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On cardinal sequences of scattered spaces. (English) Zbl 0926.54025

Summary: It was proved by A. Dow and P. Simon [Algebra Univers. 29, No. 2, 211-226 (1992; Zbl 0786.06011)] that there are \(2^{\omega_1}\) (as many as possible) pairwise nonhomeomorphic compact, \(T_2\), scattered spaces of height \(\omega_1\) and width \(\omega\). In this paper, we prove that if \(\alpha\) is an ordinal with \(\omega_1\leq \alpha< \omega_2\) and \(\theta= \langle \kappa_\xi: \xi<\alpha\rangle\) is a sequence of cardinals such that either \(\kappa_\xi= \omega\) or \(\kappa_\xi= \omega_1\) for every \(\xi< \alpha\), then there are \(2^{\omega_1}\) pairwise nonhomeomorphic compact, \(T_2\), scattered spaces whose cardinal sequence is \(\theta\).

MSC:

54G12 Scattered spaces
06E99 Boolean algebras (Boolean rings)

Citations:

Zbl 0786.06011
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References:

[1] Baumgartner, J. E.; Shelah, S., Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic, 33, 109-129 (1987) · Zbl 0643.03038
[2] Dow, A.; Simon, P., Thin-tall Boolean algebras and their automorphism groups, Algebra Universalis, 29, 211-226 (1992) · Zbl 0786.06011
[3] Juhász, I.; Weiss, W., On thin-tall scattered spaces, (Colloq. Math., 40 (1978)), 63-68 · Zbl 0416.54038
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