Proximity structures on hesitant fuzzy sets and Its application.

The present article discusses the extension of classical proximity theory to multi valued systems. Existing distance measures for hesitant fuzzy sets operate primarily at the computational level without establishing the rigorous mathematical foundations. This paper develops a comprehensive axiomatic framework for proximity spaces on hesitant fuzzy sets. First, five fundamental axioms (HFP0–HFP5) for hesitant fuzzy proximity relations are introduced, generalizing classical proximity theory to fit multi-valued membership structures. Second, it is proven that hesitant fuzzy proximity spaces induce completely regular topologies, and the Smirnov compactification theorem is established for these spaces. Third, a bijective correspondence between hesitant fuzzy proximities and totally bounded uniformities is demonstrated, along with a metrization theorem characterizing when compatible metrics exist. Fourth, a new concept of variable length proximity measure is proposed that naturally handles hesitant fuzzy elements of different cardinalities without requiring artificial data extension or reduction. Finally, the practical applicability of the framework is validated through a facial recognition case study, where the proposed discrete hesitant fuzzy proximity successfully identifies images of the same individual under varying lighting and rotation conditions while correctly discriminating different subjects. This work bridges classical proximity theory and modern uncertainty modeling, giving both theoretical foundations and applicability for decision-making under hesitancy.