Complete metrizability of topologies of strong uniform convergence on bornologies

We continue the study of topologies of strong uniform convergence on bornologies initiated by Beer and Levi (2009) [4], . In Beer and Levi (2009) [4] the metrizability of such topologies restricted to continuous functions was characterized and a compatible metric was displayed. We find necessary and sufficient conditions for completeness of the metric, for complete metrizability and for Polishness of topologies of strong uniform convergence on bornologies on continuous functions. Polishness of such topologies forces their coincidence with the compact-open topology.