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The Baire category theorem (BCT) is an important result in
general topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and
functional 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), ...
. The theorem has two forms, each of which gives
sufficient condition In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
s for a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
to be a
Baire space In mathematics, a Baire space is a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and ...
(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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s is still dense). The theorem was proved by French mathematician René-Louis Baire in his 1899 doctoral thesis.

# Statement

A
Baire space In mathematics, a Baire space is a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and ...
is a topological space with the property that for each
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
collection of
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
dense set In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), a ...
s , their intersection $\bigcap_ U_n$ is dense. * (BCT1) Every complete
pseudometric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
to an
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of a complete
pseudometric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
is a Baire space. * (BCT2) Every
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s with the metric defined below; also, any
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of infinite dimension), and there are locally compact Hausdorff spaces that are not
metrizable In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
(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 the
axiom 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, the
Baire space In mathematics, a Baire space is a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and ...
ωω, the
Cantor space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
2ω, and a separable
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
such as .

# Uses

BCT1 is used in
functional 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and the
uniform boundedness principle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. BCT1 also shows that every complete metric space with no
isolated point ] In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s is
uncountable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. (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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
cannot have countably infinite dimension.

# Proof

The following is a standard proof that a complete pseudometric space $\scriptstyle X$ 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.

* * * *

# Works cited

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