
Journal of Convex Analysis 08 (2001), No. 1, 279289 Copyright Heldermann Verlag 2001 Metric Bornologies and KuratowskiPainleve Convergence to the Empty Set Gerald Beer Dept. of Mathematics, California State University, Los Angeles, CA 90032, U.S.A. Given a sequence {T_{n}} of nonempty closed sets KuratowskiPainlevé convergent to the empty set in a noncompact metrizable space X, we show not only that there exists an admissible unbounded metric such that {T_{n}} converges to infinity in distance, but also that there must exist another such metric for which this is not the case. For such a sequence, let A consist of all subsets A of X whose closure hits T_{n} for at most finitely many indices n. We give necessary and sufficient conditions for A to be the family of bounded sets induced by some admissible metric for X, and show that all possible nontrivial metric bornologies for X arise in this manner if and only if the derived set of X is compact. Keywords: Bounded set, metric bornology, KuratowskiPainleve convergence, UC space, metric mode of convergence to infinity. MSC: 54E35; 54B20 [ Fulltextpdf (308 KB)] 