Superatomic Boolean algebras constructed from strongly unbounded functions

摘要

Using Koszmider’s strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ~(+++) ≤ λ, κ~(<κ) = κ and 2~κ = κ~+, and η is an ordinal with κ~+ ≤ η < κ~(++) and cf(η) = κ~+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra B such that ht(B) = η + 1, wd_α (B) = κ for every α < η and wd_η (B) = λ (i.e., there is a locally compact scattered space with cardinal sequence <κ>_η ~<λ>). Especially, <ω>_(ω1)~<ω_3> and <ω_1>_(ω2) ~<ω_4> can be cardinal sequences of superatomic Boolean algebras.