Tree (descriptive set theory)

In descriptive set theory, a tree on a set $X$ is a subset $T$ of $X^{<\omega }$ (that is, a set of finite sequences of elements of $X$) that is closed under subsequence (that is, if $<x_{0},x_{1},\ldots ,x_{n-1}>\in T$ and $m<n$, then $<x_{0},x_{1},\ldots ,x_{m-1}>\in T$). A branch through $T$ is an infinite sequence ${\vec {x}}\in X^{\omega }$ of elements of $X$ such that, for every natural number $n$, ${\vec {x}}|n\in T$, where ${\vec {x}}|n$ denotes the sequence of the first $n$ elements of ${\vec {x}}$. The set of all branches through $T$ is denoted $[T]$. A tree that has no branches is called wellfounded; a tree with at least one branch is illfounded.

A node (that is, element) of $T$ is terminal if there is no node of $T$ properly extending it; that is, $<x_{0},x_{1},\ldots ,x_{n-1}>\in T$ is terminal if there is no element $x$ of $X$ such that that $<x_{0},x_{1},\ldots ,x_{n-1},x>\in T$. A tree with no terminal nodes is called pruned.

Frequently trees on cartesian products $X\times Y$ are considered. In this case, by convention, the set $(X\times Y)^{\omega }$ is identified in the natural way with $X^{\omega }\times Y^{\omega }$, and $[T]$ is considered as a subset of $X^{\omega }\times Y^{\omega }$. We may then form the projection of $[T]$,

$p[T]=\{{\vec {x}}\in X^{\omega }|(\exists {\vec {y}}\in Y^{\omega })<{\vec {x}},{\vec {y}}>\in [T]\}$