Cardinal sequences of LCS spaces under GCH

Abstract

Let C ( α ) denote the class of all cardinal sequences of length α associated with compact scattered spaces. Also put C λ ( α ) = { f ∈ C ( α ) : f ( 0 ) = λ = min [ f ( β ) : β < α ] } . If λ is a cardinal and α < λ + + is an ordinal, we define D λ ( α ) as follows: if λ = ω , D ω ( α ) = { f ∈ α { ω , ω 1 } : f ( 0 ) = ω } , and if λ is uncountable, D λ ( α ) = { f ∈ α { λ , λ + } : f ( 0 ) = λ , f − 1 { λ } is < λ -closed and successor-closed in α } . We show that for each uncountable regular cardinal λ and ordinal α < λ + + it is consistent with GCH that C λ ( α ) is as large as possible, i.e. C λ ( α ) = D λ ( α ) . This yields that under GCH for any sequence f of regular cardinals of length α the following statements are equivalent: (1) f ∈ C ( α ) in some cardinal-preserving and GCH-preserving generic-extension of the ground model. (2) for some natural number n there are infinite regular cardinals λ 0 > λ 1 > ⋯ > λ n − 1 and ordinals α 0 , … , α n − 1 such that α = α 0 + ⋯ + α n − 1 and f = f 0 ⌢ f 1 ⌢ ⋯ ⌢ f n − 1 where each f i ∈ D λ i ( α i ) . The proofs are based on constructions of universal locally compact scattered spaces.