space, and explores the topological structure of related hyperspaces.

Contribution

It provides a necessary and sufficient condition on the space for its hyperspace to be homeomorphic to a non-separable Hilbert space, advancing understanding of hyperspace topology.

Findings

Characterization of spaces whose hyperspaces are homeomorphic to non-separable Hilbert spaces

Analysis of the topological structure of hyperspaces involving finite sets and their completions

Conditions linking the properties of a space to the topology of its hyperspace

Abstract

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