$Sz(\cdot )⩽{\omega }^{\xi }$ is rarely a three space property

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.