(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of size ≤ |ℝ| has an ultrafilter.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba61-1-2,
author = {Eric Hall and Kyriakos Keremedis},
title = {On BPI Restricted to Boolean Algebras of Size Continuum},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {61},
year = {2013},
pages = {9-21},
zbl = {1271.03070},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba61-1-2}
}
Eric Hall; Kyriakos Keremedis. On BPI Restricted to Boolean Algebras of Size Continuum. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 61 (2013) pp. 9-21. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba61-1-2/