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