On a characterization of complete metrizability of the Hausdorff metric topology
Volume 268 / 2025
Fundamenta Mathematicae 268 (2025), 215-223
MSC: Primary 54B20; Secondary 54E52, 91A44
DOI: 10.4064/fm230623-3-2
Published online: 20 February 2025
Abstract
It is shown, under CH, that the Hausdorff metric topology on the hyperspace $CL(X)$ of all nonempty closed subsets of a metric space $(X,d)$ is completely metrizable if and only if $(X,d)$ is completely metrizable and its completion remainder $(\tilde X\setminus X,\tilde d)$ is separable.
