For any cardinal $\alpha$ we define the space $X$ to be $\alpha$-Baire to mean every union of less than $\alpha$-many nowhere dense sets is a proper subset of $X$. The Baire category theorem states that every Banach space / topological group is $\aleph_1$-Baire.
Define the Baire Characteristic $b(X)$ to be the least cardinal $\beta$ such that $X$ fails to be $\beta$-Baire.
Obviously the Baire category theorem implies $b(\mathbb R) \ge \aleph_0$. But can anything further be said? For example is it true that $b(\mathbb R) =\mathfrak c $? Note this is the highest possible value since $\mathfrak c = |\mathbb R|$.
We always have $b(\mathbb R) = b([0,1]) $ since each contains a nice subset homeomorphic to the other. What is known more generally about the range of allowed values for $b(X)$ when $X$ is a compact metric space? For example is it possible to construct such spaces with arbitrarily large Baire characteristic? We can make this precise. . . .
Conjecture: For every cardinal $\alpha < \mathfrak c$ there exists a compact metric space $X$ with $b(X) \ge \alpha$.
Is this true?