# A question about infinite product of Baire and meager spaces

A topological space $$X$$ is

1. Baire if every sequence $$(U_{n})_{n\in\omega}$$ of dense open subsets of $$X$$ has a dense intersection in $$X$$.
2. Meager if it can be written as a countable union of closed sets with empty interior.

The $$\textsf{BM}$$ and $$\textsf{MB}$$ games. Let $$X$$ be a topological space.

1. The game $$\textsf{BM}(X)$$ is started by the Player I who selects a non-empty open set $$U_0 \subseteq X$$. Then Player II responds selecting a non-empty open set $$U_{1}\subseteq U_0$$. At the $$n$$-th inning the Player I selects a non-empty open set $$U_{2n} \subseteq U_{2n−1}$$ and the Player II responds selecting a non-empty open set $$U_{2n+1} \subseteq U_{2n}$$. At the end of the game, the Player I is declared the winner if $$\bigcap_{n\in\omega}U_{n}$$ is empty. In the opposite case the Player II wins the game $$\textsf{BM}(X)$$.

2. The game $$\textsf{MB}(X)$$ differs from the game $$\textsf{BM}(X)$$ by the order of the players. The game $$\textsf{MB}(X)$$ is started by the Player II who selects a non-empty open set $$U_{0}\subseteq X$$. Then Player I responds selecting a non-empty open set $$U_1 \subseteq U_0$$. At the $$n$$-th inning the Player II selects a non-empty open set $$U_{2n} \subseteq U_{2n−1}$$ and the Player I responds selecting a non-empty open set $$U_{2n+1} \subseteq U_{2n}$$. At the end of the game, the Player I is declared the winner if $$\bigcap_{n\in\omega} U_n$$ is empty. In the opposite case the Player II wins the game $$\textsf{MB}(X)$$.

We have the following classical characterization (Oxtoby):

1. meager if and only if the Player I has a winning strategy in the game $$\textsf{MB}(X)$$.
2. Baire if and only if the Player I has no winning strategy in the game $$\textsf{BM}(X)$$.

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

I was trying 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