Baire space

The Baire space is the set of all infinite sequences of natural numbers, denoted ω^ω or ℕ^ℕ, equipped with the product topology derived from the discrete topology on ℕ. It is one of the most important spaces in descriptive set theory, recursion theory, and the study of infinite games. Every point in Baire space is an infinite sequence ⟨n₀, n₁, n₂, …⟩, and the basic open sets are determined by finite initial segments.

Despite its name, Baire space is homeomorphic to the irrational numbers as a subspace of the real line, and it shares many properties with the real numbers while being technically more convenient for certain constructions. The space is a Polish space — separable and completely metrizable — which makes it the natural setting for descriptive set theory.

The significance of Baire space in the study of the Axiom of Determinacy is that the infinite games whose determinacy AD asserts are games on Baire space: the players build a point in ω^ω, and the payoff set is a subset of this space. The topological and combinatorial properties of Baire space are therefore inseparable from the logical properties of determinacy.