On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall ¹   ; Kyriakos Keremedis ²

¹ Department of Mathematics & Statistics College of Arts & Sciences University of Missouri – Kansas City 206 Haag Hall, 5100 Rockhill Rd Kansas City, MO 64110, USA
² Department of Mathematics University of the Aegean Karlovassi, Samos 83200, Greece

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 61 (2013) no. 1, pp. 9-21

Résumé

(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.

Détail

DOI : 10.4064/ba61-1-2

Keywords: statement mathbf omega every partition mathbb has size leq mathbb equivalent proposition mathbf omega every subspace tychonoff product mathbf mathcal omega restriction mathcal cap mathcal standard clopen base mathcal mathbf mathcal omega has size leq mathcal omega mathbf mathbf omega does imply every partition mathcal omega has choice set iii under mathbf omega following statements equivalent every boolean algebra size leq mathbb every filter extended ultrafilter every boolean algebra size leq mathbb has ultrafilter