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). Versions of Baire category theorem were first proved independently in 1897 and 1899 by Osgood and Baire respectively. This theorem says that every complete metric space is a Baire space. Statement. A Baire space is a topological space with the property that for each countable collection  of open dense sets formula_1 their intersection formula_2 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. 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 formula_3 the Cantor space formula_4 and a separable Hilbert space such as the formula_5space formula_6 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. (If formula_7 is a countable complete metric space with no isolated points, then each singleton formula_8 in formula_7 is nowhere dense, and so formula_7 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 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. Proof. The following is a standard proof that a complete pseudometric space formula_7 is a Baire  space. Let formula_17 be a countable collection of open dense subsets. It remains to show that the intersection formula_18 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 suffices to show that any nonempty open subset formula_19 of formula_7 has some point formula_14 in common with all of the formula_22. Because formula_23 is dense, formula_19 intersects formula_25 consequently, there exists a point formula_26 and a number formula_27 such that: formula_28 where formula_29 and formula_30 denote an open and closed ball, respectively, centered at formula_14  with radius formula_32 Since each formula_22 is dense, this construction can be continued recursively to find a pair of sequences formula_34 and formula_35 such that: formula_36 The sequence formula_38 is Cauchy because formula_39 whenever formula_40 and hence formula_38 converges to some limit formula_14 by completeness. If formula_13 is a positive integer then formula_44 (because this set is closed). Thus formula_45 and formula_46 for all formula_47 formula_48 There is an alternative proof by M. Baker for the proof of the theorem using Choquet's game.