Thin-very tall compact scattered spaces which are hereditarily separable

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....

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Bibliographic Details
Published in:arXiv.org 2010-05
Main Authors: Brech, Christina, Koszmider, Piotr
Format: Article
Language:English
Subjects:
Banach spaces
Online Access:Get full text
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Summary: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 \(n\in\N\). 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.
ISSN:2331-8422
DOI:10.48550/arxiv.1005.3528