Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Metrizability, Completeness and Compactness in Modular Pseudometric Topologies

Philani Rodney Majozi

TL;DR

This work addresses the relationship between modular topologies $τ(w)$ and the uniform-topology $τ(V)$ induced by the associated pseudometric, establishing necessary and sufficient conditions for their equality via a generalized $Δ_2$-condition and providing counterexamples when inclusion is strict. It develops complete, compact, and separable theories for modular spaces, introduces Orlicz–Musielak-type analogues, and connects these ideas to fuzzy metrics and categorical enrichment. Key contributions include precise equivalences of topologies under $Δ_2$, transfer principles for Cauchy and compactness properties, and a Lawvere-enriched categorical framework that situates modular spaces alongside uniform spaces and function-space theory. The results offer a unified toolkit for analysis on modular spaces, with implications for reflexivity, precompact embeddings, and structural invariants across modular, uniform, and fuzzy settings.

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 $τ(w)$ coincides with the uniform topology $τ(\mathcal{V})$ induced by the corresponding pseudometric, and characterize this coincidence in terms of a generalized $Δ$-condition. Explicit examples are given where $τ(w)\subsetneqτ(\mathcal{V})$, 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.

On Metrizability, Completeness and Compactness in Modular Pseudometric Topologies

TL;DR

This work addresses the relationship between modular topologies and the uniform-topology induced by the associated pseudometric, establishing necessary and sufficient conditions for their equality via a generalized -condition and providing counterexamples when inclusion is strict. It develops complete, compact, and separable theories for modular spaces, introduces Orlicz–Musielak-type analogues, and connects these ideas to fuzzy metrics and categorical enrichment. Key contributions include precise equivalences of topologies under , transfer principles for Cauchy and compactness properties, and a Lawvere-enriched categorical framework that situates modular spaces alongside uniform spaces and function-space theory. The results offer a unified toolkit for analysis on modular spaces, with implications for reflexivity, precompact embeddings, and structural invariants across modular, uniform, and fuzzy settings.

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 coincides with the uniform topology induced by the corresponding pseudometric, and characterize this coincidence in terms of a generalized -condition. Explicit examples are given where , 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.
Paper Structure (31 sections, 27 theorems, 48 equations)

This paper contains 31 sections, 27 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries and Definitions
  4. Modular (pseudo)metrics and modular sets
  5. The modular topology and a canonical uniformity
  6. Examples
  7. Variants and auxiliary constructions
  8. Fuzzy metrics
  9. Topology-Uniformity Comparisons
  10. Uniformities naturally attached to a modular metric
  11. Comparing $\tau(w)$ and $\tau(\mathcal{V})$
  12. Consequences imported from uniform space theory
  13. Standard examples
  14. Fuzzy metrics as uniformities
  15. Completeness and Compactness
  16. ...and 16 more sections

Key Result

Lemma 2.5

If $\varphi:(0,\infty)\to(0,\infty)$ is nondecreasing and $w$ is convex with $\lambda\mapsto \lambda\varphi(\lambda)$ nondecreasing, then for every $x\in X_w^\ast$ the set $\bigcup_{\lambda>0}B^w_{\lambda,\varphi(\lambda)}(x)$ is $\tau(w)$-open Chistyakov2015.

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • Remark 2.6
  • Example 2.7
  • Example 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 56 more