A note on \(W\)-\(I\)-continuous functions
Let \((X,T)\) be a topological space. An ideal of \(X\) is a collection \(I\) of subset of \(X\) satisfying: (1) If \(A\in I\) and \(B\subset A\), then \(B\in I\); (2) If \(A\in I\) and \(B\in I\), then \(A\cup B\in I\). An ideal topological space is a topological space \((X,T)\) with an ideal on \(X\) and is denoted \((X,T,I)\). For a subset \(A\subset X\), \(A^*(I)=\{x\in X:U\cap A\notin I\}\) for each open neighbourhood \(U\) of \(X\). For every ideal topological space \((X,T,I)\) there exists a topology \(T^*(I)\), finer than \(T\), generated by \(\{U-A:U\in T\;\text{and}\;A\in I\}\). Then \(CL^*(A)=A\cup A^*\). In 2004, \textit{A. Açikgöz, T. Noiri} and \textit{S. Yüksel} [Acta Math. Hung. 105, No.~4, 285--289 (2004; Zbl 1068.54016)] introduced the notions of weakly-\(I\)-continuous and weak\({}^*\)-\(I\)-continuous functions. A function \(f:(X,T)\to (Y,Q,I)\) is said to be weakly-\(I\)-continuous if for each \(x\in X\) and each open set \(V\) containing \(f(x)\) there exists an open set \(U\) containing \(x\) such that \(f(U)\subset Cl^*(V)\). A function \(f:(X,T)\to(Y,Q,I)\) is said to be weak\({}^*\)-\(I\)-continuous if for each open set \(V\) of \(Y\), \(f^{-1}(fr^*(V))\) is closed in \(X\), where \(fr^*(A)=A^*-\text{Int}(A)\). A function \(f:(X,T)\to(Y,Q)\) is said to be \(\theta\)-continuous at \(x\in X\), [see \textit{S. Fomin}, Ann. Math. (2)44, 471--480 (1943; Zbl 0061.39601)], if for each open set \(V\) containing \(f(x)\), there exists an open set \(U\) containing \(x\) such that \(f(Cl(U)\subset Cl(V)\). In this paper, the authors investigate the relationship of weakly-\(I\)-continuous and weak\({}^*\)-\(I\)-continuous functions with continuous and \(\theta\)-continuous functions. Some generalizations of results by \textit{V. Jeyanthi, V. R. Devi} and \textit{D. Sivaraj} [Acta Math. Hungar. 113, 319--324 (2006; Zbl 1121.54027)] are obtained.
scientific article; zbMATH DE number 5355142
scientific article