Baire Space (set Theory) - Topology and Trees

Topology and Trees

The product topology used to define the Baire space can be described more concretely in terms of trees. The definition of the product topology leads to this characterization of basic open sets:

If any finite set of natural number coordinates {c_i : i < n } is selected, and for each c_i a particular natural number value v_i is selected, then the set of all infinite sequences of natural numbers that have value v_i at position c_i for all i < n is a basic open set. Every open set is a union of a collection of these.

By moving to a different basis for the same topology, an alternate characterization of open sets can be obtained:

If a sequence of natural numbers {w_i : i < n} is selected, then the set of all infinite sequences of natural numbers that have value w_i at position i for all i < n is a basic open set. Every open set is a union of a collection of these.

Thus a basic open set in the Baire space specifies a finite initial segment τ of an infinite sequence of natural numbers, and all the infinite sequences extending τ form a basic open set. This leads to a representation of the Baire space as the set of all paths through the full tree ω<ω of finite sequences of natural numbers ordered by extension. An open set is determined by some (possibly infinite) union of nodes of the tree; a point in Baire space is in the open set if and only if its path goes through one of these nodes.

The representation of the Baire space as paths through a tree also gives a characterization of closed sets. For any closed subset C of Baire space there is a subtree T of ω<ω such that any point x is in C if and only if x is a path through T. Conversely, the set of paths through any subtree of ω<ω is a closed set.