It is well-known that if \(X\) is locally compact and Lindelöf then the compact-open topology on the set of real-valued continuous functions is completely metrizable, see \textit{R. F. Arens} [Ann. Math. (2) 47, 480--495 (1946; Zbl 0060.39704)]; the key property here is that there is a countable family of compact sets whose interiors cover~\(X\). The author generalizes this to topological spaces with a bornology (an ideal of nonempty subsets that also covers the space). Under suitable assumptions including that the bornology has a countable cofinal family consisting of closed sets, one can characterize (complete) metrizability of the topology of strong uniform convergence on members of the bornology.