Baire category theorem

From Wikipedia, the free encyclopedia

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.

Contents 1 Statement of the theorem 2 Relation to AC 3 Uses of the theorem 4 References

[edit] Statement of the theorem

• (BCT1) Every non-empty complete metric space is a Baire space. 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 non-empty locally compact Hausdorff space is a Baire space.

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.

[edit] Relation to AC

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. [1]

[edit] 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.

Various applications of BCT1 and its relations with similar phenomena are listed in the Bwatabaire (mostly in French but English is allowed)

[edit] 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).