On fuzzy soft $\beta$-continuity and $\beta$-irresoluteness: some new results

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.