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Baire category theorem

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Baire category theorem
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The Baire category theorem (BCT) is an important result 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 (a topological space such that the intersection of countably many dense open sets is still dense).The theorem was proved by French mathematician René-Louis Baire in his 1899 doctoral thesis.

Statement of the theorem

A Baire space is a topological space with the following property: for each countable collection of open dense sets {U_n}_{n=1}^infty, their intersection textstylebigcap_{n=1}^infty U_n is dense. Neither of these statements implies the other, since there are complete metric spaces which are not locally compact (the irrational numbers with the metric defined below; also, any Banach space of infinite dimension), and there are locally compact Hausdorff spaces which are not metrizable (for instance, any uncountable product of non-trivial compact Hausdorff spaces is such; also, several function spaces used in Functional Analysis; the uncountable Fort space). See Steen and Seebach in the references below.
  • (BCT3) A non-empty complete metric space, or any of its subsets with nonempty interior, is not the countable union of nowhere-dense sets.
This formulation is equivalent to 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.

Relation to the axiom of choice

The proofs of BCT1 and BCT2 for arbitrary complete metric spaces require some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.Blair 1977A restricted form of the Baire category theorem, in which the complete metric space is also assumed to be separable, is provable in ZF with no additional choice principles.Levy 1979, p. 212 This restricted form applies in particular to the real line, the Baire space ωω, the Cantor space 2ω, and a separable Hilbert space such as L2(R n).

Uses of the theorem

BCT1 is used in functional analysis 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: 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.BCT is used to prove Hartog's theorem, a fundamental result in the theory of several complex variables.

Proof

The following is a standard proof that a complete pseudometric space scriptstyle X is a Baire space.Let U_n 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 in X has a point x in common with all of the U_n. Since U_1 is dense, W intersects U_1; thus, there is a point x_1 and 0 ;

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