Locally-generic Boolean algebras and cardinal sequences

Given a sequence of cardinals of length less than , with each cardinal in the sequence being either or , we construct a -poset (see Defnition 1 below) which, with a natural topology, becomes a locally-compact, Hausdorff, scattered space with cardinal sequence . The algebra of the clopen subsets of its one-point compactification yields, in turn, a superatomic Boolean algebra with as its cardinal sequence. The posets are locallygeneric, that is, they are constructed generically over countable sets. This gives them additional chain properties, specially under Under Martin's Axiom, the construction allows any cardinals in the sequence, provided it has length Finally, we modify a forcing argument of Baumgartner-Shelah B-S], to build -posets for any given cardinal sequence of length with each cardinal in the sequence being either or . Received September 2, 1998; accepted in final form September 13, 2001.