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A tall space with a small bottom

Authors: István Juhász, Saharon Shelah, Lajos Soukup and Zoltán Szentmiklóssy
Journal: Proc. Amer. Math. Soc. 131 (2003), 1907-1916
MSC (2000): Primary 54A25, 06E05, 54G12, 03E20
Published electronically: January 2, 2003
MathSciNet review: 1955280
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Abstract: We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if $\kappa^{<\kappa} = \kappa$, then there is such a space of height $\kappa^+$ with only $\kappa$ many isolated points. This implies that there is a locally compact scattered space of height ${\omega}_2$ with $\omega_1$ isolated points in ZFC, solving an old problem of the first author.

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Additional Information

István Juhász
Affiliation: Alfréd Rényi Institute of Mathematics, P.O. Box 127, 1364 Budapest, Hungary

Saharon Shelah
Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel

Lajos Soukup
Affiliation: Alfréd Rényi Institute of Mathematics, P.O. Box 127, Budapest, Hungary

Zoltán Szentmiklóssy
Affiliation: Eötvös University of Budapest, 1117, Budapest, Pázmány Péter Sétány 1/C, Hungary

Keywords: Locally compact scattered space, superatomic Boolean algebra
Received by editor(s): April 4, 2001
Received by editor(s) in revised form: December 4, 2001
Published electronically: January 2, 2003
Additional Notes: The first, third and fourth authors were supported by the Hungarian National Foundation for Scientific Research grant no. 25745
The second author was supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Publication 714
The third author was partially supported by Grant-in-Aid for JSPS Fellows No. 98259 of the Ministry of Education, Science, Sports and Culture, Japan
Communicated by: Alan Dow
Article copyright: © Copyright 2003 American Mathematical Society