Thin-very tall compact scattered spaces which are hereditarily separable

Brech, Christina; Koszmider, Piotr

doi:10.1090/S0002-9947-2010-05149-9

Thin-very tall compact scattered spaces which are hereditarily separable
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by Christina Brech and Piotr Koszmider PDF

Trans. Amer. Math. Soc. 363 (2011), 501-519 Request permission

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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