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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

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    $\begingroup$ What is your source for Proposition 1? $\endgroup$
    – bof
    Mar 7, 2022 at 23:10
  • $\begingroup$ Lemma 4.2 in the article : A countable dense homogeneous topological vector space is a Baire space (doi.org/10.1090/proc/15271). $\endgroup$
    – Gabriel Medina
    Mar 9, 2022 at 23:41
  • $\begingroup$ My question for the proof the before lemma is : Why BCT implies that there is a non-empty open Baire subset $U$ of $X^{\kappa}$? $\endgroup$
    – Gabriel Medina
    Mar 9, 2022 at 23:42

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