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On BPI Restricted to Boolean Algebras of Size Continuum

Hall E., Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

Vol. 61, no 1

9--21

(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 2P(ω) the restriction B|Y={Y∩B:B∈B} of the standard clopen base B of 2P(ω) to Y has size ≤|P(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of P(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.