On local relative continuity
A subset \(A\) of a topological space \(X\) is \(\alpha\)-open if \(A \subset \text{Int(Cl(Int} (A)))\). The collection of \(\alpha\)-open subsets of \(X\), written \(X^\alpha\), is a topology for \(X\) [\textit{O. Njåstad}, Pac. J. Math. 15, 961-970 (1965; Zbl 0137.41903)]. A function \(f : X \to Y\) is weakly continuous at \(x \in X\) [\textit{N. Levine}, Am. Math. Mon. 68, 44-46 (1961; Zbl 0100.18601)] if for any open set \(V \subset Y\) containing \(f(x)\), there exists an open set \(U \subset X\) containing \(x\) such that \(f(U) \subset \text{Cl} (V)\). If this condition is satisfied at each \(x \in X\), then \(f\) is said to be weakly continuous. A function \(f : X \to Y\) is weakly \(\alpha\)-continuous [\textit{T. Noiri}, Int. J. Math. Math. Sci. 10, 483-490 (1987; Zbl 0638.54012)] if \(f : X \to Y\) is weakly continuous. In this paper the authors introduce a local version of the relative continuity of \textit{J. Chew} and \textit{J. Tong} [Am. Math. Mon. 98, No. 10, 931-934 (1991; Zbl 0764.54007)]. ``A function \(f : X \to Y\) is locally relatively continuous if there exists an open basis \(B\) for the topology on \(Y\) such that \(f^{-1} (V)\) is open in the subspace \(f^{-1} (\text{Cl} (V))\) for any \(V \in B\)'' and obtain the following new decompositions of continuity: Theorem 1. A function \(f : X \to Y\) is continuous if and only if it is both weakly continuous and locally relatively continuous. Theorem 2. A function \(f : X \to Y\) is continuous if and only if it is both weakly \(\alpha\)-continuous and locally relatively continuous. Theorem 1 improves Theorem 5 of [the second author, Can. Math. Bull. 21, 477-481 (1978; Zbl 0394.54004)] and Theorem 2 improves Theorem 1 of [the second author, Indian J. Pure Appl. Math. 21, No. 11, 985-987 (1990; Zbl 0726.54009)].
scientific article; zbMATH DE number 801040