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Tree (descriptive set theory)

## Summary

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.

## Definitions

### Trees

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 , 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.

### Branches and bodies

A branch through a tree ${\displaystyle T}$  is an infinite sequence of elements of ${\displaystyle X}$ , each of whose finite prefixes belongs to ${\displaystyle T}$ . The set of all branches through ${\displaystyle T}$  is denoted ${\displaystyle [T]}$  and called the body of the tree ${\displaystyle T}$ .

A tree that has no branches is called wellfounded; a tree with at least one branch is illfounded. By Kőnig's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded.

### Terminal nodes

A finite sequence that belongs to a tree ${\displaystyle T}$  is called a terminal node if it is not a prefix of a longer sequence in ${\displaystyle T}$ . Equivalently, ${\displaystyle \langle x_{0},x_{1},\ldots ,x_{n-1}\rangle \in T}$  is terminal if there is no element ${\displaystyle x}$  of ${\displaystyle X}$  such that that ${\displaystyle \langle x_{0},x_{1},\ldots ,x_{n-1},x\rangle \in T}$ . A tree that does not have any terminal nodes is called pruned.

## Relation to other types of trees

In graph theory, a rooted tree is a directed graph in which every vertex except for a special root vertex has exactly one outgoing edge, and in which the path formed by following these edges from any vertex eventually leads to the root vertex. If ${\displaystyle T}$  is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for each sequence in ${\displaystyle T}$ , and an outgoing edge from each nonempty sequence that connects it to the shorter sequence formed by removing its last element. This graph is a tree in the graph-theoretic sense. The root of the tree is the empty sequence.

In order theory, a different notion of a tree is used: an order-theoretic tree is a partially ordered set with one minimal element in which each element has a well-ordered set of predecessors. Every tree in descriptive set theory is also an order-theoretic tree, using a partial ordering in which two sequences ${\displaystyle T}$  and ${\displaystyle U}$  are ordered by ${\displaystyle T  if and only if ${\displaystyle T}$  is a proper prefix of ${\displaystyle U}$ . The empty sequence is the unique minimal element, and each element has a finite and well-ordered set of predecessors (the set of all of its prefixes). An order-theoretic tree may be represented by an isomorphic tree of sequences if and only if each of its elements has finite height (that is, a finite set of predecessors).

## Topology

The set of infinite sequences over ${\displaystyle X}$  (denoted as ${\displaystyle X^{\omega }}$ ) may be given the product topology, treating X as a discrete space. In this topology, every closed subset ${\displaystyle C}$  of ${\displaystyle X^{\omega }}$  is of the form ${\displaystyle [T]}$  for some pruned tree ${\displaystyle T}$ . Namely, let ${\displaystyle T}$  consist of the set of finite prefixes of the infinite sequences in ${\displaystyle C}$ . Conversely, the body ${\displaystyle [T]}$  of every tree ${\displaystyle T}$  forms a closed set in this topology.

Frequently trees on Cartesian products ${\displaystyle X\times Y}$  are considered. In this case, by convention, we consider only the subset ${\displaystyle T}$  of the product space, ${\displaystyle (X\times Y)^{<\omega }}$ , containing only sequences whose even elements come from ${\displaystyle X}$  and odd elements come from ${\displaystyle Y}$  (e.g., ${\displaystyle \langle x_{0},y_{1},x_{2},y_{3}\ldots ,x_{2m},y_{2m+1}\rangle }$ ). Elements in this subspace are identified in the natural way with a subset of the product of two spaces of sequences, ${\displaystyle X^{<\omega }\times Y^{<\omega }}$  (the subset for which the length of the first sequence is equal to or 1 more than the length of the second sequence). In this way we may identify ${\displaystyle [X^{<\omega }]\times [Y^{<\omega }]}$  with ${\displaystyle [T]}$  for over the product space. We may then form the projection of ${\displaystyle [T]}$ ,

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