In this paper, we first introduced the concept of $r$-fuzzy soft $\beta$-closed sets in fuzzy soft topological spaces based on the sense of Šostak and investigated some properties of them. Also, we defined the closure and interior operators with respect to the classes of $r$-fuzzy soft $\beta$-closed and $r$-fuzzy soft $\beta$-open sets and studied some of their properties. Moreover, the concept of $r$-fuzzy soft $\beta$-connected sets was introduced and characterized with the help of fuzzy soft $\beta$-closure operators. Thereafter, some properties of a fuzzy soft $\beta$-continuity were studied. Also, we introduced and studied the concepts of fuzzy soft almost (weakly) $\beta$-continuous functions, which are weaker forms of a fuzzy soft $\beta$-continuity. The relationships between these classes of functions were specified with the help of some illustrative examples. Finally, we explored new types of fuzzy soft functions called fuzzy soft $\beta$-irresolute (strongly $\beta$-irresolute, $\beta$-irresolute open, $\beta$-irresolute closed, and $\beta$-irresolute homeomorphism) functions and discussed some properties of them. Also, we showed that fuzzy soft strongly $\beta$-irresolute $\Rightarrow$ fuzzy soft $\beta$-irresolute $\Rightarrow$ fuzzy soft $\beta$-continuity, but the converse may not be true.

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.