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Abstract: We strengthen the property $\Delta$ of a function $f:\omega 2^2 ightarrow\omega 2^{\leq \omega}$ considered by Baumgartner and Shelah. This allows usto consider new types of amalgamations in the forcing used by Rabus, Juh\-aszand Soukup to construct thin-very tall compact scattered spaces. Weconsistently obtain spaces $K$ as above where $K^n$ is hereditarily separablefor each $n\in\N$. This serves as a counterexample concerning cardinalfunctions on compact spaces as well as having some applications in Banachspaces: the Banach space $CK$ is an Asplund space of density $\aleph 2$ whichhas no Fr\-echet smooth renorming, nor an uncountable biorthogonal system.



Author: Christina Brech, Piotr Koszmider

Source: https://arxiv.org/







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