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Another note on Levine’s decomposition of continuity. (English) Zbl 0845.54009
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)]. Let $$X$$ and $$Y$$ be two topological spaces. 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 $$f: X^\alpha\to Y$$ 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 J. Chew and 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.].
Reviewer: V.Popa (Bacau)
##### MSC:
 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54C08 Weak and generalized continuity
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