Baire space
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- 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.
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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 countable 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 in X if it is a union of countably many nowhere dense subsets
- of second category 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.
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.
- <math>\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty} (r_{n}-{1 \over 2^{n+m} }, r_{n}+{1 \over 2^{n+m}})<math>
- where <math> \left\{r_{n}\right\}_{n=1}^{\infty} <math> 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
- Every manifold
- Every topological space homeomorphic to 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].
- Every open subspace of a Baire space is a Baire space.
- Given a family of continuous functions fn:X→Y with 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.
See also
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.es:espacio de Baire