The isomorphism class of $ c_{0} $ is not Borel

TL;DR

This paper proves that the collection of all Banach spaces isomorphic to c0 forms a complex, non-Borel set within the space of separable Banach spaces, using advanced descriptive set theory and a recent construction.

Contribution

It demonstrates that the class of Banach spaces isomorphic to c0 is a complete analytic set, revealing its high complexity in the descriptive set-theoretic hierarchy.

Findings

The class of c0-isomorphic Banach spaces is a complete analytic set.

The proof utilizes a recent Bourgain-Delbaen construction.

The result shows the class is not Borel within the Effros Borel structure.

Abstract

We show that the class of all Banach spaces which are isomorphic to $ c_{0} $ is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. The proof employs a recent Bourgain-Delbaen construction by Argyros, Gasparis and Motakis.