an:06674857
Zbl 1413.54091
Mirmostafaee, Alireza Kamel; Piotrowski, Zbigniew
On the preservation of Baire and weakly Baire category
EN
Math. Bohem. 141, No. 4, 475-481 (2016).
00362272
2016
j
54E52 54C08 54C10
Baire space; feebly continuous; feebly open; quasi-continuous; quasi-interior continuous; weakly Baire space; fiber-completeness
Recall that a topological space is said to be a \textit{Baire space} if no non-empty open subset is of first category. \textit{G. Beer} and \textit{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 \textit{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 \textit{Z. Frol??k} 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 \textit{D. Noll} in [Proc. Am. Math. Soc. 107, No. 3, 847--854 (1989; Zbl 0687.54012)] is also obtained.
Jiling Cao (Auckland)
Zbl 0665.54019; Zbl 0104.17204; Zbl 0687.54012