On (p,q)-Fuzzy Ω-Open Sets and Associated Functions in Double Fuzzy Topological Spaces

This study introduces new classes of fuzzy open sets, namely (p,q)-FΩ-open (resp. (p,q)-FΩP open, (p,q)-FΩS-open, (p,q)-FΩα-open, and (p,q)-FΩγ-open) sets in double fuzzy topological spaces (DFTSs) in view of Šostak. We conduct a detailed investigation of the relationships among these classes of open sets, supported by carefully constructed illustrative examples. Furthermore, we propose and characterize the associated DFΩ-interior and DFΩ-closure operators. Subsequently, we define and analyze new classes of fuzzy functions based on (p,q)-FΩ-open sets, referred to as DFΩ-continuous and DFΩ-irresolute functions within the framework of DFTSs (S,ϑ,ϑ^∗) and (Z,ζ,ζ^∗). We also introduce the notions of DFΩP-continuous, DFΩS-continuous, DFΩα continuous, and DFΩγ-continuous functions, which constitute weaker forms of DFΩ-continuity. As an application, we demonstrate that these newly defined continuity concepts generalize, extend, and unify several existing results in the theory of DFTSs. Finally, we propose and discuss the concepts of DFAΩ-continuity and DFWΩ-continuity as additional weaker variants of DFΩ-continuity. Moreover, we establish new separation axioms, termed (p,q)-FΩ-normal and (p,q)-FΩ-regular spaces, formulated via (p,q)-FΩ-closed sets.