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.

In this paper, we first introduced and studied a new class of fuzzy open sets, called $j$-fuzzy $\gamma \mathcal{I}$-open ( $j$-F $\gamma \mathcal{I}$-open) sets on fuzzy ideal topological spaces ( $\mathcal{F}\mathcal{I}\mathcal{T}\mathcal{S}s$). The class of $j$-F $\gamma \mathcal{I}$-open sets is contained in the class of $j$-fuzzy strong $\beta$- $\mathcal{I}$-open ( $j$-FS $\beta \mathcal{I}$-open) sets and contains all $j$-fuzzy pre- $\mathcal{I}$-open ( $j$-FP $\mathcal{I}$-open) sets and $j$-fuzzy semi- $\mathcal{I}$-open ( $j$-FS $\mathcal{I}$-open) sets. We also defined and investigated the closure and interior operators with respect to the classes of $j$-F $\gamma \mathcal{I}$-closed sets and $j$-F $\gamma \mathcal{I}$-open sets. However, we explored and discussed novel types of fuzzy $\mathcal{I}$-separation axioms using $j$-F $\gamma \mathcal{I}$-closed sets, called $j$-F $\gamma \mathcal{I}$-regular spaces and $j$-F $\gamma \mathcal{I}$-normal spaces. Thereafter, we displayed and investigated the concept of fuzzy $\gamma \mathcal{I}$-continuity (F $\gamma \mathcal{I}$-continuity) using $j$-F $\gamma \mathcal{I}$-open sets. Also, we presented and characterized the concepts of fuzzy weak $\gamma \mathcal{I}$-continuity (FW $\gamma \mathcal{I}$-continuity) and fuzzy almost $\gamma \mathcal{I}$-continuity (FA $\gamma \mathcal{I}$-continuity), which are weaker forms of F $\gamma \mathcal{I}$-continuity. Moreover, we showed that F $\gamma$ $\mathcal{I}$-continuity $⟹$ FA $\gamma$ $\mathcal{I}$-continuity $⟹$ FW $\gamma$ $\mathcal{I}$-continuity, but the converse may not be true. Finally, we defined and studied some new fuzzy $\gamma \mathcal{I}$-mappings via $j$-F $\gamma \mathcal{I}$-open sets and $j$-F $\gamma \mathcal{I}$-closed sets, called F $\gamma \mathcal{I}$-open mappings, F $\gamma \mathcal{I}$-closed mappings, F $\gamma \mathcal{I}$-irresolute mappings, F $\gamma \mathcal{I}$-irresolute open mappings, and F $\gamma \mathcal{I}$-irresolute closed mappings. The relationships between these classes of mappings were discussed with the help of some examples.