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A decomposition of continuity. (English) Zbl 0609.54012
This paper introduces the following notions. A subset B of a topological space (X,\({\mathcal T})\) is defined to be an \({\mathcal A}\)-set if \(B=U-W\) where U is open and W is regular open. A function f: (X,\({\mathcal T})\to (Y,{\mathcal U})\) is called \({\mathcal A}\)-continuous if \(f^{-1}(V)\) is an \({\mathcal A}\)- set in X for each open set V in Y. O. Njåstad [Pac. J. Math. 15, 961-970 (1965; Zbl 0137.419)] defined a subset S of (X,\({\mathcal T})\) to be an \(\alpha\)-set if \(S\subset int(cl(int S))\). A. S. Mashhour et al [Acta Math. Hung. 41, 213-218 (1983; Zbl 0534.54006)] defined a function f: (X,\({\mathcal T})\to (Y,{\mathcal U})\) to be \(\alpha\)-continuous if \(f^{-1}(V)\) is an \(\alpha\)-set in X for each open set V in Y. The main result of this paper is that f is continuous if and only if it is \(\alpha\)-continuous and \({\mathcal A}\)-continuous.
Reviewer: I.L.Reilly

MSC:
54C05 Continuous maps
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