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Journal of Convex Analysis 08 (2001), No. 1, 279--289
Copyright Heldermann Verlag 2001

Metric Bornologies and Kuratowski-Painleve Convergence to the Empty Set

Gerald Beer
Dept. of Mathematics, California State University, Los Angeles, CA 90032, U.S.A.

Given a sequence {Tn} of nonempty closed sets Kuratowski-Painlevé convergent to the empty set in a noncompact metrizable space X, we show not only that there exists an admissible unbounded metric such that {Tn} 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 Tn 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, Kuratowski-Painleve convergence, UC space, metric mode of convergence to infinity.

MSC: 54E35; 54B20

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