Tree (descriptive set theory)

In descriptive set theory, a tree on a set ${\displaystyle X}$ is a collection of finite sequences of elements of ${\displaystyle X}$ such that every prefix of a sequence in the collection also belongs to the collection.

The collection of all finite sequences of elements of a set ${\displaystyle X}$ is denoted ${\displaystyle X^{<\omega }}$. With this notation, a tree is a nonempty subset ${\displaystyle T}$ of ${\displaystyle X^{<\omega }}$, such that if ${\displaystyle \langle x_{0},x_{1},\ldots ,x_{n-1}\rangle }$ is a sequence of length ${\displaystyle n}$ in ${\displaystyle T}$, and if ${\displaystyle 0\leq m<n}$, then the shortened sequence ${\displaystyle \langle x_{0},x_{1},\ldots ,x_{m-1}\rangle }$ also belongs to ${\displaystyle T}$. In particular, choosing ${\displaystyle m=0}$ shows that the empty sequence belongs to every tree.

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