Definitions

# Baire space (set theory)

In mathematics field of set theory, especially descriptive 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.”

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). It is often denoted B, NN, or ωω. Moschovakis denotes it $mathcal\left\{N\right\}$.

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

## 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).