# A question about infinite product of Baire and meager spaces

Proposition 1: For any space $$X$$ and an infinite cardinal $$\kappa$$, the product $$X^{\kappa}$$ is either meager or a Baire space.

Does anyone have any suggestions to demonstrate Proposition 1?

I was trying using topological games in the following way, suppose that $$X^{\kappa}$$ is not a Baire space, then $$X^{\omega}$$ is not a Baire space (Theorem 2, Barely Baire Spaces - W. Fleissner and K. Kunen), then, using the Banach-Mazur game, Player I has a winning strategy in $$\textsf{BM}(X^{\omega})$$. I was trying to show that Player I has a winning strategy in the game $$\textsf{MB}(X^{\kappa})$$, but unfortunately I still haven't got it.

Could someone tell me if I am on the right path for the proof of the proposition? Otherwise, someone has another idea for the proof.

Thanks

• What is your source for Proposition 1?
– bof
Mar 7 at 23:10
• Lemma 4.2 in the article : A countable dense homogeneous topological vector space is a Baire space (doi.org/10.1090/proc/15271). Mar 9 at 23:41
• My question for the proof the before lemma is : Why BCT implies that there is a non-empty open Baire subset $U$ of $X^{\kappa}$? Mar 9 at 23:42