Abstract

It is proved that the following conditions are equivalent: there exists a complete, atomless, $\sigma$-centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra, there exists a nowhere dense ultrafilter on $\omega$. Therefore, the existence of such algebras is undecidable in ZFC. In "forcing language" condition says that there exists a non-trivial $\sigma$-centered forcing not adding Cohen reals.