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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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