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.1. Either there exists a Maharam submeasure on B or every nonempty open set in (B, τs ) 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, 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.

The Todorcevic ordering T(X) consists of all finite families of convergent sequences in a given topological space X. Such an ordering was defined for the special case of the real line by S. Todorcevic (1991) as an example of a Borel ordering satisfying ccc that is not σ-finite cc and even need not have the Knaster property. We are interested in properties of T(X) where the space X is taken as a parameter. Conditions on X are given which ensure the countable chain condition and its stronger versions for T(X). We study the properties of T(X) as a forcing notion and the homogeneity of the generated complete Boolean algebra.

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