# 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 $A$.

*For More Information:* Greg Hjorth: G.Hjorth@ms.unimelb.edu.au