Metric Bornologies and Kuratowski-Painlevé Convergence to the Empty Set

Given a sequence

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 ⟨ 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

\mathcal{A} A

consist of all subsets A of X whose closure hits

T_n T n

for at most finitely many indices n. We give necessary and sufficient conditions for

\mathcal{A} 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.