On Quasi-Modular Pseudometric Spaces and Asymmetric Uniformities

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.