Abstract
We show that the set of all separable Banach spaces that have the π-property is a Borel subset of the set of all closed subspaces of C(Δ), where Δ is the Cantor set, equipped with the standard Effros-Borel structure. We show that if α < ω 1, the set of spaces with Szlenk index at most α which have a shrinking FDD is Borel.
Highlights
Let C( ) be the space of continuous functions on the Cantor space
It is well known that C( ) is isometrically universal for all separable Banach spaces
In [1], Bossard considered the topological complexity of the isomorphism relation and of many subsets of SE
Summary
Let C( ) be the space of continuous functions on the Cantor space. It is well known that C( ) is isometrically universal for all separable Banach spaces. In this note we show that the set of all separable Banach spaces that have the π-property is a Borel subset of SE. We show that in the set of spaces whose Szlenk index is bounded by some countable ordinal, the subset consisting of spaces which have a shrinking finite-dimensional decomposition is Borel. Lemma 2.1 Suppose (xn)∞ n=1 is a dense sequence in a Banach space X .
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