Cardinal sequences of length $ \omega_2$ under GCH

István Juhász ¹   ; Lajos Soukup ¹   ; William Weiss ²

¹ Alfréd Rényi Institute of Mathematics V. Reáltanoda utca, 13–15 H-1053 Budapest, Hungary
² Mathematics Department University of Toronto Toronto, ON, M5S 1A1, Canada

Fundamenta Mathematicae, Tome 189 (2006) no. 1, pp. 35-52

Résumé

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 $$ {\cal C}_ {\lambda}(\alpha)=\{s\in \mathcal C(\alpha): s(0)={\lambda} = \min[ s({\beta}) : \beta {\alpha}]\}. $$ 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 $f=f_0\kern-3pt\mathop{{}^{\frown}\kern-3pt} f_1\kern-3pt\mathop{{}^{\frown}\kern-3pt} \ldots \kern-3pt\mathop{{}^{\frown}\kern-3pt} f_{n-1}$ where each $f_i\in\mathcal C_{\lambda_i}(\alpha_i)$. Under GCH we prove that if $\alpha \omega_2$ then(i) $\mathcal C_{\omega}(\alpha)=\{s\in {}^{\alpha}\{{\omega},\omega_1\}: s(0)={\omega}\}$;(ii) if $\lambda > \mathop{\rm cf} (\lambda)=\omega$, $$ {\cal C}_ {\lambda}(\alpha)=\{s\in {}^{\alpha}\{{\lambda},{\lambda}^+\}: s(0)={\lambda},\ s^{-1}\{\lambda\}\hbox{ is ${\omega}_1$-closed in ${\alpha}$} \}; $$ (iii) if $\mathop{\rm cf} (\lambda)=\omega_1$, $$ {\cal C}_ {\lambda}(\alpha)=\{s\in {}^{\alpha}\{{\lambda},{\lambda}^+\}: s(0)={\lambda},\, s^{-1}\{\lambda\}\hbox{ is ${\omega}$-closed and successor-closed in ${\alpha}$} \};$$ (iv) if $\mathop{\rm cf} (\lambda)>\omega_1$, $\mathcal C_\lambda (\alpha)= {}^\alpha\{\lambda\}$.

Détail

DOI : 10.4064/fm189-1-3

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