Abstract

Given a sequence ⟨Tn⟩\langle T_n\rangle 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⟩\langle T_n\rangle 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\mathcal{A} consist of all subsets A of X whose closure hits TnT_n for at most finitely many indices n. We give necessary and sufficient conditions for A\mathcal{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.

Author information

Contact details are reproduced from the original publication and may be historical.

Gerald Beer

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

Keywords

  • Bounded set
  • metric bornology
  • Kuratowski-Painlevé convergence
  • UC space
  • metric mode of convergence to infinity

Mathematics Subject Classification

54E35; 54B20

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Suggested citation

G. Beer. “Metric Bornologies and Kuratowski-Painlevé Convergence to the Empty Set.” Journal of Convex Analysis 8 (2001), No. 1, 279–289. https://doi.org/10.68381/jca08014

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Published by Heldermann Verlag, 2001. Rights now held by Banach Press LLC.