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On the preservation of Baire and weakly Baire category. (English) Zbl 1413.54091
Recall that a topological space is said to be a Baire space if no non-empty open subset is of first category. G. Beer and L. Villar [Southeast Asian Bull. Math. 11, No. 2, 127–133 (1988; Zbl 0665.54019)] extended this notion by introducing the class of weakly Baire spaces. By a weakly Baire space, we mean a space in which no non-empty open, dense-in-itself subset is countable. Clearly, every \(T_1\) Baire space is weakly Baire.
In the paper under review, the authors study the preservation of Baire and weakly Baire spaces under images and/or preimages of special kinds of mappings. It is shown that a space \(Y\) is Baire provided that there exist a Baire space \(X\) and a surjection \(f:X\to Y\) satisfying the following conditions: (1) \(f(U)\) has non-empty interior in \(Y\) whenever \(U\) is a non-empty open subset of \(X\); and (2) if \(U\) is a non-empty open subset of \(X\) and \(W\) is a non-empty open subset of \(Y\) such that \(W\subseteq f(U)\), then there exists a non-empty open \(U'\subseteq U\) such that \(f(U')\subseteq W\). This slightly improves a result of Z. FrolĂ­k in [Czech. Math. J. 11(86), 381–385 (1961; Zbl 0104.17204)]. A similar result is obtained for weakly Baire spaces. Finally, a result on the preservation of Baireness under preimages along the framework of D. Noll in [Proc. Am. Math. Soc. 107, No. 3, 847–854 (1989; Zbl 0687.54012)] is also obtained.
MSC:
54E52 Baire category, Baire spaces
54C08 Weak and generalized continuity
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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