S-topologies and bounded convergences

Recently, Lechicki, Levi and Spakowski have studied set convergence of the Attouch–Wets type expressed in terms of truncations by members of a prescribed family of sets. When this family is either the family of metrically bounded sets or the family of all nonempty subsets, it is well known that convergence means uniform convergence of distance functions on members of the family. In this paper we give necessary and sufficient conditions for coincidence to occur when the family is a general bornology of subsets.