Thin-very tall compact scattered spaces which are hereditarily separable
Christina Brech, Piotr Koszmider

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 of a function 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 as above where is hereditarily separable for each . This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space is an Asplund space of density which has no Fr\'echet smooth renorming, nor an uncountable biorthogonal system.
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