On the inverse image of Baire spaces
\textit{Z. Frolík} [Czech. Math. J. 11(86), 381-385 (1961; Zbl 0104.17204)] proved that if \(f\) is an open and continuous function from a metrizable separable space \(X\) onto a Baire space \(Y\) and if the point inverses are Baire spaces, then \(X\) is Baire. In the paper under review it is shown that both assumptions ``open and continuous'' and ``metrizable separable'' in the above result can be weakened to ``semi-open and semi-continuous'' and ``second countable'', respectively. Three other ``pairs'' of almost continuous and almost open functions are found for which Frolík's theorem holds. Reviewer's note: \textit{I. L. Reilly} and the reviewer [Quest. Answers Gen. Topology 11, No. 1, 105-107 (1993; Zbl 0779.54025)] provide an example showing the necessity of semi-continuity of \(f\) in the author's result.
scientific article; zbMATH DE number 1054246
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