Distance functionals and suprema of hyperspace topologies

Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A --> d(x,A) is continuous for each x is-an-element-of X and each admissible metric d, We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) --> --> inf{d(a, b): a is-an-element-of A, b is-an-element-of B} is continuous on CL(X) x CL(X)