On BPI Restricted to Boolean Algebras of Size Continuum

Volume 61 / 2013

Eric Hall, Kyriakos Keremedis Bulletin Polish Acad. Sci. Math. 61 (2013), 9-21 MSC: 03E25, 03G05, 06E25. DOI: 10.4064/ba61-1-2


(i) The statement $\mathbf{P}(\omega ) = {}$“every partition of $\mathbb{R}$ has size $\leq |\mathbb{R}|$” is equivalent to the proposition $\mathbf{R}(\omega ) ={}$“for every subspace $Y$ of the Tychonoff product $\mathbf{2}^{\mathcal{P}(\omega )}$ the restriction $\mathcal{B}|Y=\{Y\cap B:B\in \mathcal{B}\}$ of the standard clopen base $\mathcal{B}$ of $\mathbf{2}^{\mathcal{P}(\omega )}$ to $Y$ has size $\leq |\mathcal{P}(\omega )|$”.

(ii) In $\mathbf{ZF}$, $\mathbf{P}(\omega )$ does not imply “every partition of $\mathcal{P}(\omega )$ has a choice set”.

(iii) Under $\mathbf{P}(\omega )$ the following two statements are equivalent:

(a) For every Boolean algebra of size $\leq |\mathbb{R}|$ every filter can be extended to an ultrafilter.

(b) Every Boolean algebra of size $\leq |\mathbb{R}|$ has an ultrafilter.


  • Eric HallDepartment of Mathematics & Statistics
    College of Arts & Sciences
    University of Missouri – Kansas City
    206 Haag Hall, 5100 Rockhill Rd
    Kansas City, MO 64110, USA
  • Kyriakos KeremedisDepartment of Mathematics
    University of the Aegean
    Karlovassi, Samos 83200, Greece

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