Thank you very much for the information Professor Taras.

Great, thank you very much, is strange because this argument is used more than once in the article. In addition, Theorem 3.1 is used to show that ccc spaces are productively Baire.

But I think that $\{[f^{\smallfrown}A_{\lambda}] \}_{\lambda<\mathfrak{c}}$ is a pairwise disjoint family of non-empty open sets, because otherwise there are $\lambda_{1}, \lambda_{2}\in\mathfrak{c}$ such that $[f^{\smallfrown}A_{\lambda_{1}}] \cap [f^{\smallfrown}A_{\lambda_{2}}]\not=\emptyset$, so there is $g\in \mathcal{K}(X) $ such that $g|_{n+1}=f^{\smallfrown}A_{\lambda_{1}}=f^{\smallfrown}A_{\lambda_{2}}$, contradiction.

Thank you so much for the idea.

Also for the part " $X$ is comeager in $\tilde{X}$ " we don't need the hypothesis about "separability".

Thanks Professor Taras, in the hypothesis of the Theorem (8.17) of Kechris's book says that " if $X$ is a nonempty separable metrizable space and $\tilde{X}$ a Polish space in which $X$ is dense. Then $X$ is Choquet iff $X$ is comeager in $\tilde{X}$ ". How can you conclude that the result is valid for any metrizable space? Thanks

Hello, excuse me, do you have any reference for the characterization of the Choquet spaces in a metrizable space? I only knew that a Choquet space is productively Baire. Also if $X$ is a Choquet space without isolated points then $X$ contains a Cantor set. Thanks

About products of Baire spaces and spaces with countable cellularity, in Baire spaces - R. C. Haworth, R. A. McCoy, I studied the following result. Theorem Let $\{X_{\alpha} : \alpha\in A \}$ be a family of Baire spaces such that the product of any countable subcollection is a Baire space and such that $\prod_{\alpha\in A} X_{\alpha}$ has the countable chain condition. Then $\prod_{\alpha\in A} X_{\alpha}$ is a Baire space.

Well on page 210 of Scheepers' article, he mentions the relationship between the box topology, Tychonoff product and the Banach-Mazur game and finally comments on Theorem 46.