Neutrosophic MR-Metric Spaces: A Topos-Theoretic Framework with Applications

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DOI: https://doi.org/10.28924/2291-8639-23-2025-333
Published: Dec 12, 2025
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Abed Al-Rahman M. Malkawi, Ayat M. Rabaiah, Neutrosophic MR-Metric Spaces: A Topos-Theoretic Framework with Applications, Int. J. Anal. Appl., 23 (2025), 333.

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Abed Al-Rahman M. Malkawi, Ayat M. Rabaiah

Abstract

This paper introduces and systematically investigates the category of Neutrosophic MR-Metric Spaces (NMR-MS), which generalizes classical metric spaces by incorporating neutrosophic logic to model truth (T), indeterminacy (I), and falsity (F). We define the category NMRMS and construct sheaves of NMR-MS over topological spaces, proving that the category Sh(X, NMRMS) forms an elementary topos. This provides a rich mathematical framework for reasoning about uncertainty, vagueness, and contextual truth in a localized manner. We develop the internal language of this topos as a neutrosophic type theory and establish its soundness and completeness. The framework is applied to diverse fields including manifold theory, dynamic systems, image processing, data fusion, functional analysis, graph theory, differential equations, machine learning, topology optimization, quantum systems, and financial modeling. Our work unifies and extends recent advances in fixed point theory, fractional calculus, and neutrosophic fuzzy metrics within a single, category-theoretic foundation.

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