Baire category theorem


The Baire category theorem 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.
The theorem was proved by French mathematician René-Louis Baire in his 1899 doctoral thesis.

Statement

A Baire space is a topological space with the property that 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, and there are locally compact Hausdorff spaces that are not metrizable.
See Steen and Seebach in the references below.
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 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 the axiom of dependent choice, a weak form of the axiom of choice.
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.
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 Lp space|.

Uses

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

Proof

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.

Citations

Works cited