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Potential Wadge classes
About this Title
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: https://doi.org/10.1090/S0065-9266-2012-00658-7
Published electronically: March 16, 2012
Keywords: Borel classes,
potentially,
products,
reduction,
Wadge classes
MSC: Primary 03E15; Secondary 54H05, 28A05, 26A21
Table of Contents
Chapters
- 1. Introduction
- 2. A condition ensuring the existence of complicated sets
- 3. The proof of Theorem 1.10 for the Borel classes
- 4. The proof of Theorem 1.11 for the Borel classes
- 5. The proof of Theorem 1.10
- 6. The proof of Theorem 1.11
- 7. Injectivity complements
Abstract
Let $\bf \Gamma$ 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 $potentially$ $in$ $\bf \Gamma$ if there is a finer Polish topology on $\mathbb R$ such that $B$ is in $\bf \Gamma$ when ${\mathbb R}^d$ is equipped with the new product topology. We give a way to recognize the sets potentially in $\bf \Gamma$. We apply this to the classes of graphs (oriented or not), quasi-orders and partial orders.- Béla Bollobás, Modern graph theory, Graduate Texts in Mathematics, vol. 184, Springer-Verlag, New York, 1998. MR 1633290
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