Tree (descriptive set theory)

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

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 $0\le m<n$ ,

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 $X ω$ with the product topology (treating X as a discrete space), then every closed subset of $X ω$ 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<y ⇔ x 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.