Thin-very tall compact scattered spaces which are hereditarily separable

TL;DR

This paper enhances a combinatorial property to construct specific compact scattered spaces with hereditary separability properties, providing counterexamples in topology and Banach space theory.

Contribution

It introduces a strengthened property for functions used in forcing, enabling the construction of new thin-very tall compact scattered spaces with hereditary separability.

Findings

Constructed spaces where all finite powers are hereditarily separable.

Provided counterexamples for cardinal functions on compact spaces.

Showed the Banach space C(K) lacks certain smooth renormings and biorthogonal systems.

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.