On Metrizability, Completeness and Compactness in Modular Pseudometric Topologies
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909856528596992 |
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| author | Majozi, Philani Rodney |
| author_facet | Majozi, Philani Rodney |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16787 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Metrizability, Completeness and Compactness in Modular Pseudometric Topologies Majozi, Philani Rodney General Topology Category Theory 54E35, 46E30, 18A40 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. |
| title | On Metrizability, Completeness and Compactness in Modular Pseudometric Topologies |
| topic | General Topology Category Theory 54E35, 46E30, 18A40 |
| url | https://arxiv.org/abs/2510.16787 |