The Baire category theorem (BCT) is an important result in

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Encyclopaedia of Mathematics article on Baire theorem

{{DEFAULTSORT:Baire Category Theorem Articles containing proofs Functional analysis General topology Theorems in topology

general topology
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and functional analysis
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. The theorem has two forms, each of which gives sufficient condition
In logic
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s for a topological space
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to be a Baire space
In mathematics, a Baire space is a topological space
In mathematics
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(a topological space such that the intersection
The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points.
In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...

of countably
In mathematics, a countable set is a Set (mathematics), set with the same cardinality (cardinal number, number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether f ...

many dense
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...

open set
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s is still dense).
The theorem was proved by French mathematician René-Louis Baire in his 1899 doctoral thesis.
Statement

ABaire space
In mathematics, a Baire space is a topological space
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is a topological space with the property that for each countable
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collection of open
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dense set
In topology
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s , their intersection $\backslash bigcap\_\; U\_n$ is dense.
* (BCT1) Every complete pseudometric space
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is a Baire space. Thus every completely metrizable topological space is a Baire space. More generally, every topological space that is homeomorphic
In the mathematical
Mathematics (from Greek
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to an open subset
Open or OPEN may refer to:
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Open is a band.
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of a complete pseudometric space
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is a Baire space.
* (BCT2) Every locally compact In mathematics
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Hausdorff space
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is a Baire space. The proof is similar to the preceding statement; the finite intersection propertyIn general topology, a branch of mathematics, a non-empty family ''A'' of subsets of a Set (mathematics), set ''X'' is said to have the finite intersection property (FIP) if the intersection (set theory), intersection over any finite subcollection o ...

takes the role played by completeness.
Neither of these statements directly implies the other, since there are complete metric spaces that are not locally compact (the irrational number
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s with the metric defined below; also, any Banach space
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of infinite dimension), and there are locally compact Hausdorff spaces that are not metrizable
In topology
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(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.
* (BCT3) A non-empty complete metric space with nonempty interior, or any of its subsets with nonempty interior, is not the countable union of nowhere-dense sets.
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 theaxiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

; and in fact BCT1 is equivalent over ZF to the axiom of dependent choice In mathematics
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, 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
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, the Baire space
In mathematics, a Baire space is a topological space
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ωCantor space In mathematics
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2Hilbert space
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such as .
Uses

BCT1 is used infunctional analysis
Functional analysis is a branch of mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...

to prove the open mapping theorem, the closed graph theorem
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and the uniform boundedness principle
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.
BCT1 also shows that every complete metric space with no isolated point
]
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s is uncountable
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. (If is a countable complete metric space with no isolated points, then each singleton in is nowhere dense, and so is of first categoryIn the Mathematics, mathematical fields of general topology and descriptive set theory, a meagre set (also called a meager set or a set of first category) is a Set (mathematics), set that, considered as a subset of a (usually larger) topological spac ...

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 of real numbers
* The irrational number
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s, 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)
* The Cantor set
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 (topology), 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
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cannot have countably infinite dimension.
Proof

The following is a standard proof that a complete pseudometric space $\backslash scriptstyle\; X$ is a Baire space. Let be a countable collection of open dense subsets. We want 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 is sufficient to show that any nonempty open set in has a point in common with all of the . Since is dense, intersects ; thus, there is a point and such that: : where and denote an open and closed ball, respectively, centered at with radius . Since each is dense, we can continue 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 .) Since when , we have that is Cauchy sequence, Cauchy, and hence converges to some limit by completeness. For any , by closedness, . Therefore, and for all . There is an alternative proof by M. Baker for the proof of the theorem using Choquet's game.See also

* * * *Notes

Citations

Works cited

* * * * * * * * Reprinted by Dover Publications, New York, 1995. (Dover edition).Further reading

*External links

Encyclopaedia of Mathematics article on Baire theorem

{{DEFAULTSORT:Baire Category Theorem Articles containing proofs Functional analysis General topology Theorems in topology