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

TL;DR

This paper develops a theory of antisymmetric connectedness in quasi-uniform and quasi-modular spaces, providing new characterizations, stability results, and applications to asymmetric normed spaces.

Contribution

It introduces the concepts of antisymmetric and local antisymmetric connectedness in quasi-uniform and quasi-modular spaces, with associated topologies and stability properties.

Findings

Characterization of local antisymmetric connectedness

Invariance under subspaces and mappings

Relation to Smyth completeness and compactness

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.