Series: Penn State Logic Seminar

 Date: Tuesday, January 31, 2006

 Time: 2:30 - 3:45 PM

 Place: 106 McAllister Building

 Speaker: John Clemens, Penn State, Mathematics

 Title: Weakly Pointed Trees and Partial Injections, part 1


   We consider some topological and recursion-theoretic questions
   motivated by the following result of Graf and Mauldin: If X and Y
   are Polish spaces and B is a Borel subset of X x Y such that for
   a.e. x the section B_x is uncountable and for a.e. y the section
   B^y is uncountable, then there is a Borel subset A of X of full
   measure and a Borel-measurable injection f: A -> Y such that the
   graph of f is contained in B.  We first consider
   recursion-theoretic results. We introduce a coding of uniformly
   branching trees and call such a tree T weakly pointed if some
   branch of the tree can compute T. We show that the set of weakly
   pointed trees is meager. More precisely, we show that no 2-generic
   tree can be weakly pointed, but give an example of a 1-generic tree
   which is weakly pointed.  We then consider a topological version of
   the Graf-Mauldin result, and show that it fails. That is, there is
   a Borel subset B of X x Y with X and Y Polish spaces such that B_x
   is uncountable for a comeager set of x and B^y is uncountable for a
   comeager set of y, but there is no comeager subset A of X and
   Baire-measurable injection f: A -> Y whose graph is contained in B.