On The Infimum of the Hausdorff Metric Topologies

Abstract

Given a metrizable space X and a compatible metric d, one defines the Hausdorff metric topology Hd and the upper and lower Hausdorff topologies corresponding to d, Ĥ and Ĥ respectively, on the collection ^(X) of all closed subsets of X. In this paper we consider the infima z, T and r~, of the topologies Hd, HJ and HJ respectively, where d runs over the set M{X) of all compatible metrics on X. These topologies are sequential, that is, they are completely characterized by convergent sequences. In particular, the topologies r and r~ are investigated in detail: a suitable topology U is defined which has the same convergent sequences as T, and the lower Vietoris topology V~ plays a similar role with respect to T~. We show that, in general, the equality T = T + V T does not hold. We also show that T is a r2-topology on ^(X) if and only if X is locally compact.