On Metrizability, Completeness and Compactness in Modular Pseudometric Topologies

Abstract

Building on the recent work of Mushaandja and Olela-Otafudu~\cite{MushaandjaOlela2025} on modular metric topologies, this paper investigates extended structural properties of modular (pseudo)metric spaces. We provide necessary and sufficient conditions under which the modular topology $\tau(w)$ coincides with the uniform topology $\tau(\mathcal{V})$ induced by the corresponding pseudometric, and characterize this coincidence in terms of a generalized $\Delta$-condition. Explicit examples are given where $\tau(w)\subsetneq\tau(\mathcal{V})$, demonstrating the strictness of inclusion. Completeness, compactness, separability, and countability properties of modular pseudometric spaces are analysed, with functional-analytic analogues identified in Orlicz-type modular settings. Finally, categorical and fuzzy perspectives are explored, revealing structural invariants distinguishing modular from fuzzy settings.