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ISSN 1088-6826(e) ISSN 0002-9939(p)

A tall space with a small bottom

Author(s): István Juhász; Saharon Shelah; Lajos Soukup; Zoltán Szentmiklóssy
Journal: Proc. Amer. Math. Soc. 131 (2003), 1907-1916.
MSC (2000): Primary 54A25, 06E05, 54G12, 03E20
Posted: 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.

References:

1.: J. E. Baumgartner and S. Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic 33 (l987), no. 2, 109-129. MR 88d:03100
2.: I. Juhász and W. Weiss, On thin-tall scattered spaces, Colloquium Mathematicum, vol. XL (1978) 63-68. MR 82k:54005
3.: W. Just, Two consistency results concerning thin-tall Boolean algebras, Algebra Universalis 20 (1985) no. 2, 135-142. MR 87c:03101
4.: K. Kunen, Set Theory, North-Holland, New York, 1980. MR 82f:03001
5.: J. C. Martínez, A forcing construction of thin-tall Boolean algebras, Fundamenta Mathematicae 159 (1999), no 2, 99-113. MR 2000g:03116
6.: Judy Roitman, Height and width of superatomic Boolean algebras, Proc. Amer. Math. Soc. 94 (1985), no. 1, 9-14. MR 86f:06024
7.: S. Shelah, On what I do not understand (and have something to say) Part I, Fundamenta Mathematicae 166 (2000), no. 1-2, pp. 1-82. MR 2002a:03091

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Additional Information:

István Juhász
Affiliation: Alfréd Rényi Institute of Mathematics, P.O. Box 127, 1364 Budapest, Hungary
Email: juhasz@renyi.hu

Saharon Shelah
Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
Email: shelah@math.huji.ac.il

Lajos Soukup
Affiliation: Alfréd Rényi Institute of Mathematics, P.O. Box 127, Budapest, Hungary
Email: soukup@renyi.hu

Zoltán Szentmiklóssy
Affiliation: Eötvös University of Budapest, 1117, Budapest, Pázmány Péter Sétány 1/C, Hungary
Email: zoli@renyi.hu

DOI: 10.1090/S0002-9939-03-06662-0
PII: S 0002-9939(03)06662-0
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
Posted: 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
Copyright of article: Copyright 2003, American Mathematical Society

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