Baire category theorem

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 theoremEdit

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

Neither of these statements directly implies the other, since there are complete metric spaces that 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 that 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 choiceEdit

The proof of BCT1 for arbitrary complete metric spaces requires 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.[1]

A 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.[2] 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(Rn).

Uses of the theoremEdit

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:

  • The space   of real numbers
  • The irrational numbers, with the metric defined by  , where   is the first index for which the continued fraction expansions of   and   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.

BCT is used to prove Hartogs's theorem, a fundamental result in the theory of several complex variables.

ProofEdit

The following is a standard proof that a complete pseudometric space   is a Baire space.

Let   be a countable collection of open dense subsets. We want to show that the intersection   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   in   has a point   in common with all of the  . Since   is dense,   intersects  ; thus, there is a point   and   such that:

 

where   and   denote an open and closed ball, respectively, centered at   with radius  . Since each   is dense, we can continue recursively to find a pair of sequences   and   such that:

 

(This step relies on the axiom of choice and the fact that a finite intersection of open sets is open and hence an open ball can be found inside it centered at  ) Since   when  , we have that   is Cauchy, and hence   converges to some limit   by completeness. For any  , by closedness,

 

Therefore,   and   for all  .

There is an alternative proof by M. Baker for the proof of the theorem using Choquet's game.[3]

See alsoEdit

NotesEdit

  1. ^ Blair 1977
  2. ^ Levy 1979, p. 212
  3. ^ "Real Numbers and Infinite Games, Part II". July 7, 2014.

ReferencesEdit

External linksEdit