Resumen de Total boundedness in metrizable spaces

Gerald Beer, Camillo Costantini, Sandro Levi

• We show that a metric space (X,d ) is separable if and only if the bornology of its d-bounded subsets agrees with the bornology of ?-totally bounded subsets with respect to some equivalent remetrization ?. We also show that the bornology of d-totally bounded subsets agrees with the bornology of ?-bounded subsets with respect to some equivalent remetrization if and only if the former bornology has a countable cofinal subfamily. Finally, we characterize those bornologies on a metrizable space that are bornologies of totally bounded sets as determined by some metric compatible with the topology