Uniform Continuity and a New Bornology for a Metric Space

Abstract

In the context of functions between metric spaces, continuity is preserved by uniform convergence on the bornology of relatively compact subsets while Cauchy continuity is preserved under uniform convergence on the bornology of totally bounded subsets. We identify a new bornology for a metric space containing the bornology of Bourbaki bounded sets on which uniform convergence preserves uniform continuity. Further, for real-valued uniformly continuous functions, the function space is a ring (with respect to pointwise multiplication) if and only if the two bornologies agree. We show that Cauchy continuity is preserved by uniform convergence on compact subsets if and only if the domain space is complete, and that uniform continuity is preserved under uniform convergence on totally bounded subsets if and only if the domain space has UC completion. Finally, uniform continuity is preserved under uniform convergence on compact subsets if and only if the domain space is a UC-space. We prove a simple omnibus density result for Lipschitz functions within a larger class of continuous functions equipped with a topology of uniform convergence on a bornology and apply that to each of our three function classes.