On Quasi-Modular Pseudometric Spaces and Asymmetric Uniformities

TL;DR

This paper explores quasi-modular pseudometric spaces, introducing directional topologies and properties, and demonstrates how symmetrization affects completeness and compactness, revealing differences between forward and backward notions.

Contribution

It introduces directional concepts in quasi-modular pseudometric spaces and analyzes their properties, highlighting differences from symmetric cases using enriched category theory.

Findings

Forward and backward completeness may differ.

Compactness of symmetrized uniformity does not imply directional compactness.

Symmetrization yields a symmetric enriched category with classical uniform completion.

Abstract

We study quasi-modular pseudometric spaces as asymmetric refinements of modular metric structures. To each such space we associate canonical forward and backward quasi-uniformities and the corresponding directional topologies. We introduce directional notions of convergence, completeness, total boundedness, and compactness, and show that these properties are not preserved under symmetrization. In particular, forward and backward completeness may differ, and compactness of the symmetrized uniformity does not imply directional compactness. Using enriched category theory as a comparison framework, we show that symmetrization yields a symmetric enriched category whose Cauchy completion coincides with the classical uniform completion, while directional notions remain invisible at this level.