Total boundedness in metrizable spaces

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 ρ-totally bounded subsets with respect to some equivalent remetrization ρ. We also show that the bornology of d-totally bounded subsets agrees with the bornology of ρ-totally 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 X that are bornologies of totally bounded sets as determined by some metric compatible with the topology of X.