Is P(ω) a subalgebra?

Abstract

We consider the question of whether P(ω) 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 βN, then it has a regular closed zero-set mapping onto βN. The second result is that if the compact space has density at most ω1, then it will map onto βN if it contains a zero-set that maps onto βN.