Hyperspaces on omega-mu-metric spaces

Abstract

The $\omega_\mu-$metric spaces, where $ \omega_\mu$ is a regular ordinal number, are sets equipped with a distance valued in a totally ordered abelian group having as character $\omega_\mu, $ but satisfying the usual formal properties of a real metric. The $\omega_\mu-$metric spaces fill a large and attractive class of peculiar uniform spaces, those with a linearly ordered base. In this paper we investigate hypertopologies associated with $\omega_\mu-$metric spaces, in particular the Hausdorff topology induced by the Bourbaki-Hausdorff uniformity associated with their natural underlying uniformity. We show that two $\omega_\mu-$metrics on the same topological space $X$ induce on the hyperspace $CL(X),$ the set of all non-empty closed sets of $X,$ the same Hausdorff topology if and only if they are uniformly equivalent. Moreover, we explore, again in the $\omega_\mu-$metric setting, the relationship between the Kuratowski and Hausdorff convergences on $CL(X)$ and prove that an $\omega_\mu-$sequence $\{A_{\alpha} \}_{\alpha < \omega_{\mu}}$ which admits $A$ as Kuratowski limit converges to $A$ in the Hausdorff topology if and only if the join of $A$ with all $A_{\alpha}$ is $\omega_\mu-$compact.