Category (topology) encyclopedia topics

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, 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

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 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

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 $Xsetminus A$ is meagre.

Examples

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} left(r_{n}-{1 over 2^{n+m} }, r_{n}+{1 over 2^{n+m}}right)

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) Every non-empty 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 non-empty 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

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 meagre 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.

References

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.