Total boundedness in metrizable spaces

We show that a metric space hX; di is separable if and only if the bornology of its d-bounded subsets agrees with the bornology of ff-totally bounded subsets with respect to some equivalent remetrization ff. We also show that the bornology of d-totally bounded subsets agrees with the bornol-ogy of ff-bounded subsets with respect to some equivalent remetrization if and only if the former bornology has a countable coffnal 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. © 2011 University of Houston.