On the Fell topology

Let 2 ^X denote the closed subsets of a Hausdorff topological space <X, {gt}>. The Fell topology _F on 2 ^X has as a subbase all sets of the form {A 2 ^X :A V 0}, whereV is an open subset ofX, plus all sets of the form {A 2 ^X :A W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for _F in terms of topological properties for . Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.