On complete metrizability of the Hausdorff metric topology

TL;DR

This paper investigates the conditions under which the hyperspace of nonempty closed subsets of a completely metrizable space, equipped with the Hausdorff metric, retains or loses complete metrizability, revealing nuanced topological properties.

Contribution

It demonstrates that the hyperspace topology can vary in metrizability properties depending on the chosen compatible metrics, providing new insights into hyperspace topology.

Findings

Existence of a completely metrizable space with non-completely metrizable hyperspace

Different compatible metrics can lead to different topological properties of hyperspaces

Hyperspaces may not inherit complete metrizability from the base space

Abstract

There exists a completely metrizable bounded metrizable space $X$ with compatible metrics $d,d'$ so that the hyperspace $CL(X)$ of nonempty closed subsets of $X$ endowed with the Hausdorff metric $H_d$, $H_{d'}$, resp. is $\alpha$-favorable, $\beta$-favorable, resp. in the strong Choquet game. In particular, there exists a completely metrizable bounded metric space $(X,d)$ such that $(CL(X),H_d)$ is not completely metrizable.