A consistency theorem for cardinal sequences of length $< ω_3$
Juan Carlos Martínez, Lajos Soukup
TL;DR
The paper proves a consistency theorem for cardinal sequences of locally compact scattered spaces of length $\delta<\omega_3$, where each entry is either $\omega$ or a fixed uncountable cardinal $\lambda$, and the set of indices mapping to $\omega$ is $\omega_2$-closed. The authors develop a two-step forcing framework under $GCH+\square_{\omega_1}$ that realizes the target sequence as the CS of a scattered space, using a decomposition into $f_0+f_1$, a tree-of-intervals construction for $f_1$, and a c.c.c. forcing ${\mathbb P}_{\delta}$ to produce an LCS poset whose associated space has the desired cardinal sequence. The approach extends previous results by enabling a wide class of length $<\omega_3$ sequences to be realized as CS(X), and furnishes a stronger subspace property that reflects the chosen cardinal parameters. The work hinges on the Δ-function, LCS posets, and detailed forcing amalgamations to preserve cardinals while shaping the Cantor-Bendixson levels. This contributes to the broader understanding of the interplay between set-theoretic forcing and the structure of superatomic Boolean algebras via scattered spaces.
Abstract
We prove that if $λ$ is a fixed uncountable cardinal and $f = \langle \ka_{\al} : \al < δ\rangle$ is a sequence of infinite cardinals where $δ< ω_3$ and $\ka_{\al}\in \{\om,λ\}$ for each $\al < δ$ in such a way that $f^{-1}\{\om\}$ is $\om_2$-closed in $δ$, then it is consistent that there is a scattered Boolean space whose cardinal sequence is $f$.