Electronic ISBN:  9780821894590 
Product Code:  MEMO/221/1038.E 
83 pp 
List Price:  $62.00 
MAA Member Price:  $55.80 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 221; 2013MSC: Primary 03; Secondary 54; 28; 26;
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. The author provides a way to recognize the sets potentially in \(\bf\Gamma\) and applies this to the classes of graphs (oriented or not), quasiorders and partial orders.

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


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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. The author provides a way to recognize the sets potentially in \(\bf\Gamma\) and applies this to the classes of graphs (oriented or not), quasiorders and partial orders.

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