Note on the Wijsman hyperspaces of completely metrizable spaces

Chaber, J. ; Pol, R.

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 827-832

Résumé

We consider the hyperspace $CL(X)$ of nonempty closed subsets of completely metrizable space $X$ endowed with the Wijsman topologies $\tau_{W_{d}}$. If $X$ is separable and $d$, $e$ are two metrics generating the topology of $X$, every countable set closed in $(CL(X), \tau_{W_{e}})$ has isolated points in $(CL(X), \tau_{W_{d}})$. For $d=e$ , this implies a theorem of Costantini on topological completeness of $(CL(X), \tau_{W_{d}})$. We show that for nonseparable $X$ the hyperspace $(CL(X), \tau_{W_{d}})$ may contain a closed copy of the rationals. This answers a question of Zsilinszky.