Baire space (set theory)

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 denoted B, N^N, ω^ω, ^ωω, or ${\displaystyle {\mathcal {N}}}$.

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.

The product topology used to define the Baire space can be described more concretely in terms of trees. The basic open sets of the product topology are cylinder sets, here characterized as:

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