Between the cofinally complete spaces and the UC spaces

Abstract. The local finiteness functional for a metric space 〈X, d 〉 intu-itively describes the radius of the ”largest ” ball about each point of the space containing at most finitely many points of the space. We give characteriza-tions of those metric spaces in which each sequence along which the func-tional tends to zero necessarily clusters, placing this class of spaces strictly between the well-studied UC metric spaces and the cofinally complete metric spaces. We produce a subtle formula for the values of the functional for the hyperspace of nonempty closed subsets equipped with Hausdorff distance. Finally, we give necessary and sufficient conditions for the hyperspace to be of this type. 1.