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Concept
Baire space (set theory)
Summary
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 NN, ωω, by the symbol or also ωω, 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.
The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits.
Countable set
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:
If any finite set of natural number coordinates I={i} is selected, and for each i a particular natural number value vi is selected, then the set of all infinite sequences of natural numbers that have value vi at position i is a basic open set. Every open set is a countable union of a collection of these.
Using more formal notation, one can define the individual cylinders as
for a fixed integer location n and integer value v. The cylinders are then the generators for the cylinder sets: the cylinder sets then consist of all intersections of a finite number of cylinders. That is, given any finite set of natural number coordinates and corresponding natural number values for each , one considers the intersection of cylinders
This intersection is called a cylinder set, and the set of all such cylinder sets provides a basis for the product topology. Every open set is a countable union of such cylinder sets.
Official source
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