Delving Into the Depths of the Properties and Behavior of Bipolar Pythagorean Neutrosophic Metric Spaces: A Theoretical Analysis

This paper advances the theory of bipolar Pythagorean neutrosophic fuzzy (BPNF) sets by establishing their formalization within a topological and metric framework, while also demonstrating their role in decision-making under uncertainty. The main contributions are as follows: (1) definition and characterization of BPNF topological spaces, providing a rigorous foundation for theoretical exploration; (2) development of two distance measures—the BPNF Hamming metric and its normalized variant—that enable precise comparison of BPNF sets under indeterminacy, Pythagorean constraints, and bipolarity; (3) introduction of a BPNF inclusion measure for capturing complex containment relationships, supported by formal proofs; (4) comparative discussion with existing models (bipolar fuzzy, Pythagorean fuzzy, and neutrosophic sets), including counterexamples that justify the need for BPNF sets; (5) demonstration of the robustness and applicability of these tools through a multicriteria decision-making (MCDM) example and sensitivity analysis. The proposed approach addresses challenges in modeling contradictory evidence, hesitation, and dual perspectives, offering advantages for pattern recognition, image processing, and uncertainty analysis. A discussion of computational complexity, scalability, advantages, and limitations is also provided. Overall, the paper delivers a unified and practical framework for reasoning with bipolar Pythagorean neutrosophic information in complex real-world environments.