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On local relative continuity. (English) Zbl 0838.54011
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 $$X$$ [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)].
Reviewer: V.Popa (Bacau)

##### MSC:
 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54C08 Weak and generalized continuity