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Local Antisymmetric Connectedness in Quasi-Uniform and Quasi-Modular Spaces

Philani Rodney Majozi

TL;DR

The paper addresses the limitation of symmetric connectedness in genuinely nonsymmetric topologies by introducing antisymmetric and local antisymmetric connectedness within a unified framework of quasi-uniform and quasi-modular pseudometric spaces. It builds a canonical bitopological structure from forward and backward modular topologies, develops completion theories (Smyth and Yoneda), and proves stability results under subspaces, uniformly continuous maps, and bicompletion. Key contributions include precise characterizations of antisymmetric connectivity, local variants with barrier criteria, and functional-analytic consequences such as compactness transfers via precompactness and Smyth completeness, alongside concrete examples in asymmetric normed spaces, directed graphs, Orlicz-modular settings, and paratopological groups. The results provide a robust directional toolkit for analysis in nonsymmetric contexts and establish connections between completion, compactness, and directional separation that can inform further study in fixed-point theory, nonlinear analysis, and directed-structure models. The work also highlights potential categorical viewpoints via Yoneda-type completions and enriched-theoretic interpretations of modular balls and formal balls.

Abstract

Directional notions in topology and analysis naturally lead to nonsymmetric structures such as quasi-metrics, quasi-uniformities, and modular spaces. In these settings, classical notions of connectedness and completion based on symmetric uniformities are often inadequate. In this paper, we study \emph{antisymmetric connectedness} and \emph{local antisymmetric connectedness} within the setting of quasi-uniform and quasi-modular pseudometric spaces. We associate to each quasi-modular pseudometric family compatible forward and backward modular topologies and quasi-uniformities, yielding a canonical bitopological structure. Using this setting, we establish characterization and stability results for local antisymmetric connectedness, including invariance under subspaces, uniformly continuous mappings, and bicompletion. We further relate these notions to Smyth completeness and Yoneda-type completions and show how precompactness combined with asymmetric completeness yields compactness in the join topology. Applications to asymmetric normed and modular spaces illustrate the theory.

Local Antisymmetric Connectedness in Quasi-Uniform and Quasi-Modular Spaces

TL;DR

The paper addresses the limitation of symmetric connectedness in genuinely nonsymmetric topologies by introducing antisymmetric and local antisymmetric connectedness within a unified framework of quasi-uniform and quasi-modular pseudometric spaces. It builds a canonical bitopological structure from forward and backward modular topologies, develops completion theories (Smyth and Yoneda), and proves stability results under subspaces, uniformly continuous maps, and bicompletion. Key contributions include precise characterizations of antisymmetric connectivity, local variants with barrier criteria, and functional-analytic consequences such as compactness transfers via precompactness and Smyth completeness, alongside concrete examples in asymmetric normed spaces, directed graphs, Orlicz-modular settings, and paratopological groups. The results provide a robust directional toolkit for analysis in nonsymmetric contexts and establish connections between completion, compactness, and directional separation that can inform further study in fixed-point theory, nonlinear analysis, and directed-structure models. The work also highlights potential categorical viewpoints via Yoneda-type completions and enriched-theoretic interpretations of modular balls and formal balls.

Abstract

Directional notions in topology and analysis naturally lead to nonsymmetric structures such as quasi-metrics, quasi-uniformities, and modular spaces. In these settings, classical notions of connectedness and completion based on symmetric uniformities are often inadequate. In this paper, we study \emph{antisymmetric connectedness} and \emph{local antisymmetric connectedness} within the setting of quasi-uniform and quasi-modular pseudometric spaces. We associate to each quasi-modular pseudometric family compatible forward and backward modular topologies and quasi-uniformities, yielding a canonical bitopological structure. Using this setting, we establish characterization and stability results for local antisymmetric connectedness, including invariance under subspaces, uniformly continuous mappings, and bicompletion. We further relate these notions to Smyth completeness and Yoneda-type completions and show how precompactness combined with asymmetric completeness yields compactness in the join topology. Applications to asymmetric normed and modular spaces illustrate the theory.
Paper Structure (31 sections, 22 theorems, 80 equations)

This paper contains 31 sections, 22 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi-Metric Spaces
  4. Quasi-Uniform Spaces
  5. Asymmetric Normed Spaces
  6. Approach Spaces
  7. Quasi-Modular Pseudometric Spaces
  8. Asymmetric modular families
  9. Induced topologies and bitopological structure
  10. Associated quasi-uniformities
  11. Completion Constructions in the Quasi-Modular Setting
  12. Smyth completeness and precompactness
  13. Yoneda completion and idempotence
  14. Symmetric and Antisymmetric Connectedness
  15. Connectedness in bitopological/quasi-uniform environments
  16. ...and 16 more sections

Key Result

Proposition 2.8

Let $(X,\mathcal{V})$ be a uniform space. If $(X,\mathcal{V})$ is complete and totally bounded, then $X$ is compact. In the quasi-uniform setting, the corresponding symmetric mechanism is: if $(X,\mathcal{U}^{*})$ is complete and totally bounded, then $(X,\mathcal{U}^{*})$ is compact as a uniform sp

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 59 more