The Baire Category Theorem and Cardinals of Countable Cofinality

Arnold W. Miller
Journal of Symbolic Logic, Vol. 47, No. 2. (Jun., 1982), pp. 275-288.


Let $\kappa_B$ be the least cardinal for which the Baire category theorem fails for the real line $\mathbf{R}$. Thus $\kappa_B$ is the least $\kappa$ such that the real line can be covered by $\kappa$ many nowhere dense sets. It is shown that $\kappa_B$ cannot have countable cofinality. On the other hand it is consistent that the corresponding cardinal for $2^{\omega_1}$ be $\aleph_\omega$. Similar questions are considered for the ideal of measure zero sets, other $\omega_1$ saturated ideals, and the ideal of zero-dimensional subsets of $\mathbf{R}^{\omega_1}$.