The Fell Topology and Kuratowski-Painlevé Convergence

Abstract

One of the most important and well-studied hit-and-miss hyperspace topologies on CL(X) is the Fell topology, where the compact subsets of the underlying space play the role of miss sets. This hyperspace topology when extended to 2^X in the natural way has a remarkable property: it is always compact, independent of the character of the underlying space! Closely linked with the Fell topology is classical Kuratowski-Painlevé convergence of sets, based on the notions of upper and lower closed limits of a net of sets. We study convergence in this sense of extended real-valued lower semicontinuous functions as identified with their epigraphs, a basic tool in one-sided optimization problems, and characterize convergence of functions in terms of sublevel sets. Finally, we study Mosco convergence of sequences of weakly closed sets in a normed linear space, which is compatible with the supremum of the Fell topologies determined by the weak and strong topologies. In the last section, we consider in detail the relationship between Wijsman and Mosco convergence of weakly closed sets, and of closed convex sets.