Metric Bornologies and Kuratowski-Painlevé Convergence to the Empty Set

Given a sequence $\langle T_n\rangle$ 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 $\langle T_n\rangle$ 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 $\mathcal{A}$ 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 $\mathcal{A}$ 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.