Local Antisymmetric Connectedness in Quasi-Uniform and Quasi-Modular Spaces
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917219344056320 |
|---|---|
| author | Majozi, Philani Rodney |
| author_facet | Majozi, Philani Rodney |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16361 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Antisymmetric Connectedness in Quasi-Uniform and Quasi-Modular Spaces Majozi, Philani Rodney General Topology Category Theory Functional Analysis 46A16, 54E15, 54D05, 46B20 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. |
| title | Local Antisymmetric Connectedness in Quasi-Uniform and Quasi-Modular Spaces |
| topic | General Topology Category Theory Functional Analysis 46A16, 54E15, 54D05, 46B20 |
| url | https://arxiv.org/abs/2601.16361 |