On BPI Restricted to Boolean Algebras of Size Continuum
Volume 61 / 2013
Bulletin Polish Acad. Sci. Math. 61 (2013), 9-21
MSC: 03E25, 03G05, 06E25.
DOI: 10.4064/ba61-1-2
Abstract
(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.
