Let B be a complete ccc Boolean algebra and let $\tau_s$ be the topology on B induced by the algebraic convergence of sequences in B.
1. Either there exists a Maharam submeasure on B or every nonempty open set in $(B,\tau_s)$ is topologically dense.
2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure.
3. The topological space $(B,\tau_s)$ 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.
