an:04040797
Zbl 0638.54012
Noiri, Takashi
Weakly \(\alpha\)-continuous functions
EN
Int. J. Math. Math. Sci. 10, 483-490 (1987).
00152819
1987
j
54C08 54C10
regular spaces; weak \(\alpha \)-continuity; almost continuity; weak quasi- continuity; almost weak continuity
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 [\textit{T. Husain}, Pr. Mat. 10, 1-7 (1966; Zbl 0138.146)], semicontinuity [\textit{N. Levine}, Am. Math. Mon. 70, 36-41 (1963; Zbl 0113.163)], weak quasi- continuity [\textit{V. Popa} and \textit{C. Stan}, Stud. Cerc. Mat. 25, 41-43 (1973; Zbl 0255.54008)] and almost weak continuity [\textit{D. S. Jankovi??}, Int. J. Math. Math. Sci. 8, 615-619 (1985; Zbl 0577.54012)].
V.Popa
Zbl 0138.146; Zbl 0113.163; Zbl 0255.54008; Zbl 0577.54012