Metric Bornologies and Kuratowski-Painleve Convergence to the Empty Set

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.