Metric Compactifications and Coarse Structures

Abstract

AbstractLet TB be the category of totally bounded, locally compact metric spaces with the C0 coarse structures. We show that if X and Y are in TB, then X and Y are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories TB → K, where K is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space X induced by some metrizable compactification is determined only by the topology of the remainder .