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The product of a Baire space with a hereditarily Baire metric space is Baire. (English) Zbl 1093.54008
A topological space \(X\) is called a Baire space if for each sequence \(\{O_n:n\in{\mathbb N}\}\) of dense open subsets of \(X\), \(\bigcap_{n\in{\mathbb N}}O_n\) is dense in \(X\). It is known that the product of two Baire spaces need not be Baire, while it was proved by J. Chaber and R. Pol [Topology Appl. 151, 132–143 (2005; Zbl 1074.54010)] that the arbitrary product of hereditarily Baire (i.e., each closed subspace of them is Baire) metrizable spaces is Baire. In this paper, answering a question by these authors, the author proves that if \(X\) is Baire and \(Y\) is hereditarily Baire and metrizable, then \(X\times Y\) is also Baire. In the proof, the game characterization of Baireness due to J. Saint Raymond [Proc. Am. Math. Soc. 87, 499–504 (1983; Zbl 0511.54007)] is used.

54E52 Baire category, Baire spaces
54B10 Product spaces in general topology
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