Thin-very tall compact scattered spaces which are hereditarily separable

Details Thin-very tall compact scattered spaces which are hereditarily separable Thin-very tall compact scattered spaces which are hereditarily separable

2010-05-19

arxiv.org/abs/1005.3528v1

Mathematics

Functional Analysis

accepted to Trans. Amer. Math. Soc.

Scientific paper

10.1090/S0002-9947-2010-05149-9

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, Juh\'asz 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 Fr\'echet smooth renorming, nor an uncountable biorthogonal system.

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