arXiv subjects

Philani Rodney Majozi

Publications and source records attributed to Philani Rodney Majozi.

5 recordsLinked to original sources

arXiv2026

Lifted Takahashi Convexity on the Isbell-Convex Hull of an Asymmetrically Normed Real Vector Space

Künzi and Yildiz introduced convexity structures in the sense of Takahashi for $T_{0}$-quasi-metric spaces. In this article, we continue this line of study on the Isbell-convex hull of an asymmetrically normed real vector space. Using the canonical hull quasi-metric and the vector-space operations on $\mathcal{E}(X,\|\cdot\|)$, we define a lifted convexity structure \[ \mathbb W(f,g,λ)=λf\oplus(1-λ)g \] and show that $(\mathcal{E}(X,\|\cdot\|),q_{\mathcal{E}},\mathbb W)$ is a convex $T_{0}$-quasi-metric space. We further prove compatibility with the canonical embedding and relate the construction to $W$-convexity of minimal function pairs.

math.GN↗

arXiv2026

On Quasi-Modular Pseudometric Spaces and Asymmetric Uniformities

We study quasi-modular pseudometric spaces as asymmetric refinements of modular metric structures. To each such space we associate canonical forward and backward quasi-uniformities and the corresponding directional topologies. We introduce directional notions of convergence, completeness, total boundedness, and compactness, and show that these properties are not preserved under symmetrization. In particular, forward and backward completeness may differ, and compactness of the symmetrized uniformity does not imply directional compactness. Using enriched category theory as a comparison framework, we show that symmetrization yields a symmetric enriched category whose Cauchy completion coincides with the classical uniform completion, while directional notions remain invisible at this level.

math.GN↗

arXiv2026

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

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.

math.GN↗

arXiv2025

On Metrizability, Completeness and Compactness in Modular Pseudometric Topologies

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.

math.GN↗

arXiv2025

Flatness and Nonforking without the Continuum Hypothesis

We investigate the structure of FN bases (Frechet-Nikodym bases) without assuming the Continuum Hypothesis (CH), refining results of Siu-Ah Ng concerning definability via flatness and nonforking. In particular, we examine the dependence of Theorem 3.4 and Corollary 3.5 of Ng's 1991 paper on CH, which guarantees the existence of nonforking primary heirs under specific ultrafilter conditions. We demonstrate that these properties can often be recovered in ZFC by analyzing the behavior of countably incomplete, good, and regular ultrafilters. By isolating model-theoretic conditions sufficient to replace CH, we establish that ultrapowers of flat FN bases satisfying Property B remain definable and nonforking. Connections are drawn to canonical bases, coheirs, and stability-theoretic ranks in superstable theories. Examples and counterexamples clarify the precise role of ultrafilter properties and reveal that the combinatorial essence of Ng's theory can be preserved within a purely ZFC framework.

math.LO↗