I have the following problem. Let A be a Boolean algebra which is uncountable and atomic (that is every element is a supremum of some set of atoms). Let C be an uncountable chain in A w.r.t standard order <=. Is it always possible to construct a subchain C' of C which is dense in C? (D is dense in B iff for all b \in B there is some d \in D such that d<= b).