Baire space

For the concept in set theory, see Baire space (set theory).

In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.

Motivation

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 in ℝ, smooth curves in the plane, and proper affine subspaces in a Euclidean space. If a topological space is a Baire space then 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

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

A Baire space is a topological space in which the union of every 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

Main article: Meagre set

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

Examples

The space $\mathbb {R}$ 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 $\mathbb {R}$ .
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 $\mathbb {R}$ with Lebesgue measure 0.

\bigcap _{m=1}^{\infty }\bigcup _{n=1}^{\infty }\left(r_{n}-({\tfrac {1}{2}})^{n+m},r_{n}+({\tfrac {1}{2}})^{n+m}\right)

where

\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

Main article: Baire category theorem

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 completely metrizable space is a Baire space.
(BCT2) Every locally compact Hausdorff space (or more generally every locally compact sober space) is a Baire space.

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

The space $\mathbb {R}$ of real numbers
The space of irrational numbers, which is homeomorphic to the Baire space ω^ω of set theory
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

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.
A closed subset of a Baire space is not necessarily Baire.
The product of two Baire spaces is not necessarily Baire. However, there exist sufficient conditions that will guarantee that a product of arbitrarily many Baire spaces is again Baire.

References

Sources

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.

External links

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