Tree (descriptive set theory)

In descriptive set theory, a tree on a set ${\displaystyle X}$ is a subset ${\displaystyle T}$ of ${\displaystyle X^{<\omega }}$ (that is, a set of finite sequences of elements of ${\displaystyle X}$) that is closed under subsequence (that is, if ${\displaystyle <x_{0},x_{1},\ldots ,x_{n-1}>\in T}$ and ${\displaystyle m<n}$, then ${\displaystyle <x_{0},x_{1},\ldots ,x_{m-1}>\in T}$). A branch through ${\displaystyle T}$ is an infinite sequence ${\displaystyle {\vec {x}}\in X^{\omega }}$ of elements of ${\displaystyle X}$ such that, for every natural number ${\displaystyle n}$, ${\displaystyle {\vec {x}}|n\in T}$, where ${\displaystyle {\vec {x}}|n}$ denotes the sequence of the first ${\displaystyle n}$ elements of ${\displaystyle {\vec {x}}}$.