Tree (descriptive set theory)
||It has been suggested that this article be merged with Tree (set theory). (Discuss) Proposed since October 2016.|
The collection of all finite sequences of elements of a set is denoted . With this notation, a tree is a nonempty subset of , such that if is a sequence of length in , and if , then the shortened sequence also belongs to . In particular, choosing shows that the empty sequence belongs to every tree.
Branches and bodies
A branch through a tree is an infinite sequence of elements of , each of whose finite prefixes belongs to . The set of all branches through is denoted and called the body of the tree .
A tree that has no branches is called wellfounded; a tree with at least one branch is illfounded. By König's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded.
A finite sequence that belongs to a tree is called a terminal node if it is not a prefix of a longer sequence in . Equivalently, is terminal if there is no element of such that that . 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 is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for each sequence in , 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 and are ordered by if and only if is a proper prefix of . 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).
The set of infinite sequences over (denoted as ) may be given the product topology, treating X as a discrete space. In this topology, every closed subset of is of the form for some pruned tree . Namely, let consist of the set of finite prefixes of the infinite sequences in . Conversely, the body of every tree forms a closed set in this topology.
Frequently trees on Cartesian products are considered. In this case, by convention, we consider only the subset of the product space, , containing only sequences whose even elements come from and odd elements come from (e.g., ). Elements in this subspace are identified in the natural way with a subset of the product of two spaces of sequences, (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 with for over the product space. We may then form the projection of ,