Hypertopologies on ω_µ-ˆ’Metric spaces

The ωµ−metric spaces, with ωµ a regular ordinal number, are sets equipped with a distance valued in a totally ordered abelian group having as character ωµ, but satisfying the usual formal properties of a real metric. The ωµ−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 ωµ−metric spaces, in particular the Hausdorff topology induced by the Bourbaki-Hausdorff uniformity associated with their natural underlying uniformity. We show that two ωµ−metrics on a 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 ωµ−metric setting, the relationship between the Kuratowski and Hausdorff convergences on CL(X) and prove that an ωµ−sequence Aαα<ωµwhich admits A as Kuratowski limit converges to A in the Hausdorff topology if and only if the join of A with all Aαis ωµ−compact.