Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-Gutiérrez¹ ; Paul J. Szeptycki¹

¹ Department of Mathematics and Statistics York University Toronto, ON M3J 1P3, Canada

Fundamenta Mathematicae, Tome 228 (2015) no. 1, pp. 63-79

Résumé

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.

Détail

DOI : 10.4064/fm228-1-5

Keywords: given metric space langle rho rangle consider its hyperspace closed sets wijsman topology tau rho known langle tau rho rangle metrizable only separable question maio meccariello whether equivalent langle tau rho rangle being normal prove weight regular uncountable cardinal locally separable langle tau rho rangle normal solve questions cao junnila moors regarding isolated points wijsman hyperspaces