Fuzzy \(b\)-open sets and fuzzy \(b\)-separation axioms (Q2724011)

A fuzzy set \(A\) in a fuzzy topological space \((X,\tau)\) is called fuzzy \(b\)-open if \(A\leq \text{pcl} (\text{pint }A)\). Fuzzy \(b\)-continuity and fuzzy weak \(b\)-continuity are defined as usually. The properties of fuzzy-\(b\)-open sets and fuzzy \(b\)-continuity are studied. Some of these properties are as follows. If \((X,\tau)\) is a fuzzy topological space, then \(\text{FSO}(\tau) \cup \text{FPO}(\tau) \subset \text{FBO}(\tau) \subset \text{FSPO}(\tau)\). A bijective mapping \(f:X \rightarrow Y\) from a fuzzy topological space \(X\) into a fuzzy topological space \(Y\) is fuzzy \(b\)-continuous iff \(\text{int }f(A) \leq f (\text{bint }A)\), for each fuzzy set \(A\) of \(X\). A mapping \(f:X \rightarrow Y\) from a fuzzy topological space \(X\) into a fuzzy topological space \(Y\) is fuzzy weakly \(b\)-continuous iff for any fuzzy singleton \(x_\alpha\) of \(X\) and any fuzzy open set \(B\) of \(Y\) containing \(f(x_\alpha)\), there exists a fuzzy \(b\)-open set \(A\) of \(X\) containing \(x_\alpha\) such that \(f(A) \leq b \& B\). Some characterizations of fuzzy \(b\)-continuity and fuzzy weak \(b\)-continuity are given. Fuzzy separation axioms \(\text{FBT}_i\) \((i= 0, 1, 2, 2{1\over 2}, 3, 4)\) are introduced and investigated.