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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

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