In this article, we first introduced new types of higher separation axioms called (r,s)-GFS-regular and (r,s)-GFS-normal spaces with the help of (r,s)-generalized fuzzy semi-closed sets (briefly, (r,s)-gfsc sets) and discussed some topological properties of them. Thereafter, we defined a stronger form of (r,s)-gfsc sets called (r,s)-g*fsc sets and investigated some of its features. Moreover, we showed that (r,s)-fsc set → (r,s)-g*fsc set → (r,s)-gfsc set, but the converse may not be true. In addition, we explored new types of fuzzy generalized mappings between double fuzzy topological spaces (U, τ, τ*) and (V, η, η*), and the relationships between these classes of mappings were examined with the help of some illustrative examples. Finally, some new types of compactness via (r,s)-gfso sets were defined and the relationships between them were introduced.

In the present study. At first, some properties of r-fuzzy soft β-closed sets are introduced on fuzzy soft topological spaces based on the paper Aygunoglu et al. (Hacet. J. Math. Stat. 2014, 43, 193-208). Also, we define the closure and interior operators with respect to the classes of r-fuzzy soft β-closed and r-fuzzy soft β-open sets, and investigate some of their properties. In addition, the concept of r-fuzzy soft β-connected sets is introduced and characterized with help of fuzzy soft β-closure operators. Thereafter, some properties of a fuzzy soft β-continuity are studied. Furthermore, we introduce and study the concepts of fuzzy soft almost (weakly) β-continuous functions, which are weaker forms of a fuzzy soft β-continuity. Also, the relationships between these classes of functions are specified with the help of some illustrative examples. Finally, we explore new types of fuzzy soft functions called, fuzzy soft β-irresolute (strongly β-irresolute, β-irresolute open, β-irresolute closed and β-irresolute homeomorphism) functions, and discuss some properties of them. Also, we show that fuzzy soft strongly β-irresolute → fuzzy soft β-irresolute → fuzzy soft β-continuity, but the converse may not be true.

In this paper, we first define the concepts of r-fuzzy soft α-open (semi-open and δ-open) sets on fuzzy soft topological spaces based on the article Aygunoglu et al. (Hacet. J. Math. Stat., 43 (2014), 193-208), and the relations of these sets with each other are established. In addition, we introduce the concepts of fuzzy soft δ-closure (δ-interior) operators, and study some properties of them. Also, the concept of r-fuzzy soft δ-connected sets is introduced and studied with help of fuzzy soft δ-closure operators. Thereafter, we define the concepts of fuzzy soft α-continuous (β-continuous, semi-continuous, pre-continuous and δ-continuous) functions, which are weaker forms of fuzzy soft continuity, and some properties of these functions along with their mutual relationships are discussed. Moreover, a decomposition of fuzzy soft α-continuity and a decomposition of fuzzy soft semi-continuity is obtained. Finally, as a weaker form of a fuzzy soft continuity, the concepts of fuzzy soft almost (weakly) continuous functions are defined, and some properties are specified. Additionally, we explore the notion of continuity in a very general setting called, fuzzy soft (L, M, N, O)-continuous functions and a historical justification is introduced.