# Tree (descriptive set theory)

12/10/2011 15:25

In descriptive set theory, a tree on a set X is a set of finite sequences of elements of X that is closed under initial segments.

More formally, it is a subset T of X < ω, such that if

$\langle x_0,x_1,\ldots,x_{n-1}\rangle \in T$

and $0\le m,

then

$\langle x_0,x_1,\ldots,x_{m-1}\rangle \in T$.

In particular, every nonempty tree contains the empty sequence.

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] and called the body of the tree 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, $\langle x_0,x_1,\ldots,x_{n-1}\rangle \in T$ is terminal if there is no element x of X such that that $\langle x_0,x_1,\ldots,x_{n-1},x\rangle \in T$. A tree with no terminal nodes is called pruned.

If we equip Xω with the product topology (treating X as a discrete space), then every closed subset of Xω is of the form [T] for some pruned tree T (namely, $T:= \{ \vec x|n: n \in \omega, x\in X\}$). Conversely, every set [T] is closed.

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 a subset of $X^{\omega}\times Y^{\omega}$, and [T] is considered as a subset of $X^{\omega}\times Y^{\omega}$. We may then form the projection of [T],

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

Every tree in the sense described here is also a tree in the wider sense, i.e., the pair (T, <), where < is defined by

x<yx is a proper initial segment of y,

is a partial order in which each initial segment is well-ordered. The height of each sequence x is then its length, and hence finite.

Conversely, every partial order (T, <) where each initial segment { y: y < x0 } is well-ordered is isomorphic to a tree described here, assuming that all elements have finite height.