# zbMATH — the first resource for mathematics

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.

##### MSC:
 54E52 Baire category, Baire spaces 54B10 Product spaces in general topology
##### Keywords:
Baire space, hereditarily Baire space; product
Full Text: