Baire space (set theory)

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". It is denoted by ${\displaystyle \mathbb {N} ^{\mathbb {N} }}$, or ^ωω, or by the symbol ${\displaystyle {\mathcal {N}}}$ or sometimes by ω^ω (not to be confused with the countable ordinal obtained by ordinal exponentiation).

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.

(This space should also not be confused with the concept of a Baire space, which is a certain kind of topological space.)

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