ABaire space is a topological space in which every countable intersection of opendense sets is dense in See the corresponding article for a list of equivalent characterizations, as some are more useful than others depending on the application.
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 spaceofinfinite 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.
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.[10]
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.[11]
This restricted form applies in particular to the real line, the Baire space the Cantor space and a separable Hilbert space such as the -space.
BCT1 also shows that every nonempty complete metric space with no isolated pointisuncountable. (If is a nonempty countable metric space with no isolated point, then each singletoninisnowhere dense, and ismeagre 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 irrational numbers, with the metric defined by where is the first index for which the continued fraction expansions of and differ (this is a complete metric space)
ByBCT2, 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.
BCT1 is used to prove that a Banach space cannot have countably infinite dimension.
(BCT1) The following is a standard proof that a complete pseudometric space is a Baire space.[6]
Let be a countable collection of open dense subsets. It remains to show that the intersection 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 of has some point in common with all of the .
Because is dense, intersects consequently, there exists a point and a number such that:
where and denote an open and closed ball, respectively, centered at with radius
Since each is dense, this construction can be continued recursively to find a pair of sequences and 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 .)
The sequence isCauchy because whenever and hence converges to some limit by completeness.
If is a positive integer then (because this set is closed).
Thus and for all