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Question. Is there a Baire subspace $X$ of a Tychonoff power $M^\kappa$ of some separable metrizable space $M$ such that for any countable subset $A\subset \kappa$ the projection $$X_A=\{x{\restriction}A:x\in X\subset M^\kappa\}\subset M^A$$ of $X$ onto $M^A$ in a meager (metrizable separable) space?

Actually, I am interested in the negative answer under the assumption that the Baire space $X$ has countable spread (i.e., does not contain an uncountable discrete subspace).

Let us recall that a topological space $X$ is

$\bullet$ meager if $X$ can be written as a countable union of closed subsets with empty interior;

$\bullet$ Baire if $X$ contains no non-empty open meager subspace.

Remark. I admit that the answer to the Question can depend on some set-theoretic assumptions (like CH or Martin's Axiom).

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  • $\begingroup$ I take it you want $X_A$ to be meager-in-itself, as $2^\kappa$ in $[0,1]^\kappa$ is Baire with nowhere dense projections. $\endgroup$
    – KP Hart
    Jan 14, 2019 at 9:36
  • $\begingroup$ @KPHart Exactly, becase of that I wrote that $X_A$ is a meager space, not a meager subset of $M^A$. $\endgroup$
    – Taras Banakh
    Jan 14, 2019 at 10:56

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