Wijsman hyperspaces of non-separable metric spaces (Q2928548)

Let \((X,\rho)\) be a metric space and \((CL(X),\tau_{W(\rho)}) \) the hyperspace \(CL(X)\) of all closed non-empty subsets of \(X\) with the Wijsman topology. In 1998, Di Maio and Meccariello asked the followingNEWLINENEWLINE{ Question.} Let \((X,\rho)\) be a metric space. If \((CL(X),\tau_{W(\rho)}) \) is normal, is it true that \(X\) is separable\(?\)NEWLINENEWLINEThe main result of this paper is the following theorem.NEWLINENEWLINE{Theorem.} Let \(\kappa\) be a regular uncountable cardinal. If \((X,\rho)\) is a locally separable metric space of weight \(\kappa\), then \((CL(X),\tau_{W(\rho)})\) is not normal.