On $r$-fuzzy soft $\gamma$-open sets and fuzzy soft $\gamma$-continuous functions with some applications

In this paper, we defined and discussed a new class of fuzzy soft open (FS-open) sets, called $r$-fuzzy soft $\gamma$-open ( $r$-FS- $\gamma$-open) sets in fuzzy soft topological spaces (FSTSs) based on fuzzy topologies in the sense of Šostak. The class of $r$-FS- $\gamma$-open sets is contained in the class of $r$-FS- $\beta$-open sets, and contains all $r$-FS-semi-open and $r$-FS-pre-open sets. However, we introduced the closure and interior operators with respect to the classes of $r$-FS- $\gamma$-closed and $r$-FS- $\gamma$-open sets, and studied some of their properties. Thereafter, we defined and studied some new FS-functions using $r$-FS- $\gamma$-open and $r$-FS- $\gamma$-closed sets, called FS- $\gamma$-continuous (respectively (resp. for short) FS- $\gamma$-irresolute, FS- $\gamma$-open, FS- $\gamma$-irresolute open, FS- $\gamma$-closed, and FS- $\gamma$-irresolute closed) functions. The relationships between these classes of functions were discussed with the help of some illustrative examples. We also explored and established the notions of FS-weakly (resp. FS-almost) $\gamma$-continuous functions, which are weaker forms of FS- $\gamma$-continuous functions. We showed that FS- $\gamma$-continuity $⟹$ FS-almost $\gamma$-continuity $⟹$ FS-weak $\gamma$-continuity, but the converse may not be true. After that, we presented some new types of FS-separation axioms, called $r$-FS- $\gamma$-regular and $r$-FS- $\gamma$-normal spaces using $r$-FS- $\gamma$-closed sets, and investigated some properties of them. Finally, we introduced a new type of FS-connectedness, called $r$-FS- $\gamma$-connected sets using $r$-FS- $\gamma$-closed sets.