Universally uniformly continuous metric spaces

TL;DR

This paper characterizes metric spaces where every continuous function is uniformly continuous, extending previous results and applying to infinite-dimensional spaces.

Contribution

It provides a new, generalized characterization theorem for such metric spaces, improving upon earlier work by Levine and Saunders.

Findings

Characterization of metric spaces with all continuous functions uniformly continuous

Generalization of previous results to infinite-dimensional spaces

Applicable to a broad class of metric spaces

Abstract

We answer the question: "on which metric spaces $(M,d)$ are all continuous functions uniformly continuous?" Our characterization theorem improves and generalizes a previous result due to Levine and Saunders, and in particular is applicable to metric spaces which are "infinite dimensional."