# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Is ${\cal P}(\omega )$ a subalgebra?

### Tom 183 / 2004

Fundamenta Mathematicae 183 (2004), 91-108 MSC: Primary 54A35. DOI: 10.4064/fm183-2-1

#### Streszczenie

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 $\beta {\mathbb N}$.

#### Autorzy

• Alan DowDepartment of Mathematics
UNC-Charlotte
9201 University City Blvd.
Charlotte, NC 28223-0001, U.S.A.
e-mail
• Ilijas FarahDepartment of Mathematics and Statistics
York University
4700 Keele Street
North York, Ontario, Canada, M3J 1P3
and
Matematicki Institut
Kneza Mihaila 35