TheInfoList

OR: 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, geometric ...
, compactness is a property that seeks to generalize the notion of a closed and bounded subset of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 rati ...
s $\mathbb$ 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 inte ...
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 ...
s $\mathbb$ is not compact either, because it excludes the two limiting values $+\infty$ and $-\infty$. 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 sett ...
, but may not be equivalent 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 poi ...
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 calle ...
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 $\mathbb^1$ (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 ...
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 inte ...
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 th ...
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 sett ...
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 (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
s that " cover" 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 ...
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. ...
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 poi ...
.

# 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 li ...
(
1817 Events January–March * January 1 – Sailing through the Sandwich Islands, Otto von Kotzebue discovers New Year Island. * January 19 – An army of 5,423 soldiers, led by General José de San Martín, starts crossing t ...
) 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 cont ...
. 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 mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematic ...
. 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 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 conti ...
. 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 inte ...
, 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 va ...
s, the precise conclusion of which was that it was possible to extract a uniformly convergent 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 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 ; u(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 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 — 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 naturally ...
. 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 c ...
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 ...
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, which was seen as fundamental for the rigorous formulation of analysis. In 1870, Eduard Heine 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 va ...
defined on a closed and bounded interval was in fact uniformly continuous. 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 (
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 o ...
( 1904). 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, geometr ...
, 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 ...
, 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 poi ...
. 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 ...
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 conta ...
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 points of the interval, since the limit points 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 geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
in particular is called compact if it is closed and bounded. 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 call ...
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 sett ...
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 poi ...
s, and the most useful notion of compactness — originally called ''bicompactness'' — is defined using covers consisting of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
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 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; 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 poi ...
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 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 marke ...
subcover. That is, is compact if for every collection of open subsets of such that :$X = \bigcup_x$, there is a finite subcollection ⊆ such that :$X = \bigcup_ x\ .$ 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 :$K \subseteq \bigcup_ c\ ,$ there is a finite subcollection ⊆ 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 .

## 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\ ...
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 compact. # Any collection of closed subsets of with the finite intersection property has nonempty intersection. # Every net 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 on converges to at least one point. # Every infinite subset of has a complete accumulation point. # For every topological space , the projection $X \times Y \to Y$ is a closed mapping (see proper map). 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 ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...
of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, is compact if and only if it is closed and bounded; 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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 bounded (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 uni ...
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 call ...
in has a convergent subsequence whose limit is in (this is also equivalent to compactness for first-countable
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 uni ...
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 cont ...
in . # is countably compact; 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. Th ...
. # 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: 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, 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 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 number to a po ...
equipped with the discrete metric 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 $\operatorname_p\colon C\left(X\right)\to \mathbb$ 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 learni ...
of is a maximal ideal, since the residue field 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 fo ...
. A topological space is pseudocompact if and only if every maximal ideal in has residue field the real numbers. For completely regular spaces, 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 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 of ).

### Hyperreal definition

A space is compact if its 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 of compact sets is compact. * A 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 of any collection of compact spaces is compact. (This is
Tychonoff's theorem In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is tran ...
, which is equivalent to the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection o ...
.) * In a metrizable space, a subset is compact if and only if it is
sequentially compact In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X. Every metric space is naturally a topological space, and for metric spaces, the notio ...
(assuming 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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS), a compact subset is 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 In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
. * A compact Hausdorff space is normal and 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 image of a compact space is compact, the
extreme value theorem In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> su ...
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 is compact.

## Compactifications

Every topological space is an open dense subspace of a compact space having at most one point more than , by the Alexandroff one-point compactification. By the same construction, every
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
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 ...
s has a greatest element and a least element. Let be a simply ordered set endowed with the order topology. Then is compact if and only if is a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' S ...
(i.e. all subsets have suprema and infima).

# Examples

* Any finite topological space, including the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
, 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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
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 $\mathbb$ is homeomorphic to the circle ; the one-point compactification of $\mathbb^2$ 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 In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
is compact. In particular, Sierpiński space is compact. * No
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
with an infinite number of points is compact. The collection of all singletons of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact. * In $\mathbb$ 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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
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\ ...
$\left( \frac, 1 - \frac \right)$ for does not have a finite subcover. Similarly, the set of ''
rational numbers 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 rat ...
'' in the closed interval is not compact: the sets of rational numbers in the intervals
extended real number line In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
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 In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
to ��1, 1of 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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
, the -sphere is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
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 In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
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. Th ...
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 In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
. Then is a compact topological space; this follows from the Tychonoff theorem. * Consider the set of all functions satisfying the
Lipschitz condition In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ex ...
for all . Consider on the metric induced by the uniform distance $d\left(f, g\right) = \sup_ , f\left(x\right) - g\left(x\right), .$ 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 inte ...
the space is compact. * The
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of any
bounded linear operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
on a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
is a nonempty compact subset of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s $\mathbb$. Conversely, any compact subset of $\mathbb$ arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space $\ell^2$ may have any compact nonempty subset of $\mathbb$ as spectrum.

## Algebraic examples

*
Compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
s such as an
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
are compact, while groups such as a
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
are not. * Since the -adic integers are
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to the Cantor set, they form a compact set. * The
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of any
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
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 schemes, "quasi" referring to the non-Hausdorff nature of the topology. * The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem. Stone spaces, compact 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 In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
is a compact Hausdorff space. * The Hilbert cube is compact, again a consequence of Tychonoff's theorem. * A profinite group (e.g.
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
) is compact.

*
Compactly generated space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subsp ...
* Compactness theorem * Eberlein compactum * Exhaustion by compact sets *
Lindelöf space In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' sub ...
* Metacompact space * Noetherian topological space * Orthocompact space * Paracompact space * Precompact set - also called '' totally bounded'' * Relatively compact subspace * Totally bounded

# 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''). * * * * * * * * * * * * * * * . * *