Locally-generic Boolean algebras and cardinal sequences

Abstract

Given a sequence of cardinals \( \theta \) of length less than \( \scr w_2 \), with each cardinal in the sequence being either \( \scr w \) or \( \scr w_1 \), we construct a \( \theta \)-poset (see Defnition 1 below) which, with a natural topology, becomes a locally-compact, Hausdorff, scattered space with cardinal sequence \( \theta \). The algebra of the clopen subsets of its one-point compactification yields, in turn, a superatomic Boolean algebra with \( \theta \) 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 \( \diamondsuit \) Under Martin's Axiom, the construction allows any cardinals \( \leq2^w \) in the sequence, provided it has length \( \leq{w_1} \) Finally, we modify a forcing argument of Baumgartner-Shelah [B-S], to build \( \theta \)-posets for any given cardinal sequence \( \theta \) of length \( w_2 \) with each cardinal in the sequence being either \( w \) or \( w_1 \).