Fuzzy topological approaches via <em>r</em>-fuzzy <em>γ</em>-open sets in the sense of Sostak

In the present article, we define and investigate the notion of r-fuzzy &gamma;-open (r-F-&gamma;-open) sets as a generalized novel class of fuzzy open (F-open) sets on fuzzy topological spaces (F T Ss) in the sense of &Scaron;ostak. This class is contained in the class of r-F-&beta;-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-&gamma;-open and r-F-&gamma;-closed sets, and study some of their properties. After that, we define and discuss the notions of F-&gamma;-continuous (respectively (resp. for short) F-&gamma;-irresolute) functions between F T Ss (M, &image;) and (N, F). Also, we display and investigate the notions of F-almost (resp. F-weakly) &gamma;-continuous functions, which are weaker forms of F-&gamma;-continuous functions. We also showed that F-&gamma;-continuity &rArr; F-almost &gamma;-continuity &rArr; F-weakly &gamma;-continuity, but the converse may not be true. Next, we present and characterize new F-functions via r-F-&gamma;-open and r-F-&gamma;-closed sets, called F-&gamma;-open (resp. F-&gamma;-irresolute open, F-&gamma;-closed, F-&gamma;-irresolute closed, and F-&gamma;-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-&gamma;-regular (resp. r-F-&gamma;-normal) spaces via r-F-&gamma;-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) &gamma;-compact sets using r-F-&gamma;-open sets.