Baire category theorem

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 ∞ , their intersection ⋂ n = 1 ∞ U n is dense.

Note that 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.

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.

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 ω^ω, the Cantor space 2^ω, 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.

Proof

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

Let U n be a countable collection of open dense subsets. We want to show that the intersection ⋂ 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 < r 1 < 1 such that:

where B ( x , r ) and B ¯ ( x , r ) denote an open and closed ball, respectively, centered at x with radius r . Since each U n is dense, we can continue recursively to find a pair of sequences x n and 0 < r n < 1 n such that:

(This step relies on the axiom of choice.) Since x n ∈ B ( x m , r m ) when n > m , we have that x n is Cauchy, and hence x n converges to some limit x by completeness. For any n , by closedness,

Therefore x ∈ W and x ∈ U n for all n .

See also this blog post [1] by M. Baker for the proof of the theorem using Choquet's game.