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A decomposition of continuity and \(\alpha\)-continuity. (English) Zbl 0820.54006

A subset of a topological space \(X\) is said to be preopen \((\alpha\)-open) if \(A\subset \mathrm{Int} (\mathrm{Cl} (A))\) (resp. \(A\subset \mathrm{Int} (\mathrm{Cl} ((\mathrm{Int} (A)))\). The union of all preopen (resp. \(\alpha\)-open) sets contained in \(A\) is called the pre-interior (resp. \(\alpha\)-interior) of \(A\) and is denoted by \(\mathrm{pInt} (A)\) (resp. \(\alpha \mathrm{Int} (A))\). In this paper and in [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 33, 153–157 (1993; Zbl 0798.54017)] the author introduces the following collections of subsets of \(X\) \[ \begin{aligned} D(c,\alpha) &= \{B: \mathrm{Int}(B)= \alpha \mathrm{Int}(B)\},\\ D(p,c) &= \{B: \mathrm{Int}(B) = \mathrm{pInt}(B)\},\\ D(\alpha, p) &= \{B: \alpha \mathrm{Int}(B) = \mathrm{pInt}(B)\}. \end{aligned} \] A function \(f: X\to Y\) is said to be precontinuous (\(\alpha\)-continuous, \(D(c, \alpha)\)-continuous, \(D(c,p)\)-continuous, \(D(\alpha, p)\)-continuous) if \(f^{-1} (V)\) is preopen (\(\alpha\)-open, \(f^{-1} (V)\in D(c, \alpha)\), \(f^{-1} (V)\in D(c,p)\), \(f^{-1} (V)\in D(\alpha, p)\)) for every open set \(V\) of \(Y\). The author obtains, as in his paper cited above the following decompositions of continuity:
1. A mapping \(f\) is continuous if and only if is both \(\alpha\)-continuous and \(D(c, \alpha)\)-continuous.
2. A mapping \(f\) is continuous if and only if it is both precontinuous and \(D(c,p)\)-continuous.
3. A mapping \(f\) is continuous if and only if it is both precontinuous and \(D(\alpha, p)\)-continuous.
A subset \(B\subset X\) is said to be simply open if \(B= U\cup K\), where \(U\) is an open set and \(K\) is nowhere dense. A mapping \(f: X\to Y\) is said to be simply continuous if for every open set \(V\) of \(Y\), \(f^{-1} (V)\) is simply open [A. Neubrunnová, Acta Fac. Rer. Natur. Univ. Comenian., Math. 30, 121–126 (1975; Zbl 0308.26003)].
The author obtains the following new decomposition of \(\alpha\)- continuity: a mapping \(f\) is \(\alpha\)-continuous if and only if is both simply continuous and precontinuous.
Reviewer: V.Popa (Bacau)

MSC:

54C08 Weak and generalized continuity
54C05 Continuous maps
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References:

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