Quasi-metric spaces on which real-valued continuous functions are uniformly continuous

TL;DR

This paper explores quasi-metric spaces where all real-valued continuous functions are uniformly continuous, providing characterizations, examples, and highlighting differences from metric spaces.

Contribution

It systematically characterizes UC quasi-metric spaces and demonstrates key differences from metric spaces through various examples and proof techniques.

Findings

Characterizations of UC quasi-metric spaces

Examples illustrating UC quasi-metric spaces

Differences between UC quasi-metric and metric spaces

Abstract

The concept of a quasi-metric space arises by relaxing the requirement of the symmetry axiom in the definition of a metric. This small variation alters several structural properties possessed by a standard metric space. This article aims to investigate the notion of UC quasi-metric spaces in a systematic manner. A quasi-metric space (X, d) is called a UC space if every real-valued continuous function on (X, d) is uniformly continuous. In the context of metric spaces, UC spaces help in bridging the gap between compactness and completeness. These spaces also play an important role in the theory of hyperspaces of closed sets and fixed point theory. In this article, we present several characterizations of UC quasi-metric spaces and provide various examples of such spaces. At several instances, our proof techniques highlight key differences between UC quasi-metric spaces and their metric counterparts.