$Sz(\cdot)\leqslant \omega^\xi$ is rarely a three space property

TL;DR

The paper investigates when the property of having Szlenk index bounded by certain ordinal powers is preserved in subspaces and quotients, showing it is rarely a three space property for non-additively indecomposable ordinals.

Contribution

It demonstrates that for non-zero, non-additively indecomposable countable ordinals, the Szlenk index property is not a three space property, complementing previous results.

Findings

For non-additively indecomposable ordinals, the property fails to be a three space property.

The result complements earlier work on additively indecomposable ordinals.

Provides a classification of Szlenk index properties in relation to three space property status.

Abstract

We prove that for any non-zero, countable ordinal $\xi$ which is not additively indecomposable, the property of having Szlenk index not exceeding $\omega^\xi$ is not a three space property. This complements a result of Brooker and Lancien, which states that if $\xi$ is additively indecomposable, then having Szlenk index not exceeding $\omega^\xi$ is a three space property.