Thin-very tall compact scattered spaces which are hereditarily separable

Abstract

We strengthen the property $\Delta$ of a function $f:[{\omega }_2{]}^2\to [{\omega }_2{]}^{\le \omega }$ 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.

accepted to Trans. Amer. Math. Soc.

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