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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Thin-very tall compact scattered spaces which are hereditarily separable


Authors: Christina Brech and Piotr Koszmider
Journal: Trans. Amer. Math. Soc. 363 (2011), 501-519
MSC (2010): Primary 54G12; Secondary 03E35, 46B26
Published electronically: August 30, 2010
MathSciNet review: 2719691
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Abstract: We strengthen the property $ \Delta$ of a function $ f:[\omega_2]^2\rightarrow [\omega_2]^{\leq \omega}$ considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juhász and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces $ K$ as above where $ K^n$ is hereditarily separable for each $ n\in\mathbb{N}$. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space $ C(K)$ is an Asplund space of density $ \aleph_2$ which has no Fréchet smooth renorming, nor an uncountable biorthogonal system.


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Additional Information

Christina Brech
Affiliation: Institute of Mathematics, Statistics and Scientific Computing, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, SP, Brazil
Address at time of publication: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05314-970, São Paulo, SP, Brazil
Email: christina.brech@gmail.com

Piotr Koszmider
Affiliation: Instytut Matematyki, Politechnika Łódzka, ul. Wólczańska 215; 90-924 Łódź, Poland
Email: pkoszmider.politechnika@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05149-9
PII: S 0002-9947(2010)05149-9
Received by editor(s): August 26, 2008
Received by editor(s) in revised form: June 24, 2009
Published electronically: August 30, 2010
Additional Notes: The research was part of Thematic Project FAPESP (2006/02378-7). The first author was supported by scholarships from CAPES (3804/05-4) and CNPq (140426/2004-3 and 202532/2006-2). She would like to thank Stevo Todorcevic and the second author, her Ph.D. advisors at the University of São Paulo and at the University of Paris 7, under whose supervision the results of this paper were obtained.
The second author was partially supported by Polish Ministry of Science and Higher Education research grant N N201 386234.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.