# A selection theorem for treeable sets

#### by Professor Greg Hjorth

Institution: Department of Mathematics and Statistics, The University of Melbourne
Date: Wed 23rd May 2007
Time: 4:30 PM
Location: Room 107, Richard Berry Building

Abstract: Let $A\subset {\mathbb R} \times {\mathbb R}$ be a Borel set
in the plane. Suppose $R\subset A\times A$ is an acyclic, symmetric
Borel relation whose connected components are the fibres $A_x={y: (x, y)\in A}$ -- in other words, suppose we can assign in a uniform Borel
manner the structure of a tree to each fibre.

Then $p[A]=\{x:\exists y (x, y)\in A\}$ is Borel and we can find a
Borel function uniformizing the set $A$.