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

Abstract: In this paper, we defined and discussed a new class of fuzzy soft open (FS-open) sets, called r-fuzzy soft-open (r-FS--open) sets in fuzzy soft topological spaces (FSTSs) based on fuzzy topologies in the sense of ˇ Sostak. The class of r-FS--open sets is contained in the class of r-FS--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--closed and r-FS--open sets, and studied some of their properties. Thereafter, we defined and studied some new FS-functions using r-FS--open and r-FS--closed sets, called FS--continuous (respectively (resp. for short) FS--irresolute, FS--open, FS--irresolute open, FS--closed, and FS--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)-continuous functions, which are weaker forms of FS--continuous functions. We showed that FS--continuity = FS-almost-continuity = FS-weak-continuity, but the converse may not be true. After that, we presented some new types of
FS-separation axioms, called r-FS--regular and r-FS--normal spaces using r-FS--closed sets, and investigated some properties of them. Finally, we introduced a new type of FS-connectedness, called r-FS--connected sets using r-FS--closed sets.