The Wijsman and Attouch-Wets topologies on hyperspaces revisited

In this paper we will prove that, for an arbitrary metric space X and a fairly arbitrary collection Σ of subsets of X, it is possible to endow the hyperspace CL(X) of all nonempty closed subsets of X (to be identified with their distance functionals) with a canonical distance function having the topology of uniform convergence on members of Σ as topological coreflection and the Hausdorff metric as metric coreflection. For particular choices of Σ, we obtain canonical distance functions overlying the Wijsman and Attouch-Wets topologies. Consequently we apply the general theory of spaces endowed with a distance function and compare the results with those obtained for the classical hyperspace topologies. In all cases we are able to prove results which are both stronger and more general than the classical ones.