Journal of Convex Analysis 17 (2010), No. 3&4, 805--826
Copyright Heldermann Verlag 2010
Topologies Associated with Kuratowski-Painlevé Convergence of Closed Sets
Gerald Beer
Dept. of Mathematics, California State University, 5151 State University Drive, Los Angeles, CA 90032, U.S.A.
gbeer@cslanet.calstatela.edu
Jesús Rodríguez-López
Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica, 46022 Valencia, Spain
jrlopez@mat.upv.es
The main purpose of this paper is to identify topologies on the closed subsets C(X) of a Hausdorff space X that are sequentially equivalent to classical Kuratowski-Painlevé convergence K. This reduces to a study of upper topologies sequentially equivalent to upper Kuratowski-Painlevé convergence K+, where we are of course led to consider the sequential modification of upper Kuratowski-Painlevé convergence. We characterize those miss topologies induced by a cobase of closed sets that are sequentially equivalent to K+, with special attention given to X first countable. Separately in the final section, we revisit Mrowka's theorem on the compactness of Kuratowski-Painlevé convergence.
Keywords: Fell topology, Kuratowski-Painleve convergence, hyperspace, hit-and-miss topology, modification, sequential modification, subsequential selector
MSC: 54B20; 54A20, 54E35
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