## Journal of Symbolic Logic

### The Baire Category Theorem and Cardinals of Countable Cofinality

Arnold W. Miller

#### Abstract

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 $\mathbf{R}^{\omega_1}$.

#### Article information

Source
J. Symbolic Logic, Volume 47, Issue 2 (1982), 275-288.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183740998

Mathematical Reviews number (MathSciNet)
MR654788

Zentralblatt MATH identifier
0487.03026

JSTOR