Hyperspaces with the Attouch-Wets topology homeomorphic to $l_2$

TL;DR

This paper characterizes when the hyperspace of nonempty closed subsets of a separable metric space, with the Attouch-Wets topology, is homeomorphic to a separable Hilbert space, based on properties of the space.

Contribution

It provides necessary and sufficient conditions for the hyperspace to be homeomorphic to a Hilbert space, linking topological properties of the base space to the hyperspace topology.

Findings

Hyperspace is homeomorphic to a Hilbert space if the completion is proper, locally connected, and has no bounded connected component.

The space must be topologically complete and not locally compact at infinity.

Conditions are both necessary and sufficient for the homeomorphism.

Abstract

It is shown that the hyperspace of all nonempty closed subsets $\Cld_{AW}(X)$ of a separable metric space $X$ endowed with the Attouch-Wets topology is homeomorphic to a separable Hilbert space if and only if the completion of $X$ is proper, locally connected and contains no bounded connected component, $X$ is topologically complete and not locally compact at infinity.