Wijsman hyperspaces of non-separable metric spaces
Volume 228 / 2015
Fundamenta Mathematicae 228 (2015), 63-79
MSC: 54B20, 54D15.
DOI: 10.4064/fm228-1-5
Abstract
Given a metric space $\langle X,\rho\rangle$, consider its hyperspace of closed sets ${\rm CL}(X)$ with the Wijsman topology $\tau_{W(\rho)}$. It is known that $\langle{{\rm CL}(X),\tau_{W(\rho)}}\rangle$ is metrizable if and only if $X$ is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $\langle{{\rm CL}(X), \tau_{W(\rho)}}\rangle$ being normal. We prove that if the weight of $X$ is a regular uncountable cardinal and $X$ is locally separable, then $\langle{{\rm CL}(X),\tau_{W(\rho)}}\rangle$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.
