In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is said to be a Baire space if countable unions of
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
s with empty
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
also have empty interior.
According to the
Baire category theorem
The Baire category theorem (BCT) is an important result 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 (a topological space such that the ...
,
compact Hausdorff space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
s and
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
s are examples of Baire spaces.
The Baire category theorem combined with the properties of Baire spaces has numerous applications in
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
,
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
, in particular
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.
Bourbaki introduced the term "Baire space" in honor of
René Baire
René (''born again'' or ''reborn'' in French) is a common first name in French-speaking, Spanish-speaking, and German-speaking countries. It derives from the Latin name Renatus.
René is the masculine form of the name (Renée being the feminine ...
, who investigated the Baire category theorem in the context of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
in his 1899 thesis.
Definition
The definition that follows is based on the notions of
meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and
nonmeagre (or second category) set (namely, a set that is not meagre). See the corresponding article for details.
A topological space
is called a Baire space if it satisfies any of the following equivalent conditions:
# Every countable intersection of
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s is dense.
# Every countable union of closed sets with empty interior has empty interior.
# Every meagre set has empty interior.
# Every nonempty open set is nonmeagre.
[As explained in the ]meagre set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
article, for an open set, being nonmeagre in the whole space is equivalent to being nonmeagre in itself.
# Every
comeagre
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called ...
set is dense.
# Whenever a countable union of closed sets has an interior point, at least one of the closed sets has an interior point.
The equivalence between these definitions is based on the associated properties of complementary subsets of
(that is, of a set
and of its
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...
) as given in the table below.
Baire category theorem
The Baire category theorem gives sufficient conditions for a topological space to be a Baire space.
* (BCT1) Every
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
pseudometric space
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric ...
is a Baire space. In particular, every
completely metrizable In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' ind ...
topological space is a Baire space.
* (BCT2) Every
locally compact regular In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
space is a Baire space. In particular, every
locally compact Hausdorff In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
space is a Baire space.
BCT1 shows that the following are Baire spaces:
* The space
of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s.
* The space of
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
s, which is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to the
Baire space of set theory.
* Every
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named bec ...
.
BCT2 shows that the following are Baire spaces:
* Every compact Hausdorff space; for example, the
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Thr ...
(or
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
).
* Every
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, even if it is not
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
(hence not
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
), like the
long line Long line or longline may refer to:
*''Long Line'', an album by Peter Wolf
*Long line (topology), or Alexandroff line, a topological space
*Long line (telecommunications), a transmission line in a long-distance communications network
*Longline fish ...
.
One should note however that there are plenty of spaces that are Baire spaces without satisfying the conditions of the Baire category theorem, as shown in the Examples section below.
Properties
* Every nonempty Baire space is nonmeagre. In terms of countable intersections of dense open sets, being a Baire space is equivalent to such intersections being dense, while being a nonmeagre space is equivalent to the weaker condition that such intersections are nonempty.
* Every open subspace of a Baire space is a Baire space.
* Every dense
''G''δ set in a Baire space is a Baire space. The result need not hold if the G
δ set is not dense. See the Examples section.
* Every comeagre set in a Baire space is a Baire space.
* A subset of a Baire space is comeagre if and only if it contains a dense G
δ set.
* A closed subspace of a Baire space need not be Baire. See the Examples section.
* If a space contains a dense subspace that is Baire, it is also a Baire space.
* A space that is locally Baire, in the sense that each point has a neighborhood that is a Baire space, is a Baire space.
* Every
topological sum
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a Indexed family, family of topological spaces is a space formed by equipping the disjoi ...
of Baire spaces is Baire.
* The product of two Baire spaces is not necessarily Baire.
* An arbitrary product of complete metric spaces is Baire.
* Every
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
sober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible closed subset of ''X'' is the closure of exactly one point of ''X'': that is, every irreducible closed subset has a unique generic point.
Definition ...
is a Baire space.
* Every finite topological space is a Baire space (because a finite space has only finitely many open sets and the intersection of two open dense sets is an open dense set).
* A
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is a Baire space if and only if it is nonmeagre, which happens if and only if every closed balanced absorbing subset has non-empty interior.
* Given a sequence of
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
functions
with pointwise limit
If
is a Baire space then the points where
is not continuous is in
and the set of points where
is continuous is dense in
A special case of this is the
uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis.
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
.
Examples
* The empty space is a Baire space. It is the only space that is both Baire and meagre.
* The space
of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s with the usual topology is a Baire space.
* The space
of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s (with the topology induced from
) is not a Baire space, since it is meagre.
* The space of
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
s (with the topology induced from
) is a Baire space, since it is comeagre in
* The space
(with the topology induced from
) is nonmeagre, but not Baire. There are several ways to see it is not Baire: for example because the subset