On complete metrizability of the Hausdorff metric topology

Abstract

There exists a completely metrizable bounded metrizable space $X$ with compatible metrics $d,d^{\prime }$ so that the hyperspace $CL(X)$ of nonempty closed subsets of $X$ endowed with the Hausdorff metric $H_d$, $H_{d^{\prime }}$, 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.