This is an announcement for the paper "Thin-very tall compact scattered spaces which are hereditarily separable" by Christina Brech and Piotr Koszmider.

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, 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.

Archive classification: math.FA math.GN

Remarks: accepted to Trans. Amer. Math. Soc.

Submitted from: christina.brech@gmail.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1005.3528

or

http://arXiv.org/abs/1005.3528