Is ${\cal P}(\omega )$ a subalgebra?
Volume 183 / 2004
Fundamenta Mathematicae 183 (2004), 91-108
MSC: Primary 54A35.
DOI: 10.4064/fm183-2-1
Abstract
We consider the question of whether ${\mathcal P}(\omega )$ is a subalgebra whenever it is a quotient of a Boolean algebra by a countably generated ideal. This question was raised privately by Murray Bell. We obtain two partial answers under the open coloring axiom. Topologically our first result is that if a zero-dimensional compact space has a zero-set mapping onto $\beta {\mathbb N}$, then it has a regular closed zero-set mapping onto $\beta {\mathbb N}$. The second result is that if the compact space has density at most $\omega _1$, then it will map onto $\beta {\mathbb N}$ if it contains a zero-set that maps onto $\beta {\mathbb N}$.
