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The Banach-Mazur game on a nonempty space $X$ is defined as follows: two players, $I$ and $II$, alternately choose nonempty open sets \begin{matrix} I & U_0 && U_1 && \cdots \\ II & & V_0 && V_1 && \cdots \end{matrix}

such that $U_0 \supseteq V_0 \supseteq U_1 \supseteq V_1 \dots$. Player $II$ wins the run $\langle U_0, V_0, U_1, V_1, \dots \rangle$ of the Banach-Mazur game on $X$ iff $\bigcap_{n\in\omega}V_n \not = \emptyset$.

Is there a description of the class of spaces in which $II$ has a winning strategy in the Banach-Mazur game, in terms of continuous maps? For example: $II$ has a winning strategy in the Banach-Mazur game on $X$ iff $X$ is the image of space from $\mathsf{P}$ under map from $\mathsf{L}$. Where $\mathsf{P}$ and $\mathsf{L}$ are some classes of spaces and maps, respectively.

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    $\begingroup$ Isn’t it the case that II has a winning strategy if and only if $X$ is a Baire space? $\endgroup$
    – Emil Jeřábek
    Commented May 6, 2021 at 15:27
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    $\begingroup$ Oh, I see: $X$ is a Baire space iff player I does not have a winning strategy, which is a weaker condition. Wikipedia tells me that spaces where II has a winning strategy are called weakly $\alpha$-favourable, if it helps. $\endgroup$
    – Emil Jeřábek
    Commented May 6, 2021 at 15:33
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    $\begingroup$ I think those spaces are also called Choquet for more googlable terms $\endgroup$
    – Alessandro Codenotti
    Commented May 6, 2021 at 15:50
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    $\begingroup$ According to mathoverflow.net/a/320503, the class of weakly $\alpha$-favourable spaces is closed under images by open continuous maps, so presumably this would be the right choice for L. $\endgroup$
    – Emil Jeřábek
    Commented May 6, 2021 at 15:52

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