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

Alan Dow ¹   ; Ilijas Farah ²

¹ Department of Mathematics UNC-Charlotte 9201 University City Blvd. Charlotte, NC 28223-0001, U.S.A.
² Department of Mathematics and Statistics York University 4700 Keele Street North York, Ontario, Canada, M3J 1P3 and Matematicki Institut Kneza Mihaila 35 Beograd, Serbia and Montenegro

Fundamenta Mathematicae, Tome 183 (2004) no. 2, pp. 91-108

Résumé

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 $\beta {\mathbb N}$.

Détail

DOI : 10.4064/fm183-2-1

Mots-clés : consider question whether mathcal omega subalgebra whenever quotient boolean algebra countably generated ideal question raised privately murray bell obtain partial answers under coloring axiom topologically first result zero dimensional compact space has zero set mapping beta mathbb has regular closed zero set mapping beta mathbb second result compact space has density omega map beta mathbb contains zero set maps beta mathbb