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S-topologies and bounded convergences

The study of convergences of closed subsets of a topological space has played an important role in several mathematical branches: functional analysis, optimization theory, variational analysis, etc. (see [1]). Traditionally, given a topological space (X, τ) the topologies defined on a family of subsets of X have been called hypertopologies. The origin of hyperspace theory is due to Hausdorff when he defined the so-called Hausdorff distance. Given a metric space (X, d) the family CLB0(X) of all nonempty closed bounded subsets of X can be endowed with the following metric Hd(A,B) = max{sup a∈A d(a,B), sup b∈B d(A, b)}.