Baire category theorem

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

The theorem was proved by René-Louis Baire in his 1899 doctoral thesis.[1]

Contents

[edit] Statement of the theorem

A Baire space is a topological space with the following property: for each countable collection of open dense sets Un, their intersection ∩ Un is dense.

Note that neither of these statements implies the other, since there is a complete metric space which is not locally compact (the irrational numbers with the metric defined below), and there is a locally compact Hausdorff space which is not metrizable (uncountable Fort space). See Steen and Seebach in the references below.

  • (BCT3) A non-empty complete metric space is NOT the countable union of nowhere-dense sets (i.e, sets whose closure has dense complement).

This formulation is a consequence of BCT1 and is sometimes more useful in applications. Also: if a non-empty complete metric space is the countable union of closed sets, then one of these closed sets has non empty interior.

[edit] Relation to the axiom of choice

The proofs of BCT1 and BCT2 require some form of the axiom of choice; and in fact BCT1 is (over ZF) equivalent to a weaker version of the axiom of choice called the axiom of dependent choices.[1]

[edit] Uses of the theorem

BCT1 is used to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle.

BCT1 also shows that every complete metric space with no isolated points is uncountable. (If X is a countable complete metric space with no isolated points, then each singleton {x} in X is nowhere dense, and so X is of first category in itself.) In particular, this proves that the set of all real numbers is uncountable.

BCT1 shows that each of the following is a Baire space:

  • The space R of real numbers
  • The irrational numbers, with the metric defined by d(x, y) = 1 / (n + 1), where n is the first index for which the continued fraction expansions of x and y differ (this is a complete metric space)
  • The Cantor set

By BCT2, every finite-dimensional Hausdorff manifold is a Baire space, since it is locally compact and Hausdorff. This is so even for non-paracompact (hence nonmetrizable) manifolds such as the long line.

[edit] Proof

The following is a standard proof that a complete metric space X is a Baire space.

Let Un be a countable collection of open dense subsets. We want to show that the intersection \bigcap U_n is dense. A subset is dense if and only if every nonempty open subset intersects it. Thus, to show that the intersection is dense, it is sufficient to show that any nonempty open set W has a point x in common with all of the Un. Since U1 is dense, W intersects U1; thus, there is a point x1 and r1 > 0 such that:

\overline{B}(x_1, r_1) \subset W \cap U_1.

(B(x,r) and \overline{B}(x, r) denote an open ball centered at x with radius r and its closure, respectively.) Since Un are dense, in a recursive manner, we find a pair of sequences xn and rn > 0 such that:

\overline{B}(x_n, r_n) \subset B(x_{n-1}, r_{n-1}) \cap U_n as well as rn < 1 / n.

Since x_n \in B(x_m, r_m) when n > m, we have that xn is Cauchy, and xn converges to some limit x by completeness. For any n, by closedness,

x \in \overline{B}(x_{n+1}, r_{n+1}) \subset B(x_n, r_n).

Hence, x \in W and x \in U_n for all n. \square

[edit] Notes

  1. ^ R. Baire. Sur les fonctions de variables réelles. Ann. di Mat., 3:1–123, 1899.

[edit] References

Retrieved from "http://en.wikipedia.org/wiki/Baire_category_theorem"