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A note on the invariance of Baire spaces under mappings. (English) Zbl 0535.54005
A surjective mapping f:\(X\to Y\) between topological spaces X and Y is called feebly continuous if \(Int(f^{-1}(V))\neq \emptyset\) for each nonempty open set V in Y. A space is said to be a Baire space provided each of its nonempty open subsets is of the second category, or, equivalently, if the intersection of each countable family of open dense sets is dense [Z. Frolik, Czechosl. Math. J. 11(86), 237-248 (1961; Zbl 0149.403)]. Generalizing results of Z. Frolik (ibid. 11(86), 381-385 (1961; Zbl 0104.172)] and of T. Neubrunn [Math. Slovaca 27, 173-176 (1977; Zbl 0371.54023)] the author proves that if a feebly continuous mapping f:\(X\to Y\) is defined on a Baire space X and if \(f^{-1}(E)\) is nowhere dense in X for each nowhere dense subset E of Y, then Y is a Baire space.
Reviewer: J.J.Charatonik

MSC:
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54E52 Baire category, Baire spaces
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