# 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 spaceXthat satisfies one (and therefore all) of the following equivalent conditions:

- Every intersection of countably many dense open sets is dense.
- The interior of every union of countably many nowhere dense sets is empty.
- Whenever the union of countably many closed subsets of
Xhas 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).

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

Definition:A subset of a topological spaceXismeagreinX(orof first categoryinX) if it is a union of countably many nowhere dense subsets ofX. A subset ofXwhich is not meagre is calledof second categoryinX.

(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:

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.