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Weakly $$\alpha$$-continuous functions. (English) Zbl 0638.54012
A subset S of a topological space $$(X,\tau)$$ is said to be $$\alpha$$-open if $$S\subset Int Cl(Int S)$$. The family of $$\alpha$$-open sets of (X,$$\tau)$$ is denoted $$\tau^{\alpha}$$ and is a topology for X.
The author introduces a new class of functions called weakly $$\alpha$$- continuous. A function $$f: (X,\tau)\to (Y,\sigma)$$ is said to be weakly $$\alpha$$-continuous if for each $$x\in X$$ and each $$V\in \sigma$$ containing f(x), there exists $$U\in \tau$$ containing x such that $$f(U)\subset Cl V$$. The author proved that weakly $$\alpha$$-continuous surjections preserve connected spaces and that weakly $$\alpha$$-continuous functions into regular spaces are continuous.
In the last section of the paper the author investigates the interrelation among weak $$\alpha$$-continuity, almost continuity [T. Husain, Pr. Mat. 10, 1-7 (1966; Zbl 0138.146)], semicontinuity [N. Levine, Am. Math. Mon. 70, 36-41 (1963; Zbl 0113.163)], weak quasi- continuity [V. Popa and C. Stan, Stud. Cerc. Mat. 25, 41-43 (1973; Zbl 0255.54008)] and almost weak continuity [D. S. Janković, Int. J. Math. Math. Sci. 8, 615-619 (1985; Zbl 0577.54012)].
Reviewer: V.Popa

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