On Todorcevic orderings

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 sigma-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.