Fuzzy Topological Continuity: A Bipolar Hesitant Set-Theoretic Approach

Abstract

This paper investigates bipolar-valued hesitant fuzzy generalized semi-precontinuous mappings in bipolar-valued hesitant fuzzy topological spaces. It extends fuzzy topology theory to hesitant fuzzy information so that more flexible uncertainty treatment in decision-making, AI, and modeling can be achieved. The bipolar-valued hesitant fuzzy set can represent positive and negative membership grades without hesitation and thereby more capable of handling uncertainty and fuzzy data in real-world problems.We establish bipolar-valued hesitant fuzzy generalized semi-precontinuous mappings and their relation to other continuity forms such as semi-continuity and pre-continuity. New theorems shed light on these mappings, with the help of numerical examples. We also suggest applications in fuzzy decision analysis, computational intelligence, medical diagnosis, and engineering optimization.

Introduction

L.A. Zadeh’s fuzzy set theory (1965) laid the foundation for managing uncertainty in many fields. Since then, extensions such as vague sets (Gau & Buehrer, 1993), bipolar-valued fuzzy sets (Zhang, 1994), and hesitant fuzzy sets (Torra, 2010) have been developed to better handle imprecision, dual criteria, and hesitation in decision-making.

Combining these ideas results in bipolar-valued hesitant fuzzy sets, which generalize fuzzy logic by incorporating both positive and negative membership degrees with multiple possible values. This framework is useful in fuzzy decision making, machine learning, medical diagnosis, and engineering optimization.

Although fuzzy topology has advanced, continuity concepts in bipolar-valued hesitant fuzzy topological spaces, especially generalized semi-precontinuous mappings, remain underexplored. This paper introduces these mappings, establishes their properties, and compares them with other continuity types, supported by theorems and examples.

The paper also revisits fundamental concepts: fuzzy sets, hesitant fuzzy sets, bipolar-valued hesitant fuzzy sets, and their topologies—extending classical topology notions such as closure, interior, neighborhoods, compactness, separation axioms, limit points, and derived sets to the bipolar-valued hesitant fuzzy context.

Key results include the definition of bipolar-valued hesitant fuzzy generalized semi-precontinuous mappings, showing that the image of generalized semiprecompact sets preserves compactness under such mappings. The study highlights the engineering and AI significance of these concepts in uncertainty modeling and expert systems, providing a theoretical foundation for future research.

Conclusion

The research on bipolarvalued hesitant fuzzy topology was carried out with a focus on its theoretical basis and practical application. The article provided a comprehensive discussion of generalized semiprecontinuous mappings, semipreopen sets, and semipreconnected spaces, thus enhancing the properties typically defined under the fuzzy topology framework. Its practical application with bipolarvalued hesitant fuzzy sets greatly improves decisionmaking by effectively handling uncertainty and hesitation. In addition, some basic properties like semipreregular spaces and semiprehomeomorphisms were defined and studied, and an illustration of their role in enhancing the theoretical framework was given. The research has implications in a number of fields such as decisionmaking, medical diagnosis, artificial intelligence, robotics, cybersecurity, and realworld transportation systems.

References

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Copyright

Copyright © 2025 Dr. K. Kavitha , Mrs. C. Yogitha, Dr. A. Saranya , Mrs. D. Lakshmi, Dr. P. Revathi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.