![Compact](https://upload.wikimedia.org/wikipedia/commons/7/7c/Compact.svg)
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 ...
, specifically
general topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, compactness is a property that seeks to generalize the notion of a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
subset 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 ...
by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any
''limiting values'' of points. For example, the open
interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval
,1would be compact. Similarly, 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
is not compact, because it has infinitely many "punctures" corresponding to the
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, and 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
is not compact either, because it excludes the two limiting values
and
. However, the
''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, but may not be
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
in other
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 ...
s.
One such generalization is that a topological space is
''sequentially'' compact if every
infinite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
of points sampled from the space has an infinite
subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
that converges to some point of the space.
The
Bolzano–Weierstrass theorem
In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n. The theorem states that each ...
states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded.
Thus, if one chooses an infinite number of points in the closed
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
, some of those points will get arbitrarily close to some real number in that space.
For instance, some of the numbers in the sequence accumulate to 0 (while others accumulate to 1).
The same set of points would not accumulate to any point of the open unit interval , so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded.
For example, considering
(the real number line), the sequence of points has no subsequence that converges to any real number.
Compactness was formally introduced by
Maurice Fréchet Maurice may refer to:
People
*Saint Maurice (died 287), Roman legionary and Christian martyr
*Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor
*Maurice (bishop of London) (died 1107), Lord Chancellor and Lo ...
in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to
spaces of functions. The
Arzelà–Ascoli theorem
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded inter ...
and the
Peano existence theorem
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees t ...
exemplify applications of this notion of compactness to classical analysis. Following its initial introduction, various equivalent notions of compactness, including
sequential compactness and
limit point compact In mathematics, a topological space ''X'' is said to be limit point compact or weakly countably compact if every infinite subset of ''X'' has a limit point in ''X''. This property generalizes a property of compact spaces. In a metric space, limit ...
ness, were developed in general
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s.
In general topological spaces, however, these notions of compactness are not necessarily equivalent. The most useful notion — and the standard definition of the unqualified term ''compactness'' — is phrased in terms of the existence of finite families of
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 that "
cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of co ...
" the space in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by
Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
and
Pavel Urysohn
Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are f ...
in 1929, exhibits compact spaces as generalizations of
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. Th ...
s. In spaces that are compact in this sense, it is often possible to patch together information that holds
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points).
P ...
— that is, in a neighborhood of each point — into corresponding statements that hold throughout the space, and many theorems are of this character.
The term compact set is sometimes used as a synonym for compact space, but also often refers to a
compact subspace of 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 ...
.
Historical development
In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand,
Bernard Bolzano
Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liber ...
(
1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
.
Bolzano's proof relied on the
method of bisection
In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and t ...
: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected.
The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts — until it closes down on the desired limit point.
The full significance of
Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
.
In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for
spaces of functions rather than just numbers or geometrical points.
The idea of regarding functions as themselves points of a generalized space dates back to the investigations of
Giulio Ascoli
Giulio Ascoli (20 January 1843, Trieste – 12 July 1896, Milan) was a Jewish-Italian mathematician. He was a student of the Scuola Normale di Pisa, where he graduated in 1868.
In 1872 he became Professor of Algebra and Calculus of the Politecn ...
and
Cesare Arzelà
Cesare Arzelà (6 March 1847 – 15 March 1912) was an Italian mathematician who taught at the University of Bologna and is recognized for his contributions in the theory of functions, particularly for his characterization of sequences of continu ...
.
The culmination of their investigations, the
Arzelà–Ascoli theorem
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded inter ...
, was a generalization of the Bolzano–Weierstrass theorem to families of
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s, the precise conclusion of which was that it was possible to extract a
uniformly convergent
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
sequence of functions from a suitable family of functions.
The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point".
Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of
integral equation
In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...
s, as investigated by
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
and
Erhard Schmidt
Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (german: link=no, Dorpat), in the Govern ...
.
For a certain class of
Green's functions
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differential ...
coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of
mean convergence
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
— or convergence in what would later be dubbed a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
.
This ultimately led to the notion of a
compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...
as an offshoot of the general notion of a compact space.
It was
Maurice Fréchet Maurice may refer to:
People
*Saint Maurice (died 287), Roman legionary and Christian martyr
*Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor
*Maurice (bishop of London) (died 1107), Lord Chancellor and Lo ...
who, in
1906
Events
January–February
* January 12 – Persian Constitutional Revolution: A nationalistic coalition of merchants, religious leaders and intellectuals in Persia forces the shah Mozaffar ad-Din Shah Qajar to grant a constitution, ...
, had distilled the essence of the Bolzano–Weierstrass property and coined the term ''compactness'' to refer to this general phenomenon (he used the term already in his 1904 paper which led to the famous 1906 thesis).
However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the
continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
, which was seen as fundamental for the rigorous formulation of analysis.
In 1870,
Eduard Heine
Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician.
Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legen ...
showed that a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
defined on a closed and bounded interval was in fact
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it.
The significance of this lemma was recognized by
Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Math ...
(
1895
Events
January–March
* January 5 – Dreyfus affair: French officer Alfred Dreyfus is stripped of his army rank, and sentenced to life imprisonment on Devil's Island.
* January 12 – The National Trust for Places of Histor ...
), and it was generalized to arbitrary collections of intervals by
Pierre Cousin (1895) and
Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
(
1904
Events
January
* January 7 – The distress signal ''CQD'' is established, only to be replaced 2 years later by ''SOS''.
* January 8 – The Blackstone Library is dedicated, marking the beginning of the Chicago Public Library system.
* ...
). The
Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.
This property was significant because it allowed for the passage from
local information about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function).
This sentiment was expressed by , who also exploited it in the development of the
integral now bearing his name.
Ultimately, the Russian school of
point-set topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, under the direction of
Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
and
Pavel Urysohn
Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are f ...
, formulated Heine–Borel compactness in a way that could be applied to the modern notion of 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 ...
. showed that the earlier version of compactness due to Fréchet, now called (relative)
sequential compactness, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers.
It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.
Basic examples
Any
finite space is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed)
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
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. If one chooses an infinite number of distinct points in the unit interval, then there must be some
accumulation point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
in that interval.
For instance, the odd-numbered terms of the sequence get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1.
The given example sequence shows the importance of including the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
points of the interval, since the
limit points
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
must be in the space itself — an open (or half-open) interval of the real numbers is not compact.
It is also crucial that the interval be
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
, since in the interval , one could choose the sequence of points , of which no sub-sequence ultimately gets arbitrarily close to any given real number.
In two dimensions, closed
disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary.
However, an open disk is not compact, because a sequence of points can tend to the boundary — without getting arbitrarily close to any point in the interior.
Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point ''within'' the space.
Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.
Definitions
Various definitions of compactness may apply, depending on the level of generality.
A subset 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 particular is called compact if it is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
.
This implies, by the
Bolzano–Weierstrass theorem
In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n. The theorem states that each ...
, that any infinite
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
from the set has a
subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
that converges to a point in the set.
Various equivalent notions of compactness, such as
sequential compactness and
limit point compact In mathematics, a topological space ''X'' is said to be limit point compact or weakly countably compact if every infinite subset of ''X'' has a limit point in ''X''. This property generalizes a property of compact spaces. In a metric space, limit ...
ness, can be developed in general
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s.
In contrast, the different notions of compactness are not equivalent in general
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 ...
s, and the most useful notion of compactness — originally called ''bicompactness'' — is defined using
cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of co ...
s consisting of
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 (see ''Open cover definition'' below).
That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the
Heine–Borel theorem.
Compactness, when defined in this manner, often allows one to take information that is known
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points).
P ...
— in a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of each point of the space — and to extend it to information that holds globally throughout the space.
An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
; here, continuity is a local property of the function, and uniform continuity the corresponding global property.
Open cover definition
Formally, 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 called ''compact'' if each of its
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
s has a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
subcover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
. That is, is compact if for every collection of open subsets of such that
:
,
there is a finite subcollection ⊆ such that
:
Some branches of mathematics such as
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, typically influenced by the French school of
Bourbaki, use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both
Hausdorff and ''quasi-compact''.
A compact set is sometimes referred to as a ''compactum'', plural ''compacta''.
Compactness of subsets
A subset of a topological space is said to be compact if it is compact as a subspace (in the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
).
That is, is compact if for every arbitrary collection of open subsets of such that
:
there is a finite subcollection ⊆ such that
:
Compactness is a "topological" property. That is, if
, with subset equipped with the subspace topology, then is compact in if and only if is compact in .
Characterization
If is a topological space then the following are equivalent:
# is compact; i.e., every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
of has a finite
subcover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
.
# has a sub-base such that every cover of the space, by members of the sub-base, has a finite subcover (
Alexander's sub-base theorem).
# is
Lindelöf and
countably compact In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
Equivalent definitions
A topological space ''X'' is called countably compact if it satisfies any of the following equivalent conditi ...
.
# Any collection of closed subsets of with the
finite intersection property In general topology, a branch of mathematics, a non-empty family ''A'' of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite inters ...
has nonempty intersection.
# Every
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
on has a convergent subnet (see the article on
nets for a proof).
# Every
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
on has a convergent refinement.
# Every net on has a cluster point.
# Every filter on has a cluster point.
# Every
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
on converges to at least one point.
# Every infinite subset of has a
complete accumulation point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
.
# For every topological space , the projection
is a
closed mapping
In mathematics, more specifically in topology, an open map is a function (mathematics), function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the Image (mathem ...
(see
proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
There are several competing definitio ...
).
Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above).
Euclidean space
For any
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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 ...
, is compact if and only if it is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
; this is the
Heine–Borel theorem.
As a
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 ...
is a metric space, the conditions in the next subsection also apply to all of its subsets.
Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed
interval or closed -ball.
Metric spaces
For any metric space , the following are equivalent (assuming
countable choice):
# is compact.
# is
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 ...
and
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
(this is also equivalent to compactness for
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s).
# is sequentially compact; that is, every
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
in has a convergent subsequence whose limit is in (this is also equivalent to compactness for
first-countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s).
# is
limit point compact In mathematics, a topological space ''X'' is said to be limit point compact or weakly countably compact if every infinite subset of ''X'' has a limit point in ''X''. This property generalizes a property of compact spaces. In a metric space, limit ...
(also called weakly countably compact); that is, every infinite subset of has at least one
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
in .
# is
countably compact In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
Equivalent definitions
A topological space ''X'' is called countably compact if it satisfies any of the following equivalent conditi ...
; that is, every countable open cover of has a finite subcover.
# is an image of a continuous function from 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 ...
.
# Every decreasing nested sequence of nonempty closed subsets in has a nonempty intersection.
# Every increasing nested sequence of proper open subsets in fails to cover .
A compact metric space also satisfies the following properties:
#
Lebesgue's number lemma
In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states:
:If the metric space (X, d) is compact and an open cover of X is given, then there exists a number \delta > 0 such ...
: For every open cover of , there exists a number such that every subset of of diameter < is contained in some member of the cover.
# is
second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
,
separable and
Lindelöf – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.
# is closed and bounded (as a subset of any metric space whose restricted metric is ). The converse may fail for a non-Euclidean space; e.g. the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
equipped with the
discrete metric
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
is closed and bounded but not compact, as the collection of all
singletons of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.
Ordered Spaces
For an ordered space (i.e. a totally ordered set equipped with the order topology), the following are equivalent:
# is compact.
# Every subset of has a supremum (i.e. a least upper bound) in .
# Every subset of has an infimum (i.e. a greatest lower bound) in .
# Every nonempty closed subset of has a maximum and a minimum element.
An ordered space satisfying (any one of) these conditions is called a complete lattice.
In addition, the following are equivalent for all ordered spaces , and (assuming
countable choice) are true whenever is compact. (The converse in general fails if is not also metrizable.):
# Every sequence in has a subsequence that converges in .
# Every monotone increasing sequence in converges to a unique limit in .
# Every monotone decreasing sequence in converges to a unique limit in .
# Every decreasing nested sequence of nonempty closed subsets ⊇ ⊇ ... in has a nonempty intersection.
# Every increasing nested sequence of proper open subsets ⊆ ⊆ ... in fails to cover .
Characterization by continuous functions
Let be a topological space and the ring of real continuous functions on .
For each , the evaluation map
given by is a ring homomorphism.
The
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
of is a
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
, since the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
is the field of real numbers, by the
first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist f ...
.
A topological space is
pseudocompact
In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of ps ...
if and only if every maximal ideal in has residue field the real numbers.
For
completely regular space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
s, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism. There are pseudocompact spaces that are not compact, though.
In general, for non-pseudocompact spaces there are always maximal ideals in such that the residue field is a (
non-Archimedean)
hyperreal field
In mathematics, the system of hyperreal numbers is a way of treating Infinity, infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an Field extension, extension of the real numbe ...
.
The framework of
non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
allows for the following alternative characterization of compactness: a topological space is compact if and only if every point of the natural extension is
infinitely close to a point of (more precisely, is contained in the
monad
Monad may refer to:
Philosophy
* Monad (philosophy), a term meaning "unit"
**Monism, the concept of "one essence" in the metaphysical and theological theory
** Monad (Gnosticism), the most primal aspect of God in Gnosticism
* ''Great Monad'', an ...
of ).
Hyperreal definition
A space is compact if its hyperreal number, hyperreal extension (constructed, for example, by the ultrapower construction) has the property that every point of is infinitely close to some point of .
For example, an open real interval is not compact because its hyperreal extension contains infinitesimals, which are infinitely close to 0, which is not a point of .
Sufficient conditions
* A closed subset of a compact space is compact.
* A finite Union (set theory), union of compact sets is compact.
* A continuous function (topology), continuous image of a compact space is compact.
* The intersection of any non-empty collection of compact subsets of a Hausdorff space is compact (and closed);
** If is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).
* The product topology, product of any collection of compact spaces is compact. (This is Tychonoff's theorem, which is equivalent to the axiom of choice.)
* In a metrizable space, a subset is compact if and only if it is sequentially compact (assuming axiom of countable choice, countable choice)
* A finite set endowed with any topology is compact.
Properties of compact spaces
* A compact subset of a Hausdorff space is closed.
** If is not Hausdorff then a compact subset of may fail to be a closed subset of (see footnote for example).
** If is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).
* In any topological vector space (TVS), a compact subset is complete space, complete. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are ''not'' closed.
* If and are disjoint compact subsets of a Hausdorff space , then there exist disjoint open set and in such that and .
* A continuous bijection from a compact space into a Hausdorff space is a homeomorphism.
* A compact Hausdorff space is Normal space, normal and Regular space, regular.
* If a space is compact and Hausdorff, then no finer topology on is compact and no coarser topology on is Hausdorff.
* If a subset of a metric space is compact then it is -bounded.
Functions and compact spaces
Since a continuous function (topology), continuous image of a compact space is compact, the extreme value theorem holds for such spaces: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.
(Slightly more generally, this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of a compact space under a
proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
There are several competing definitio ...
is compact.
Compactifications
Every topological space is an open dense topological subspace, dense subspace of a compact space having at most one point more than , by the Compactification (mathematics), Alexandroff one-point compactification.
By the same construction, every locally compact Hausdorff space is an open dense subspace of a compact Hausdorff space having at most one point more than .
Ordered compact spaces
A nonempty compact subset of the
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 has a greatest element and a least element.
Let be a total order, simply ordered set endowed with the order topology.
Then is compact if and only if is a complete lattice (i.e. all subsets have suprema and infima).
Examples
* Any finite topological space, including the empty set, is compact. More generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.
* Any space carrying the cofinite topology is compact.
* Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification. The one-point compactification of
is homeomorphic to the circle ; the one-point compactification of
is homeomorphic to the sphere . Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
* The right order topology or left order topology on any bounded totally ordered set is compact. In particular, Sierpiński space is compact.
* No discrete space with an infinite number of points is compact. The collection of all singleton (mathematics), singletons of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.
* In
carrying the lower limit topology, no uncountable set is compact.
* In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not locally compact but is still
Lindelöf.
* The closed
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
is compact. This follows from the
Heine–Borel theorem. The open interval is not compact: the
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
for does not have a finite subcover. Similarly, the set of ''rational numbers'' in the closed interval is not compact: the sets of rational numbers in the intervals
cover all the rationals in [0, 1] for but this cover does not have a finite subcover. Here, the sets are open in the subspace topology even though they are not open as subsets of
.
* The set
of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals , where takes all integer values in , cover
but there is no finite subcover.
* On the other hand, the extended real number line carrying the analogous topology ''is'' compact; note that the cover described above would never reach the points at infinity and thus would ''not'' cover the extended real line. In fact, the set has the homeomorphism to [−1, 1] of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homeomorphisms preserve covers, the Heine-Borel property can be inferred.
* For every natural number , the n-sphere, -sphere is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.
* On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. (Alaoglu's theorem)
* 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 ...
is compact. In fact, every compact metric space is a continuous image of the Cantor set.
* Consider the set of all functions from the real number line to the closed unit interval, and define a topology on so that a sequence
in converges towards if and only if
converges towards for all real numbers . There is only one such topology; it is called the topology of pointwise convergence or the product topology. Then is a compact topological space; this follows from the Tychonoff theorem.
* Consider the set of all functions satisfying the Lipschitz condition for all . Consider on the metric induced by the uniform convergence, uniform distance
Then by
Arzelà–Ascoli theorem
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded inter ...
the space is compact.
* The spectrum of an operator, spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers
. Conversely, any compact subset of
arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space sequence spaces#ℓp spaces,
may have any compact nonempty subset of
as spectrum.
Algebraic examples
* Compact groups such as an orthogonal group are compact, while groups such as a general linear group are not.
* Since the p-adic numbers, -adic integers are homeomorphic to the Cantor set, they form a compact set.
* The spectrum of a ring, spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact, but never
Hausdorff (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact scheme (mathematics), schemes, "quasi" referring to the non-Hausdorff nature of the topology.
* The spectrum of a boolean algebra, spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem. Stone spaces, compact totally disconnected space, totally disconnected Hausdorff spaces, form the abstract framework in which these spectra are studied. Such spaces are also useful in the study of profinite groups.
* The structure space of a commutative unital Banach algebra is a compact Hausdorff space.
* The Hilbert cube is compact, again a consequence of Tychonoff's theorem.
* A profinite group (e.g. Galois group) is compact.
See also
* Compactly generated space
* Compactness theorem
* Eberlein compactum
* Exhaustion by compact sets
* Lindelöf space
* Metacompact space
* Noetherian topological space
* Orthocompact space
* Paracompact space
* Totally bounded space, Precompact set - also called ''
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
''
* Relatively compact subspace
* Totally bounded
Notes
References
Bibliography
*
*.
*.
* (''Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation'').
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* .
*
*
External links
*
*
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{{DEFAULTSORT:Compact Space
Compactness (mathematics)
General topology
Properties of topological spaces
Topology