# A Baire space with meager projections

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).

• 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. – KP Hart Jan 14 at 9:36
• @KPHart Exactly, becase of that I wrote that $X_A$ is a meager space, not a meager subset of $M^A$. – Taras Banakh Jan 14 at 10:56