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

Tom 183 / 2004

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


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 $\beta {\mathbb N}$.


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

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