Cauchy Metrizability of Bornological Universes

We call a bornology on a metric space (X, d) d-Cauchy metrizable if there exists a metric ρ on X, Cauchy equivalent to d, such that the family of ρ-bounded subsets coincides with the bornology. Recall that two metrics on a set are said to be Cauchy equivalent if the collections of Cauchy sequences with respect to both the metrics are same. In this paper we give necessary and sufficient conditions for a bornology on a metric space (X, d) to be d-Cauchy metrizable. We solve this problem for two different approaches, one given by S.-T. Hu [Boundedness in a topological space, J. Math. Pures Appl. 28 (1949) 287-320; Introduction to General Topology, Holden-Day, San Francisco (1966)] and the other given by G. Beer [On metric boundedness structures, Set-Valued Anal. 7 (1999) 195-208]. Furthermore, we investigate the same for some most common bornologies.