Baire category theorem

The Baire category theorem is an important tool 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.

Statement of the theorem

*(BCT1) Every complete metric space is a Baire space, i.e. for each countable collection of open and dense sets "U_n", their intersection ∩ "U_n" is dense. More generally, every topological space which is homeomorphic to an open subset of a complete pseudometric space is a Baire space. Thus every completely metrizable topological space is a Baire space.
*(BCT2) Every locally compact Hausdorff space is a Baire space. The proof is similar to the preceding statement; the finite intersection property takes the role played by completeness.

Note that neither of these statements implies the other, since there is a complete metric space which is not locally compact (the irrational numbers with the metric defined below), and there is a locally compact Hausdorff space which is not metrizable (uncountable Fort space). See Steen and Seebach in the references below.

*(BCT3) A non-empty complete metric space is NOT the countable union of nowhere-dense sets. Notice that this formulation is the contrapositive of BCT1. This version is sometimes more useful.

Relation to the axiom of choice

The proofs of BCT1 and BCT2 require some form of the axiom of choice; and in fact BCT1 is (over ZF) equivalent to a weaker version of the axiom of choice called the axiom of dependent choices. [http://www.math.vanderbilt.edu/~schectex/ccc/excerpts/equivdc.gif]

Uses of the theorem

BCT1 is used 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 R of real numbers
* The irrational numbers, with the metric defined by "d"("x", "y") = 1 / ("n" + 1), where "n" is the first index for which the continued fraction expansions of "x" and "y" differ (this is a complete metric space)
* The Cantor set

By BCT2, every 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.

References

*Schechter, Eric, "Handbook of Analysis and its Foundations", Academic Press, ISBN 0-12-622760-8
*Lynn Arthur Steen and J. Arthur Seebach, Jr., "Counterexamples in Topology", Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).