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Baire space

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In topology and related branches of 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 this concept.

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.

The property of being a Baire space is a topological property, i.e., it is preserved by homeomorphisms.

Definition

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.

Alternative characterizations are

• Every intersection of countable dense open sets is dense.
• The interior of every union of countably many nowhere dense sets is empty set.
• 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

The definition for a Baire space can then be stated as

A topological space X is called a Baire space if every non-empty open set is of second category in X.

Examples

[itex]\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty} (r_{n}-{1 \over 2^{n+m} }, r_{n}+{1 \over 2^{n+m}})[itex]
where [itex] \left\{r_{n}\right\}_{n=1}^{\infty} [itex] 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.

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.

Baire category theorem

The Baire category theorem is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in topology and functional analysis.

1. Every complete metric space is a Baire space.
2. Every locally compact Hausdorff space is a Baire space.

The proof of the Baire category theorem uses the axiom of choice; and in fact it is logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice.

The Baire category theorem is used in the proof of the open mapping theorem and the uniform boundedness principle. It also gives a quick proof that the reals are uncountable (since the reals are a complete metric space, and hence cannot be the countable union of points).

In set theory and related branches of mathematics, Baire space is the set of all infinite sequences of natural numbers. Baire space is often denoted B, NN, or ωω.

B has the same cardinality as the set R of real numbers, and can be used as a convenient substitute for R in some set-theoretical contexts.

B is also of independent, but minor, interest in real analysis, where it is considered as a uniform space: the product of countably many copies of the discrete space N. This is a Baire space in the above topological sense. As a topological space, B is homeomorphic to the set Ir of irrational numbers carrying their standard topology inherited from the reals. The homeomorphism between B and Ir can be constructed using continued fractions. The uniform structures of B and Ir are different however: B is complete and Ir is not.

Baire space should be contrasted with Cantor space, the set of infinite sequences of binary digits.

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.