TITLE: Products of Baire spaces
QUESTION [13 upvotes]: 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

REPLY [2 votes]: There are two metric Baire spaces, whose product is not Baire; a counterexample is given in the slides An absolute barely Baire space, L.F. Aurichi and G.A.A. Medina, 2020. We present Fleissner and Kunen's proof with applications of Clubs and stationary sets in $\omega_{1}$.
Fleissner, W. G.; Kunen, K., Barely Baire spaces, Fundam. Math. 101, 229-240 (1978). ZBL0413.54036.