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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-2-1,
author = {Alan Dow and Ilijas Farah},
title = {Is P(o) a subalgebra?},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {91-108},
zbl = {1074.54004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-2-1}
}
Alan Dow; Ilijas Farah. Is 𝓟(ω) a subalgebra?. Fundamenta Mathematicae, Tome 184 (2004) pp. 91-108. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-2-1/