Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on any nonempty open set), $f^{-1}[A]$ is meager in $Y$.

In this paper, Todorcevic proves the following results under the assumption of the existence of arbitrarily large Woodin cardinals.

  1. There is no universally meager, uncountable, separable metric space. More specifically, if $X$ is a separable metric space of size a regular $\kappa > \aleph_0$, then there is a Baire space $Y$ and a continuous nowhere-constant surjection $f : Y \to X$.

  2. A metric space is universally meager if and only if it is the union of countably many discrete subspaces.

There are several other interesting results in the paper, but these are the simplest to state.

Question: Can these statements consistently fail? Is there a counterexample in $L$?


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