On cardinal sequences of length

Abstract

We prove the following consistency result for cardinal sequences of length <ω3: if GCH holds and λ≥ω2 is a regular cardinal, then in some cardinal-preserving generic extension 2ω=λ and for every ordinal η<ω3 and every sequence f=〈κα:α<η〉 of infinite cardinals with κα≤λ for α<η and κα=ω if cf(α)=ω2, we have that f is the cardinal sequence of some LCS space.Also, we prove that for every specific uncountable cardinal λ it is relatively consistent with ZFC that for every α,β<ω3 with cf(α)<ω2 there is an LCS space Z such that CS(Z)=〈ω〉α⌢〈λ〉β.