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), quasi-orders and partial orders.

Table of Contents

• Introduction
• A condition ensuring the existence of complicated sets
• The proof of Theorem 1.10 for the Borel classes
• The proof of Theorem 1.11 for the Borel classes
• The proof of Theorem 1.10
• The proof of Theorem 1.11
• Injectivity complements
• Bibliography