Baire space

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For the concept in set theory, 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 honor of René-Louis Baire who introduced the concept.

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[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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 shows that each of the following is a Baire 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.

[edit] 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: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.

[edit] See also

[edit] 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.
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