On $$\alpha$$-continuity in topological spaces.(English)Zbl 0576.54014

In this paper $$\alpha$$-continuity, $$\alpha$$-open functions, and $$\alpha$$- closed functions, which were introduced in 1983 by A. Mashhour, I. Hasanein, and S. El-Deeb [Acta Math. Hung. 41, 213-218 (1983; Zbl 0534.54006)], and $$\alpha$$-irresolute functions, which were introduced in 1980 by S. Maheshwari and S. Thakur [Tamkang J. Math. 11, 209-214 (1981; Zbl 0485.54009)] are further investigated. The authors retopologize the domain space and/or range space to characterize $$\alpha$$-continuity using continuity, $$\alpha$$-open using open, $$\alpha$$- closed using closed, and $$\alpha$$-irresolute using continuity and these new characterizations are used to quickly establish many known properties of $$\alpha$$-continuous, $$\alpha$$-open, $$\alpha$$-closed, and $$\alpha$$- irresolute functions.
Reviewer: Ch.Dorsett

MSC:

 54C10 Special maps on topological spaces (open, closed, perfect, etc.)

Citations:

Zbl 0534.54006; Zbl 0485.54009
Full Text:

References:

 [1] N. Levine, Semi-open sets and semi-continuity in topological spaces,Amer. Math. Monthly,70 (1963), 36–41. · Zbl 0113.16304 [2] S. N. Maheshwari and S. S. Thakur, On {$$\alpha$$}-irresolute mappings,Tamkang J. Math.,11 (1980), 209–214. [3] 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). [4] A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, {$$\alpha$$}-continuous and {$$\alpha$$}-open mappings,Acta Math. Hung.,41 (1983), 213–218. · Zbl 0534.54006 [5] O. Njåstad, On some classes of nearly open sets,Pacific J. Math.,15 (1965), 961–970. · Zbl 0137.41903
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