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In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called “reals.” It is often denoted B, NN, or ωω. Moschovakis denotes it $$\mathcal{N}.$$

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 Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits.

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 {ci : i < n } is selected, and for each ci a particular natural number value vi is selected, then the set of all infinite sequences of natural numbers that have value vi at position ci 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 {wi : i < n} is selected, then the set of all infinite sequences of natural numbers that have value wi 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 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 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 is a closed set.

Properties

The Baire space has the following properties:

1. It is a perfect Polish space, which means it is a completely metrizable second countable space with no isolated points. As such, it has the same cardinality as the real line and is a Baire space in the topological sense of the term.
2. It is zero dimensional and totally disconnected.
3. It is not locally compact.
4. It is universal for Polish spaces in the sense that it can be mapped continuously onto any non-empty Polish space.
5. The Baire space is homeomorphic to the product of any finite or countable number of copies of itself.

Relation to the real line

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.

From the point of view of descriptive set theory, the fact that the real line is connected causes technical difficulties. For this reason, it is more common to study Baire space. Because every Polish space is the continuous image of Baire space, it often possible to prove results about arbitrary Polish spaces by showing these properties hold for Baire space and showing they are preserved by continuous functions.

B is also of independent, but minor, interest in real analysis, where it is considered as a uniform space. The uniform structures of B and Ir (the irrationals) are different however: B is complete in its usual metric while Ir is not (although these spaces are homeomorphic).
References

Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9.
Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.