Fuzzy Topological Approaches via r-Fuzzy γ-Open Sets in the Sense of Sostak

In the present article, we define and investigate the notion of r-fuzzy γ-open (r-F-γ-open) sets as a generalized novel class of fuzzy open (F-open) sets on fuzzy topological spaces (F T Ss) in the sense of Šostak. This class is contained in the class of r-F-β-open sets and contains all r-F-pre-open and r-F-semi-open sets. However, we introduce the interior and closure operators with respect to the classes of r-F-γ-open and r-F-γ-closed sets, and study some of their properties. After that, we define and discuss the notions of F-γ-continuous (respectively (resp. for short) F-γ-irresolute) functions between F T Ss (M, ℑ) and (N, F). Also, we display and investigate the notions of F-almost (resp. F-weakly) γ-continuous functions, which are weaker forms of F-γ-continuous functions. We also showed that F-γ-continuity ⇒ F-almost γ-continuity ⇒ F-weakly γ-continuity, but the converse may not be true. Next, we present and characterize new F-functions via r-F-γ-open and r-F-γ-closed sets, called F-γ-open (resp. F-γ-irresolute open, F-γ-closed, F-γ-irresolute closed, and F-γ-irresolute homeomorphism) functions. The relationships between these classes of functions were discussed with the help of some examples. We also introduce some new types of F-separation axioms, called r-F-γ-regular (resp. r-F-γ-normal) spaces via r-F-γ-closed sets, and study some properties of them. Lastly, we explore and discuss some new types of F-compactness, called r-F-almost (resp. r-F-nearly) γ-compact sets using r-F-γ-open sets.