Some Characterizations of k-Fuzzy γ-Open Sets and Fuzzy γ-Continuity with Further Selected Topics

In the present paper, we first introduced the notion of k-fuzzy γ-open (k-F-γ-open) sets as a generalized novel class of fuzzy open (F-open) sets on fuzzy topological spaces (FTSs) in the sense of Šostak. The class of k-F-γ-open sets is contained in the class of k-F-β-open sets and contains all k-F-semi-open and k-F-pre-open sets. Also, we introduced the closure and interior operators with respect to the classes of k-F-γ-closed and k-F-γ-open sets and discussed some of their properties. After that, we defined and studied the notions of F-γ-continuous (resp. F-γ-irresolute) functions between FTSs(M,ℑ) and (N,Ϝ). However, we displayed and investigated the notions of F-almost (resp. F-weakly) γ-continuous functions, which are weaker forms of F-γ-continuous functions. Next, we presented and characterized some new F-functions via k-F-γ-open and k-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 investigated with the help of some examples. We also introduced some new types of F-separation axioms called k-F-γ-regular (resp. k-F-γ-normal) spaces via k-F-γ-closed sets and discussed some properties of them. Lastly, we explored and studied some new types of F-compactness called k-F-almost (resp. k-F-nearly) γ-compact sets.