On complete metrizability of the Hausdorff metric topology

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 α-favorable, β-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.2010 Mathematics Subject Classification. Primary 54B20; Secondary 54E50, 54E52, 91A44.