Some Characterizations of \((r,s)\)-Fuzzy \(b\)-Open Sets with Applications in Double Fuzzy Topological Spaces

In this paper, we displayed and characterized a novel class of fuzzy open sets ($\mathcal{F}$-open sets) in double fuzzy topological spaces ($\mathcal{DFTS}s$) based on \v{S}ostak$^{,}$s sense, called $(r,s)$-fuzzy $b$-open sets ($(r,s)$-$\mathcal{F}$-$b$-open sets). This class is contained in the class of $(r,s)$-$\mathcal{F}$-$\beta$-open sets and contains all $(r,s)$-$\mathcal{F}$-$\alpha$-open sets, $(r,s)$-$\mathcal{F}$-pre-open sets, and $(r,s)$-$\mathcal{F}$-semi-open sets. Next, we explored and studied the notion of $\mathcal{DF}$-$b$-continuity between $\mathcal{DFTS}s$ $(G, \Im, \Im^*)$ and $(Z, \digamma, \digamma^*)$. We also defined and discussed the notions of $\mathcal{DF}$-almost $b$-continuity and $\mathcal{DF}$-weakly $b$-continuity, which are weaker forms of $\mathcal{DF}$-$b$-continuity. Thereafter, we presented and investigated novel $\mathcal{DF}$-mappings via $(r,s)$-$\mathcal{F}$-$b$-open and $(r,s)$-$\mathcal{F}$-$b$-closed sets. Finally, we introduced some novel types of $\mathcal{DF}$-separation axioms, called $(r,s)$-$\mathcal{F}$-$b$-regular and $(r,s)$-$\mathcal{F}$-$b$-normal spaces, and studied some properties of them.