On q`-Rung Orthopair Neutrosophic Metric Spaces and Topological Properties

In this paper, we introduce the novel structure of the q`-rung orthopair neutrosophic metric space (q`-RˇONM^S), which unifies and extends the frameworks of q`-rung orthopair fuzzy sets and neutrosophic metric spaces. The proposed model incorporates membership, nonmembership, and indeterminacy degrees within a q`-rung orthopair framework, thereby offering a more flexible and expressive structure to capture uncertainty, inconsistency, and indeterminacy in data. Fundamental topological properties of the space are rigorously developed, including notions of open balls, induced topology, compactness, completeness, convergence, and Hausdorffness. Several illustrative examples are provided to support theoretical developments. Furthermore, the relevance of the proposed structure is highlighted through applications in decision-making under uncertainty, artificial intelligence, multicriteria decision-making (MCDM), image processing, and clustering. This framework lays the foundation for further exploration in modeling real-world problems involving complex, vague, and imprecise information.