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 $\gamma$-open (k-$\mathcal{F}$-$\gamma$-open) sets as a generalized novel class of fuzzy open ($\mathcal{F}$-open) sets on fuzzy topological spaces ($\mathcal{FTS}s$) in the sense of Šostak. The class of k-$\mathcal{F}$-$\gamma$-open sets is contained in the class of k-$\mathcal{F}$-$\beta$-open sets and contains all k-$\mathcal{F}$-semi-open and k-$\mathcal{F}$-pre-open sets. Also, we introduced the closure and interior operators with respect to the classes of k-$\mathcal{F}$-$\gamma$-closed and k-$\mathcal{F}$-$\gamma$-open sets and discussed some of their properties. After that, we defined and studied the notions of $\mathcal{F}$-$\gamma$-continuous (resp. $\mathcal{F}$-$\gamma$-irresolute) functions between $\mathcal{FTS}s$$(M,\Im )$ and $(N,ϝ)$. However, we displayed and investigated the notions of $\mathcal{F}$-almost (resp. $\mathcal{F}$-weakly) $\gamma$-continuous functions, which are weaker forms of $\mathcal{F}$-$\gamma$-continuous functions. Next, we presented and characterized some new $\mathcal{F}$-functions via k-$\mathcal{F}$-$\gamma$-open and k-$\mathcal{F}$-$\gamma$-closed sets, called $\mathcal{F}$-$\gamma$-open (resp. $\mathcal{F}$-$\gamma$-irresolute open, $\mathcal{F}$-$\gamma$-closed, $\mathcal{F}$-$\gamma$-irresolute closed, and $\mathcal{F}$-$\gamma$-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 $\mathcal{F}$-separation axioms called k-$\mathcal{F}$-$\gamma$-regular (resp. k-$\mathcal{F}$-$\gamma$-normal) spaces via k-$\mathcal{F}$-$\gamma$-closed sets and discussed some properties of them. Lastly, we explored and studied some new types of $\mathcal{F}$-compactness called k-$\mathcal{F}$-almost (resp. k-$\mathcal{F}$-nearly) $\gamma$-compact sets.