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
Best Answer
See:
Cohen, Paul E.
Products of Baire spaces.
Proc. Amer. Math. Soc. 55 (1976), no. 1, 119--124.
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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.