Abstract
This study introduces new classes of fuzzy open sets, namely (p,q)-FG-open (resp. (p,q)-FGP-open, (p,q)-FGS-open, (p,q)-FGα-open, and (p,q)-FGγ-open) sets in double fuzzy topological spaces (DFTSs) based on the sense of Šostak. We conduct a detailed investigation of the relationships among these classes of open sets, supported by carefully constructed illustrative examples. Also, we propose and characterize the associated DFG-interior and DFG-closure operators. Subsequently, we define and analyze new classes of fuzzy functions based on (p,q)-FG-open sets, referred to as DFG-continuous and DFG-irresolute functions within the framework of DFTSs (S ,ϑ,ϑ∗) and (Z ,ζ,ζ∗). We also introduce the notions of DFGP-continuous, DFGS-continuous, DFGα-continuous, and DFGγ-continuous functions, which constitute weaker forms of DFG-continuity. As an application, we demonstrate that these newly defined continuity concepts generalize, extend, and unify several existing results in the theory of DFTSs. Moreover, we propose and discuss the concepts of DFAG-continuity and DFWG-continuity as additional weaker variants of DFG-continuity. Finally, we establish new separation axioms, termed (p,q)-FG-normal and (p,q)-FG-regular spaces, formulated via (p,q)-FG-closed sets.