The main results of this work can be formulated in such an elementary way that it is likely to attract mathematicians from a broad spectrum of specialties, though its main audience will likely be combinatorialists, set-theorists, and topologists. The central question is this: Suppose one is given an at most countable family of algebras of subsets of some fixed set such that, for each algebra, there exists at least one set that is not a member of that...