Cofinal completeness vis-á-vis hyperspaces

Abstract

A metric space is called cofinally complete if every cofinally Cauchy sequence in it has a cluster point. For a metric space (X, d), we consider the set AC(X) of almost nowhere locally compact sets (Beer and Di Maio in Acta Math Hung 134(3):322–343, 2012), which is a subset of the set of closed subsets in X. We consider some hyperspace topologies on the set AC(X) which it inherits from the set of non-empty closed subsets in X as subspace topologies and characterize cofinally complete metric spaces in terms of relations to Hausdorff metric topology, proximal topology, Vietoris topology and locally finite topology on AC(X). We characterize cofinally complete metric spaces in terms of some function space topologies as well. Furthermore, we characterize the class of metric spaces for which the corresponding space AC(X) equipped with the Hausdorff metric topology is cofinally complete.