On local relative continuity.

*(English)*Zbl 0838.54011A 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 \(X\) [O. Njåstad, Pac. J. Math. 15, 961-970 (1965; Zbl 0137.41903)]. A function \(f : X \to Y\) is weakly continuous at \(x \in X\) [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 [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 J. Chew and 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)].

In this paper the authors introduce a local version of the relative continuity of J. Chew and 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)].

Reviewer: V.Popa (Bacau)