Cardinal Sequences of length $< \omega_2$ under GCH

István Juhász, Lajos Soukup and William Weiss

Let ${\mathcal C}(\alpha)$ denote the class of all cardinal sequences of length $\alpha$ associated with compact scattered spaces (or equivalently, superatomic boolean algebras). Also put

$${\mathcal C}_{\lambda}(\alpha)=\{s\in {\mathcal C}(\alpha): s(0)={\lambda} = \min[ s({\beta}) : \beta < {\alpha}]\}.$$

We show that $f\in {\mathcal C}(\alpha)$ iff for some natural number $n$ there are infinite cardinals $\lambda_0>\lambda_1>\dots>\lambda_{n-1}$ and ordinals ${\alpha}_0,\dots {\alpha}_{n-1}$ such that ${\alpha}={\alpha}_0+\cdots+{\alpha}_{n-1}$ and $f=f_0\mathop{{}^{\frown}\makebox[-3pt]{}}\ f_1\mathop{{}^{\frown}\makebox[-3pt]{}}\cdots \ \mathop{{}^{\frown}\makebox[-3pt]{}}f_{n-1}$ where each $f_i\in{\mathcal C}_{\lambda_i}(\alpha_i)$.

Under GCH we prove that if $\alpha < \omega_2$ then

• 
• if $\lambda > \operatorname{cf}(\lambda)=\omega$, then is ${\omega}_1$-closed in ${\alpha}$$$\};$$
• if ,
• if , .

This yields a complete characterization of the classes ${\mathcal C}(\alpha)$ for all $\alpha < \omega_2$, under GCH.

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2000 Mathematics Subject Classification:

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