Weak$^*$ dentability index of spaces $C([0,\alpha])$

TL;DR

This paper calculates the weak$^*$-dentability index for spaces of continuous functions on countable compact spaces, providing explicit ordinal values based on the structure of the space.

Contribution

It determines the weak$^*$-dentability index for $C(K)$ spaces where $K$ is a countable scattered compact space, extending previous understanding of these indices.

Findings

Explicit formula for $Dz(C([0,eta]))$ as $eta$ varies.

Generalization to scattered compact spaces with specific height.

Connection between space structure and dentability index.

Abstract

We compute the weak$^*$-dentability index of the spaces $C(K)$ where $K$ is a countable compact space. Namely ${Dz}(C([0,\omega^{\omega^\alpha}])) = \omega^{1+\alpha+1}$, whenever $0\le\alpha<\omega_1$. More generally, ${Dz}(C(K))=\omega^{1+\alpha+1}$ if $K$ is a scattered compact whose height $\eta(K)$ satisfies $\omega^\alpha<\eta(K)\leq \omega^{\alpha+1}$ with an $\alpha$ countable.