Another note on Levine's decomposition of 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)]. Let \(X\) and \(Y\) be two topological spaces. 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 \(f: X^\alpha\to Y\) is weakly continuous. Two new function conditions \((*)\) and \((**)\) are introduced each strictly weaker than local relative continuity [the first and the second author, Real Anal. Exch. 20, 823-830 (1995; Zbl 0838.54011)]\ and \((**)\) being strictly weaker than \((*)\). Definition 1. A function \(f: X\to Y\) satisfies condition \((*)\) if for each \(x\in X\) and for each open set \(V\) containing \(f(x)\), there exists an open set \(V_x \subset V\) with \(f(x)\in V_x\) such that \(x\not\in \text{Cl} (f^{-1} (\text{Cl} (V_x)- f^{-1} (V)))\). Definition 2. A function \(f: X\to Y\) satisfies condition \((**)\) if for each open set \(V\) containing \(f(x)\), there exists an open set \(V_x \subset V\) with \(f(x)\in V_x\) such that \(x\not\in \text{Cl(Int} (f^{-1} (\text{Cl} (V_x)))- f^{-1} (V))\). It is shown that for any function \(f: X\to Y\), the following are equivalent: (a) \(f\) is continuous; (b) \(f\) is weakly continuous and satisfies \((**)\); (c) \(f\) is weakly \(\alpha\)-continuous and satisfies \((*)\). Decomposition (b) improves a result of \textit{J. Chew} and \textit{Tong Jingcheng} [Am. Math. Mon. 98, 931-934 (1991; Zbl 0764.54007)]\ and Theorem 5 of [the first and the second author, loc. cit.]. Decomposition (c) improves a result of Noiri and Theorem 6 of [the first and the second author, loc. cit.].
scientific article; zbMATH DE number 874315