Cofinal completion vis-á-vis Cauchy continuity and total boundedness

Abstract

A function f from a metric space (X,d) to another metric space (Y,ρ) is said to be Cauchy-continuous if for every Cauchy sequence (xn) in (X,d), (f(xn)) is Cauchy in (Y,ρ). It is well known that a metric space is complete if and only if every real-valued continuous function defined on it is Cauchy-continuous. Here we consider a well-studied intermediate class of metric spaces which lies between the class of compact metric spaces and that of complete metric spaces called cofinally complete metric spaces. In this paper, we use Cauchy-continuous functions and some Lipschitz-type functions to study a more general case, that is, to study the metric spaces which have a cofinal completion. We also study a geometric functional that measures the local total boundedness of a metric space at its each point and characterize the aforesaid metric spaces using it.