Quasi-metric spaces on which real-valued continuous functions are uniformly continuous
Om Dev Singh, Anubha Jindal
TL;DR
This work develops a systematic treatment of UC (Atsuji-type) properties in quasi-metric spaces, where asymmetry of the distance $d$ complicates the transfer of metric results. It defines UC quasi-metric spaces and shows that spaces like $(\mathbb{R},d_{l})$ and $(\mathbb{R},d_{u})$ are UC, while forward continuity alone does not guarantee uniform continuity or left K-completeness in general. The paper provides a suite of characterizations—via isolation functionals, uniform discreteness, the Lebesgue-number property, and pseudo left K-Cauchy sequences—and proves Cantor-intersection-type criteria that parallel classical compactness results. It also develops inheritance properties for closed subsets and reciprocal-function criteria under suitable regularity, illuminating the delicate balance between topology and asymmetry in UC quasi-metric spaces. These results deepen understanding of when continuous real-valued maps are automatically uniformly continuous in asymmetric settings, with implications for hyperspaces and fixed-point theory in quasi-metric contexts.
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.