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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 their changes (cal ...
, specifically
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 ...
, compactness is a property that generalizes the notion of a subset of
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
being closed (containing all its
limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
s) and bounded (having all its points lie within some fixed distance of each other). Examples of compact spaces include a closed real interval, a union of a finite number of closed intervals, a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
, or a finite set of points. This notion is defined for more general
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 ...
s in various ways, which are usually equivalent in Euclidean space but may be inequivalent in other spaces. One such generalization is that a topological space is ''sequentially'' compact if every
infinite sequence 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 gene ...
of points sampled from the space has an infinite
subsequence 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 ...
that converges to some point of the space. The
Bolzano–Weierstrass 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 h ...
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 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). I ...
, 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 \mathbb^1, the entire real number line, the sequence of points , has no subsequence that converges to any real number. Compactness was formally introduced by
Maurice FréchetMaurice may refer to: People *Saint Maurice Saint Maurice (also Moritz, Morris, or Mauritius; ) was the leader of the legendary Roman Theban Legion in the 3rd century, and one of the favorite and most widely venerated saints of that group. He w ...
in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions.
Arzelà–Ascoli theorem The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, in ...
and the
Peano existence 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 ...
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 compactIn mathematics, a topological space 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 (mat ...
ness, were developed in general
metric 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 t ...
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 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 that "
cover Cover or covers may refer to: Packaging, science and technology * A covering, usually - but not necessarily - one that completely closes the object ** Cover (philately), generic term for envelope or package ** Housing (engineering), an exterior ...
" 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 The Soviet Union,. officially the Union of Soviet ...
and
Pavel Urysohn Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a federal socialist state in Northern Eurasia Eurasia () is the large ...
in 1929, exhibits compact spaces as generalizations of
finite 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 t ...
s. In spaces that are compact in this sense, it is often possible to patch together information that holds
locallyIn 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'' neighbourhood (mathematics), ne ...
—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 often refers to a compact subspace of a topological space as well.


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 Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian A Bohemian () is a resident of Bohemia Bohemia ( ; cs, Čechy ; ; hsb, Čěska; szl, Czechy) is the westernmost a ...

Bernard Bolzano
(
1817 Events January–March * January 1 – Sailing through the Sandwich Islands, Otto von Kotzebue Otto von Kotzebue (russian: О́тто Евста́фьевич Коцебу́, Romanization of Russian, tr. ;  – ) was a Russian of ...
) 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 (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
. Bolzano's proof relied on the method of bisection: 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 mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

Karl Weierstrass
. 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 Trieste ( , ; sl, Trst ; german: Triest, ) is a city and a seaport File:PorticcioloCedas.jpg, The Porticciolo del Cedas port in Barcola near Trieste, a small local port A port is a maritime ...
and
Cesare Arzelà
Cesare Arzelà
. 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 In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, in ...
, was a generalization of the Bolzano–Weierstrass theorem to families of
continuous function 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 gen ...
s, the precise conclusion of which was that it was possible to extract a
uniformly convergentIn the mathematics, mathematical field of mathematical analysis, analysis, uniform convergence is a Modes of convergence, mode of Limit of a sequence, convergence of functions stronger than pointwise convergence. A sequence of Function (mathematics), ...
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 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 ...
s, as investigated by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
and
Erhard Schmidt Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German The Baltic Germans (german: Deutsch-Balten or , later ; and остзейцы ''ostzeitsy'' 'Balters' in Russian) are ethnic German inhabitants of the eastern shore ...

Erhard Schmidt
. For a certain class of
Green's functions In mathematics, a Green's function is the impulse response of an inhomogeneous ordinary differential equation, inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that ...
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 Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressi ...
—or convergence in what would later be dubbed a
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 ...
. This ultimately led to the notion of a
compact operator In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional a ...
as an offshoot of the general notion of a compact space. It was
Maurice FréchetMaurice may refer to: People *Saint Maurice Saint Maurice (also Moritz, Morris, or Mauritius; ) was the leader of the legendary Roman Theban Legion in the 3rd century, and one of the favorite and most widely venerated saints of that group. He w ...
who, in 1906, 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) Continuum theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, categorical theories or models explain variatio ...
, which was seen as fundamental for the rigorous formulation of analysis. In 1870,
Eduard Heine Heinrich Eduard Heine (16 March 1821 – October 1881) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of German ...
showed that a
continuous function 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 gen ...
defined on a closed and bounded interval was in fact
uniformly continuous 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 h ...
. 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 people, French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biograph ...
(
1895 Events January–March * January 5 Events Pre-1600 *1477 – Battle of Nancy: Charles the Bold is defeated and killed in a conflict with René II, Duke of Lorraine; Duchy of Burgundy, Burgundy subsequently becomes part of ...
), 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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (a ...
(
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 sy ...
). The
Heine–Borel theorem In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space R''n'', the following two statements are equivalent: *''S'' is closed set, closed and bounded set, bounded *''S'' i ...
, 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 , a useful example in point-set topology. It is connected but not path-connected. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
, 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 The Soviet Union,. officially the Union of Soviet ...
and
Pavel Urysohn Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a federal socialist state in Northern Eurasia Eurasia () is the large ...
, formulated Heine–Borel compactness in a way that could be applied to the modern notion of 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 ...
. 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 trivially compact. A non-trivial example of a compact space is the (closed)
unit interval 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). I ...
of
real 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 g ...
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 (or cluster point or accumulation point) of a Set (mathematics), 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 (mathematics), neighbourhood ...
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 point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
s 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, 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 classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
in particular is called compact if it is closed and bounded. This implies, by the
Bolzano–Weierstrass 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 h ...
, that any infinite
sequence 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 ...
from the set has a
subsequence 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 ...
that converges to a point in the set. Various equivalent notions of compactness, such as sequential compactness and
limit point compactIn mathematics, a topological space 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 (mat ...
ness, can be developed in general
metric 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 t ...
s. In contrast, the different notions of compactness are not equivalent in general
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 ...
s, and the most useful notion of compactness—originally called ''bicompactness''—is defined using
cover Cover or covers may refer to: Packaging, science and technology * A covering, usually - but not necessarily - one that completely closes the object ** Cover (philately), generic term for envelope or package ** Housing (engineering), an exterior ...
s consisting of
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 (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 In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space R''n'', the following two statements are equivalent: *''S'' is closed set, closed and bounded set, bounded *''S'' i ...
. Compactness, when defined in this manner, often allows one to take information that is known
locallyIn 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'' neighbourhood (mathematics), ne ...
—in a
neighbourhood A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...
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 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 h ...
; 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 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 ...
is called ''compact'' if each of its
open cover 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 h ...
s has a
finite Finite is the opposite of Infinity, 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 ...
subcover. That is, is compact if for every collection of open subsets of such that :X = \bigcup_x, there is a finite subset of such that :X = \bigcup_x. Some branches of mathematics such as
algebraic geometry Algebraic geometry is a branch of 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 thei ...

algebraic geometry
, 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 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 topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

subspace topology
). That is, is compact if for every arbitrary collection of open subsets of such that :K \subseteq \bigcup_ c, there is a finite subset of such that :K \subseteq \bigcup_ c. Compactness is a "topological" property. That is, if K \subset Z \subset Y, with subset equipped with the subspace topology, then is compact in if and only if is compact in .


Equivalent definitions

If is a topological space then the following are equivalent: # is compact. # Every
open cover 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 h ...
of has a finite subcover. # 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 compactIn 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 conditio ...
. # Any collection of closed subsets of with 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 ...
has nonempty intersection. # Every
net Net or net may refer to: Mathematics and physics * Net (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
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 Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
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 (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 pr ...
on converges to at least one point. # Every infinite subset of has a
complete accumulation point 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 ...
. #Every open cover of that is totally ordered by subset inclusion contains itself.


Euclidean space

For any
subset 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). ...

subset
of
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, is compact if and only if it is closed and bounded; this is the
Heine–Borel theorem In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space R''n'', the following two statements are equivalent: *''S'' is closed set, closed and bounded set, bounded *''S'' i ...
. As a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
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 and
totally boundedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...
(this is also equivalent to compactness for
uniform space In the mathematical 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 a ...
s). # is sequentially compact; that is, every
sequence 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 ...

sequence
in has a convergent subsequence whose limit is in (this is also equivalent to compactness for
first-countable In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
uniform space In the mathematical 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 a ...
s). # is
limit point compactIn mathematics, a topological space 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 (mat ...
(also called weakly countably compact); that is, every infinite subset of has at least one
limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
in . # is
countably compactIn 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 conditio ...
; that is, every countable open cover of ''X'' 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 remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

Cantor set
. # Every decreasing sequence of closed sets in has a nonempty intersection. # is closed and totally bounded. A compact metric space also satisfies the following properties: # Lebesgue's number lemma: For every open cover of , there exists a number such that every subset of of diameter < ''δ'' 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 (topology), base. More explicitly, a topological space T is second-countable if there exists some countable ...
, 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 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 ...
equipped with the
discrete metric Discrete in science is the opposite of continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random vari ...
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.


Characterization by continuous functions

Let be a topological space and the ring of real continuous functions on . For each , the evaluation map \operatorname_p\colon C(X)\to \mathbf 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 idealIn 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 ha ...
, since the
residue fieldIn 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 (mathematics), field. Fre ...
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 for ...

first isomorphism theorem
. A topological space is
pseudocompactIn 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 set, bounded. Many authors include the requirement that the space be Tychonoff space, completely regu ...
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. Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, ...
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. The framework of non-standard analysis allows for the following alternative characterization of compactness: a topological space is compact if and only if every point of the natural extension is infinitesimal, infinitely close to a point of (more precisely, is contained in the monad (non-standard analysis), monad 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).Let , , and . Endow with the topology generated by the following basic open sets: every subset of is open; the only open sets containing are and ; and the only open sets containing are and . Then and are both compact subsets but their intersection, which is , is not compact. Note that both and are compact open subsets, neither one of which is closed. * 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).Let and endow with the topology . Then is a compact set but it is not closed. ** If is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).Let be the set of non-negative integers. We endow with the particular point topology by defining a subset to be open if and only if . Then is compact, the closure of is all of , but is not compact since the collection of open subsets does not have a finite subcover. * 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: 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 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 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 g ...
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 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). I ...
is compact. This follows from the
Heine–Borel theorem In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space R''n'', the following two statements are equivalent: *''S'' is closed set, closed and bounded set, bounded *''S'' i ...
. The open interval is not compact: the
open cover 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 h ...
\left( \frac, 1 - \frac \right) 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 \left[0, \frac - \frac\right]\text\left[\frac + \frac, 1\right] 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. 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 remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

Cantor set
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 d(f, g) = \sup_ , f(x) - g(x), . Then by
Arzelà–Ascoli theorem The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, in ...
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, \ell^2 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 (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 boundedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...
'' * 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

* * ---- {{DEFAULTSORT:Compact Space Compactness (mathematics) General topology Properties of topological spaces Topology