I could not find any references about this fact. I apologize if this is completely trivial, but is the product of two Baire spaces, or for that matter of finitely many of them a Baire space? Now is a countable product of Baire spaces a Baire space? 


What about an uncountable product of Baire space? This fact seems to be treated in an article I can't access.


It seems to work for the Sorgenfrey line: $S$ is a Baire space and $SxS$ is a Baire space since if you consider the diagonal $A$={($-x$,$x$): $x\in S$ and $x\in$ℚ} then this is a closed discrete subspaces which is the union of countably many closed nowhere dense sets but its interior is empty.


Is that true?


Thx



Products of Baire spaces
See:


Cohen, Paul E.  

Products of Baire spaces.  

Proc. Amer. Math. Soc. 55 (1976), no. 1, 119--124.


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> MathSciNet review by Douglas Censer: A topological space is said to be Baire if any countable intersection of its dense open sets is dense. Assuming the continuum hypothesis (CH), J. C. Oxtoby [Fund. Math. 49 (1960/61), 157--166; MR0140638 (25 #4055); errata, MR 26, p. 1543] constructed a Baire space whose square is not Baire. The author shows that the assumption of CH is unnecessary here. The spaces considered in this paper are partially ordered $(P,\leq)$ with topology generated by the initial segments. The construction of the desired Baire space with non-Baire product is based on the study of $P$ as a set of forcing conditions, as outlined by R. M. Solovay [Ann. of Math. 92 (1970), 1--56; MR0265151 MR0265151 (42 #64)].
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Note that googling "products of Baire spaces" returns Cohen's article as the second hit. (The first hit is this question!)


For a positive result, see


Zsilinszky, László  

Products of Baire spaces revisited.  

Fund. Math. 183 (2004), no. 2, 115--121.



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> Excerpted from the MathSciNet review by Paul Bankston: Without extra assumptions, the product of two Baire spaces need not be Baire [see, e.g., J. C. Oxtoby, Fund. Math. 49 (1960/1961), 157--166; MR0140638 (25 #4055); P. E. Cohen, Proc. Amer. Math. Soc. 55 (1976), no. 1, 119--124; MR0401480 (53 #5307)]. This brings us to the notion of a $\pi$-base; i.e., a collection of open sets such that every nonempty open set in the space contains a member of the collection. A $\pi$-base each of whose members contains only countably many members of the $\pi$-base is called countable-in-itself.
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> Oxtoby's theorem, from his 1961 paper, states that any Tikhonov (resp., finite) product of Baire spaces with countable (resp., countable-in-itself) $\pi$-bases is a Baire space. The main result of the present paper is a significant strengthening of this; in particular it implies that arbitrary Tikhonov products of Baire spaces with countable-in-itself $\pi$-bases are Baire spaces.
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> A space $X$ is called universally Kuratowski-Ulam (uK-U for short, first considered in [C. Kuratowski and S. Ulam, Fund. Math. 19 (1932), 247--251; Zbl 0005.18301]) if whenever $Y$ is a space and $E$ is a meager subset of $X\times Y$, the set $ Y \setminus \{y\in Y\:\ \{x\in X\:\ (x,y)\in E\}\ \text{is meager in}\ X\} $ is meager in $Y$. The author now defines a space $X$ to be almost locally uK-U if the set $\{x\in X\:\ x \text{has an open uK-U neighborhood} \}$ is dense in $X$.
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> After showing that the property of being almost locally uK-U is a proper generalization of having a countable-in-itself $\pi$-base, the author proves his main theorem: only a Tikhonov (or countable box) product of Baire spaces that are almost locally uK-U is a Baire space.
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> The proof for the box product case is a variant of that for the Tikhonov case, and partially answers a question raised in [W. G. Fleissner, in General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976), Part B, 125--126, Soc. Czechoslovak Mathematicians and Physicists, Prague, 1977; MR0464181 (57 #4116)].
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