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
DOI:
https://doi.org/10.1090/S0002-9939-03-06662-0
Published electronically:
January 2, 2003
MathSciNet review:
1955280
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- James E. Baumgartner and Saharon Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic 33 (1987), no. 2, 109–129. MR 874021, DOI https://doi.org/10.1016/0168-0072%2887%2990077-7
- I. Juhász and W. Weiss, On thin-tall scattered spaces, Colloq. Math. 40 (1978/79), no. 1, 63–68. MR 529798, DOI https://doi.org/10.4064/cm-40-1-63-68
- Winfried Just, Two consistency results concerning thin-tall Boolean algebras, Algebra Universalis 20 (1985), no. 2, 135–142. MR 806609, DOI https://doi.org/10.1007/BF01278592
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- Juan Carlos Martínez, A forcing construction of thin-tall Boolean algebras, Fund. Math. 159 (1999), no. 2, 99–113. MR 1670099, DOI https://doi.org/10.4064/fm-159-2-99-113
- Judy Roitman, Height and width of superatomic Boolean algebras, Proc. Amer. Math. Soc. 94 (1985), no. 1, 9–14. MR 781045, DOI https://doi.org/10.1090/S0002-9939-1985-0781045-0
- Saharon Shelah, On what I do not understand (and have something to say). I, Fund. Math. 166 (2000), no. 1-2, 1–82. Saharon Shelah’s anniversary issue. MR 1804704
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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
MR Author ID:
160185
ORCID:
0000-0003-0462-3152
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
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
