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