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).
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.
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.
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 L2(ℝn).
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.
BCT3 is used to prove that a Banach space cannot have countably infinite dimension.
Let Un be a countable collection of open dense subsets. We want to show that the intersection ∩Un 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 Un. Since U1 is dense, W intersects U1; thus, there is a point x1 and 0 < r1 < 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 Un is dense, we can continue recursively to find a pair of sequences xn and 0 < rn < 1/n 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 xn.) Since xn ∈ B(xm, rm) when n > m, we have that xn is Cauchy, and hence xn converges to some limit x by completeness. For any n, by closedness, x ∈ B(xn, rn).