COMPLETE CCC BOOLEAN ALGEBRAS, THE ORDER SEQUENTIAL TOPOLOGY, AND A PROBLEM OF VON NEUMANN

Abstract

Let B be a complete ccc Boolean algebra and let τs be the topology on B induced by the algebraic convergence of sequences in B. Either there exists a Maharam submeasure on B or every nonempty open set in (B, τs) is topologically dense. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure. The topological space(B, τs) is sequentially compact if and only if the generic extension by B does not add independent reals. Examples are also given of ccc forcings adding a real but not independent reals. 2000 Mathematics Subject Classification 28A60, 06E10 (primary); 03E55, 54A20, 54A25 (secondary).