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Dominique Lecomte, Université Paris 6, Institut de Mathématiques de Jussieu, Projet Analyse Fonctionnelle, Couloir 16-26, 4ème étage, Case 247, 4, place Jussieu, 75 252 Paris Cedex 05, France

Publication: Memoirs of the American Mathematical Society
Publication Year 2013: Volume 221, Number 1038
ISBNs: 978-0-8218-7557-5 (print); 978-0-8218-9459-0 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00658-7
Published electronically: March 16, 2012
Keywords:Borel classes, potentially, products, reduction, Wadge classes
MSC (2010): Primary 03E15; Secondary 54H05, 28A05, 26A21

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Chapters

• Chapter 1. Introduction
• Chapter 2. A condition ensuring the existence of complicated sets
• Chapter 3. The proof of Theorem 1.10 for the Borel classes
• Chapter 4. The proof of Theorem 1.11 for the Borel classes
• Chapter 5. The proof of Theorem 1.10
• Chapter 6. The proof of Theorem 1.11
• Chapter 7. Injectivity complements

### Abstract

Let $\Gamma$ be a Borel class, or a Wadge class of Borel sets, and $2\leq d\leq \omega$ be a cardinal. A Borel subset $B$ of ${\mathbb R}^d$ is $potentially$ $in$ $\Gamma$ if there is a finer Polish topology on $\mathbb R$ such that $B$ is in $\Gamma$ when ${\mathbb R}^d$ is equipped with the new product topology. We give a way to recognize the sets potentially in $\Gamma$. We apply this to the classes of graphs (oriented or not), quasi-orders and partial orders.