On the infimum of the Hausdorff metric topologies

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, Hd+ and Hd- respectively, on the collection C(X) of all closed subsets of X. In this paper we consider the infima τ, τ+ and τ−, of the topologies Hd, Hd+ and Hd- 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 τ+ and τ− are investigated in detail: a suitable topology U+ is defined which has the same convergent sequences as τ+, and the lower Vietoris topology V− plays a similar role with respect to τ−. We show that, in general, the equality τ = τ+ ∨ τ− does not hold. We also show that τ is a T2-topology on C(X) if and only if X is locally compact.