\(\alpha\)-continuous and \(\alpha\)-open mappings. (English) Zbl 0534.54006

Mashhour, A. S.; Hasanein, I. A.; El-Deeb, S. N.

Acta Math. Hung. 41, 213-218 (1983).

A subset S of a topological X is called an \(\alpha\)-set if \(S\subset int(cl(int S))\). A mapping f:\(X\to Y\) is called \(\alpha\)-continuous if the inverse image of each open set in Y is an \(\alpha\)-set in X and \(\alpha\)-open if the image of each open set in X is an \(\alpha\)-set in Y. This paper studies these classes of mappings, and especially their relationships with other classes of mappings.

Reviewer: I.L.Reilly

MSC:

54C10	Special maps on topological spaces (open, closed, perfect, etc.)
54A05	Topological spaces and generalizations (closure spaces, etc.)

Keywords:

\(\alpha\)-set; semi-open set; preopen set; \(\alpha\)-continuity; \(\alpha\)- open mappings

References:

[1]	S. V. Fomin, Extensions of topological spaces,Ann. of Math.,44 (1943), 471–480. · Zbl 0061.39601
[2]	T. Husain, Almost continuous mappings,Prace Mat.,10 (1966), 1–7. · Zbl 0138.17601
[3]	N. Levine, A decomposition of continuity in topological spaces,Amer. Math. Monthly,68 (1961), 44–46. · Zbl 0100.18601
[4]	N. Levine, Semi-open sets and semi-continuity in topological spaces,Amer. Math. Monthly,70 (1963), 36–41. · Zbl 0113.16304
[5]	A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak precontinuous mappings,Proc. Math. and Phys. Soc. Egypt,51 (1981).
[6]	O. Njåstad, On some classes of nearly open sets,Pacific J. Math.,15 (1965), 961–970. · Zbl 0137.41903
[7]	T. Noiri, A generalization of closed mappings,Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur, [8]54 (1973), 412–415. · Zbl 0283.54001
[8]	M. K. Singal and A. R. Singal, Almost continuous functions,Yokohama Math. J.,16 (1968), 63–73. · Zbl 0191.20802