Baire space

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

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 honour of René-Louis Baire who introduced the concept.

Motivation

In a topological space we can think of closed sets with empty interior as points in the space. Ignoring spaces with isolated points, which are their own interior, a Baire space is "large" in the sense that it cannot be constructed as a countable union of its points. A concrete example is a 2-dimensional plane with a countable collection of lines. No matter what lines we choose we cannot cover the space completely with the lines.

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 topological space is called a Baire space if the countable union of any 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 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

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 (meager) in X if it is a union of countably many nowhere dense subsets
• of second category or nonmeagre (nonmeager) 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 (comeager) if its complement $X\setminus A$ is meagre.

Examples

• In 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.
$\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty} (r_{n}-{1 \over 2^{n+m} }, r_{n}+{1 \over 2^{n+m}})$
where $\left\{r_{n}\right\}_{n=1}^{\infty}$ is a sequence that counts 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

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 shows that each of the following is a Baire space:

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].
• Given a family of continuous functions fn:XY with limit f:XY. If X is a Baire space then the points where f is not continuous is meagre in X and the set of points where f is continuous is dense in 'X.