In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.

Motivation[link]

In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries of dense open sets. These sets are, in a certain sense, "negligible". Some examples are finite sets, smooth curves in the plane, and proper affine subspaces in a Euclidean space. A topological space is a Baire space if it is "large", meaning that it is not a countable union of negligible subsets. For example, the three dimensional Euclidean space is not a countable union of its affine planes.

Definition[link]

The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. First, we give the usual modern definition, and then we give a historical definition which is closer to the definition originally given by Baire.

Modern definition[link]

A topological space is called a Baire space if the union of any countable collection of closed sets with empty interior has empty interior.

This definition is equivalent to each of the following conditions:

• Every intersection of countably many dense open sets is dense.
• The interior of every union of countably many closed nowhere dense sets is empty.
• 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.

Historical definition[link]

In his original definition, Baire defined a notion of category (unrelated to category theory) as follows.

A subset of a topological space X is called

• nowhere dense in X if the interior of its closure is empty
• of first category or meagre in X if it is a union of countably many nowhere dense subsets
• of second category or nonmeagre in X if it is not of first category in X

The definition for a Baire space can then be stated as follows: a topological space X is a Baire space if every non-empty open set is of second category in X. This definition is equivalent to the modern definition.

A subset A of X is comeagre if its complement Failed to parse (Missing texvc executable; please see math/README to configure.): X\setminus A

is meagre. A topological space X is a Baire space if and only if every comeager subset of X is dense.

Examples[link]

• The space R of real numbers with the usual topology, is a Baire space, and so is of second category in itself. The rational numbers are of first category and the irrational numbers are of second category in R.
• The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval [0, 1] with the usual topology.
• Here is an example of a set of second category in R with Lebesgue measure 0.

Failed to parse (Missing texvc executable; please see math/README to configure.): \bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty} \left(r_{n}-{1 \over 2^{n+m} }, r_{n}+{1 \over 2^{n+m}}\right)

where Failed to parse (Missing texvc executable; please see math/README to configure.): \left\{r_{n}\right\}_{n=1}^{\infty}

is a sequence that enumerates the rational numbers.

• 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 singletons.

Baire category theorem[link]

The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. It is an important tool in topology and functional analysis.

• (BCT1) Every complete metric space is a Baire space. More generally, every topological space which is homeomorphic to an open subset of a complete pseudometric space is a Baire space. In particular, every topologically complete space is a Baire space.
• (BCT2) Every locally compact Hausdorff space is a Baire space.

BCT1 shows that each of the following is a Baire space:

• The space R of real numbers
• The space of irrational numbers
• The Cantor set
• Indeed, every Polish space

BCT2 shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable. For example, the long line is of second category.

Properties[link]

• 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].

• Every open subspace of a Baire space is a Baire space.

• Given a family of continuous functions f_n:X→Y with pointwise limit f:X→Y. If X is a Baire space then the points where f is not continuous is a meagre set in X and the set of points where f is continuous is dense in X. A special case of this is the uniform boundedness principle.

See also[link]

• Banach–Mazur game
• Descriptive set theory
• Baire space (set theory)

References[link]

• Munkres, James, Topology, 2nd edition, Prentice Hall, 2000.
• Baire, René-Louis (1899), Sur les fonctions de variables réelles, Annali di Mat. Ser. 3 3, 1--123.