Baire space

In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are "enough" points for certain limit processes.

Definition: A Baire space is a topological space X that satisfies one (and therefore all) of the following equivalent conditions:

1. Every intersection of countably many dense open sets is dense.
2. The interior of every union of countably many nowhere dense sets is empty.
3. Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets must have an interior point.

A common proof technique in analysis is the following: one first shows that the given space X is Baire (typically using general theorems mentioned below), and then one applies condition 3 in order to show that certain interior points must exist.

Examples of Baire spaces:

• All complete metric spaces are Baire spaces (this is the Baire category theorem).
• Every space that is homeomorphic to an open subset of a complete pseudometric space is a Baire space (this includes the irrational numbers with their standard topology, as well as the Cantor set).
• Every locally compact Hausdorff space is a Baire space (this includes all manifolds).

Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space, since it is the union of countably many closed sets without interior, the singletonss.

Two closely related definitions often appear, especially in older literature:

Definition: A subset of a topological space X is meagre in X (or of first category in X) if it is a union of countably many nowhere dense subsets of X. A subset of X which is not meagre is called of second category in X.

(Note that this notion of "category" has nothing to do with category theory.)

In this language, a topological space X is a Baire space if and only if every non-empty open set is of second category in X. In particular, every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval [0,1].

See also:

• Banach-Mazur game

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In set theory and related branches of mathematics, Baire space is the set of all infinite sequences of natural numbers. Baire space is often denoted B, N^N, or ω^ω.

B has the same cardinality as the set R of real numbers, and can be used as a convenient substitute for R in some set-theoretical contexts.

B is also of independent, but minor, interest in real analysis, where it is considered as a uniform space: the product of countably many copies of the discrete space N. This is a Baire space in the above topological sense. As a topological space, B is homeomorphic to the set Ir of irrational numbers carrying their standard topology inherited from the reals. The homeomorphism between B and Ir can be constructed using continued fractions. The uniform structures of B and Ir are different however: B is complete and Ir is not.

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