In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology, called the product topology.
This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals".
The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of natural numbers, and is given the product topology (where each copy of the set of natural numbers is given the discrete topology).
The Baire space is often represented using the tree of finite sequences of natural numbers.
The product topology used to define the Baire space can be described in one of two equivalent ways: in terms of a basis consisting of cylinder sets, or of a basis of trees.
That is, given any finite set of natural number coordinates
, one considers the finite intersection of cylinders This intersection is called a cylinder set, and the set of all such cylinder sets provides a basis for the product topology.
An alternative basis for the product topology can be given in terms of trees.
The basic open sets can be characterized as: Thus a basic open set in the Baire space is the set of all infinite sequences of natural numbers extending a common finite initial segment σ.
This leads to a representation of the Baire space as the set of all infinite paths passing through the full tree ω<ω of finite sequences of natural numbers ordered by extension.
Each open set is determined by a countable union S of nodes of that tree.
A point in Baire space is in an open set if and only if its path goes through one of the nodes in its determining union.
Conversely, each open set corresponds to a subtree S of the full tree ω<ω, consisting of at most a countable number of nodes.
The representation of the Baire space as paths through a tree also gives a characterization of closed sets as complements of subtrees defining the open sets.
This defines a subtree T of the full tree ω<ω, in which the nodes of S defining the open set are missing.
The subtree T consists of all nodes in ω<ω that are not in S. This subtree T defines a closed subset C of Baire space such that any point x is in C if and only if x is a path through T. Conversely, for any closed subset C of Baire space there is a subtree T which consists of all of ω<ω with at most a countable number of nodes removed.
Since the full tree ω<ω is itself countable, this implies the closed sets correspond to any subtree of the full tree, including finite subtrees.
This implies that the Baire space is zero-dimensional with respect to the small inductive dimension (as are all spaces whose base consists of clopen sets.)
The above definition of the Baire space generalizes to one where the elements
With this metric, the basic open sets of the tree basis are balls of radius
[2][3] The Baire space has the following properties: The Baire space is homeomorphic to the set of irrational numbers when they are given the subspace topology inherited from the real line.
A homeomorphism between Baire space and the irrationals can be constructed using continued fractions.
, we can assign a corresponding irrational number greater than 1 Using
From the point of view of descriptive set theory, Baire spaces are more flexible than the real line in the following sense.
This difference makes the real line "slightly awkward to use", despite the focus of descriptive set theory on sets of reals.
Instead, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Baire space and are preserved by continuous functions.
[4] ωω is also of independent, but minor, interest in real analysis, where it is considered as a uniform space.
The uniform structures of ωω and Ir (the irrationals) are different, however: ωω is complete in its usual metric while Ir is not (although these spaces are homeomorphic).
This space of maps inherits a topology from the product topology on Baire space; for example, one may consider functions having uniform convergence.
The shift map, acting on this space of functions, is then the GKW operator.