Baire category theorem
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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.
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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. In particular, every topologically complete 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 Baire space of irrational numbers), and there is a locally compact Hausdorff space which is not metrizable (uncountable Fort space). See Steen and Seebach in the references below.
Relation to AC
The proofs of BCT1 and BCT2 require some form of the axiom of choice; and in fact the statement that every complete pseudometric space is a Baire space is logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice. [1] (http://www.math.vanderbilt.edu/~schectex/ccc/excerpts/equivdc.gif)
![[1]](http://www.math.vanderbilt.edu/~schectex/ccc/excerpts/equivdc.gif){kind=link}
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 space of irrational numbers
- The Cantor set
- Every manifold
- Every topological space homeomorphic to a Baire space
References
- Schechter, Eric, Handbook of Analysis and its Foundations, Academic Press, ISBN 0-126-22760-8
- 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).