Weakly pointed trees and partial injections

We define the notion of a weakly pointed tree, and characterize the amount of genericity necessary to prevent a uniformly branching tree being weakly pointed. We use these ideas to show there is no topological analogue of a measure-theoretic selection theorem of Graf and Mauldin. We consider some topological and recursion-theoretic questions motivated by the following measure-theoretic result of Graf and Mauldin (Theorem 4.4 of [1]): Theorem. Let X and Y be analytic spaces, λ a probability measure on X, μ a probability measure on Y , and R ⊆ X × Y a Borel set such that Rx is uncountable for λ-a.e. x ∈ X and R is uncountable for μ-a.e. y ∈ Y . Then there exists a Borel set A ⊆ X with λ(A) = 1, a Borel set B ⊆ Y with μ(B) = 1, and a Borel isomorphism f from A onto B whose graph is contained in R. This says that a sufficiently thick plane set admits a selector defined almost everywhere which is injective. We can consider the following topological analogue of this result: Question. Suppose X and Y are Polish spaces, and R ⊆ X×Y is a Borel set such that Rx is uncountable for a comeager set of x ∈ X and R is uncountable for a comeager set of y ∈ Y . Mush there exist a comeager Borel set A ⊆ X, a comeager Borel set B ⊆ Y , and a Borel isomorphism f from A to B whose graph is contained in R? This turns out to be false, as we will show in Section 2. A counterexample is the set {(x, y) ∈ 3×2 : ∀n (x(n) 6= 2⇒ y(n) = x(n))}. In the next section we introduce the notion of a weakly pointed tree, and draw a connection between this set and such trees. We then show that a generic tree is not weakly pointed, and characterize the precise amount of genericity necessary to prevent weak pointedness. A suitable generalization will then provide the counterexample to the above question. I would like to thank the referee for several helpful suggestions for clarifying the proofs of Theorems 7 and 9. c © 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$00.00