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, compactness is a property that seeks to generalize the notion of a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and
bounded 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 Euclidea ...
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 ra ...
s
is not compact, because it has infinitely many "punctures" corresponding to the
irrational numbers, 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
is not compact either, because it excludes the two limiting values
and
. However, the
''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, 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 poin ...
s.
One such generalization is that a topological space is
''sequentially'' compact if every
infinite sequence of points sampled from the space has an infinite
subsequence that converges to some point of the space.
The
Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded.
Thus, if one chooses an infinite number of points in the closed
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
, some of those points will get arbitrarily close to some real number in that space.
For instance, some of the numbers in the sequence accumulate to 0 (while others accumulate to 1).
The same set of points would not accumulate to any point of the open unit interval , so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded.
For example, considering
(the real number line), the sequence of points has no subsequence that converges to any real number.
Compactness was formally introduced by
Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to
spaces of functions. The
Arzelà–Ascoli theorem and the
Peano existence theorem 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 compactness, 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 setti ...
s.
In general topological spaces, however, these notions of compactness are not necessarily equivalent. The most useful notion — and the standard definition of the unqualified term ''compactness'' — is phrased in terms of the existence of finite families of
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s that "
cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of copy ...
" 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 ar ...
in 1929, exhibits compact spaces as generalizations of
finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds
locally — 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 poin ...
.
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) 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.
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.
In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for
spaces of functions rather than just numbers or geometrical points.
The idea of regarding functions as themselves points of a generalized space dates back to the investigations of
Giulio Ascoli
Giulio Ascoli (20 January 1843, Trieste – 12 July 1896, Milan) was a Jewish-Italian mathematician. He was a student of the Scuola Normale di Pisa, where he graduated in 1868.
In 1872 he became Professor of Algebra and Calculus of the Pol ...
and
Cesare Arzelà.
The culmination of their investigations, the
Arzelà–Ascoli theorem, 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 val ...
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 equations, 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 ...
and
Erhard Schmidt.
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 natural ...
.
This ultimately led to the notion of a
compact operator as an offshoot of the general notion of a compact space.
It was
Maurice Fréchet 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 val ...
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
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability.
Biography
Borel was ...
(
1895), and it was generalized to arbitrary collections of intervals by
Pierre Cousin (1895) and
Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
(
1904). The
Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.
This property was significant because it allowed for the passage from
local information about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function).
This sentiment was expressed by , who also exploited it in the development of the
integral now bearing his name.
Ultimately, the Russian school of
point-set topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, under the direction of
Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
and
Pavel Urysohn
Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which ar ...
, 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 poin ...
. 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 contai ...
in that interval.
For instance, the odd-numbered terms of the sequence get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1.
The given example sequence shows the importance of including the
boundary 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 Euclidea ...
in particular is called compact if it is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and
bounded.
This implies, by the
Bolzano–Weierstrass theorem, 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 called ...
from the set has a
subsequence that converges to a point in the set.
Various equivalent notions of compactness, such as
sequential compactness and
limit point compactness, 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 setti ...
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 poin ...
s, and the most useful notion of compactness — originally called ''bicompactness'' — is defined using
cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of copy ...
s consisting of
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
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 a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
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 poin ...
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_\alp ...
s has a
finite subcover. That is, is compact if for every collection of open subsets of such that
:
,
there is a finite subcollection ⊆ such that
:
Some branches of mathematics such as
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, typically influenced by the French school of
Bourbaki, use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both
Hausdorff and ''quasi-compact''.
A compact set is sometimes referred to as a ''compactum'', plural ''compacta''.
Compactness of subsets
A subset of a topological space is said to be compact if it is compact as a subspace (in the
subspace topology).
That is, is compact if for every arbitrary collection of open subsets of such that
:
there is a finite subcollection ⊆ such that
:
Compactness is a "topological" property. That is, if
, with subset equipped with the subspace topology, then is compact in if and only if is compact in .
Characterization
If is a topological space then the following are equivalent:
# is compact; i.e., every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
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 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
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 of ...
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 Euclidea ...
, is compact if and only if it is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and
bounded; 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 Euclidea ...
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
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function ''A'' with domain N (where ...
):
# is compact.
# is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and
totally bounded (this is also equivalent to compactness for
uniform spaces).
# 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 called ...
in has a convergent subsequence whose limit is in (this is also equivalent to compactness for
first-countable uniform spaces).
# is
limit point compact (also called weakly countably compact); that is, every infinite subset of has at least one
limit point 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.
T ...
.
# 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
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \ma ...
,
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
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
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
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function ''A'' with domain N (where ...
) are true whenever is compact. (The converse in general fails if is not also metrizable.):
# Every sequence in has a subsequence that converges in .
# Every monotone increasing sequence in converges to a unique limit in .
# Every monotone decreasing sequence in converges to a unique limit in .
# Every decreasing nested sequence of nonempty closed subsets ⊇ ⊇ ... in has a nonempty intersection.
# Every increasing nested sequence of proper open subsets ⊆ ⊆ ... in fails to cover .
Characterization by continuous functions
Let be a topological space and the ring of real continuous functions on .
For each , the evaluation map
given by is a ring homomorphism.
The
kernel of is a
maximal ideal, since the
residue field is the field of real numbers, by the
first isomorphism theorem.
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 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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
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, 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 ...
.)
* In a
metrizable space, a subset is compact if and only if it is
sequentially compact (assuming
countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function ''A'' with domain N (where ...
)
* A finite set endowed with any topology is compact.
Properties of compact spaces
* A compact subset of a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
is closed.
** If is not Hausdorff then a compact subset of may fail to be a closed subset of (see footnote for example).
** If is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).
* In any
topological vector space (TVS), a compact subset is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
. 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 isom ...
.
* 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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
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> s ...
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 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.'' ...
(i.e. all subsets have suprema and infima).
Examples
* Any
finite topological space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements.
Finite topological spaces are often used to provide example ...
, 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 othe ...
, 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
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 top ...
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
carrying the
lower limit topology, no uncountable set is compact.
* In the
cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not
locally compact but is still
Lindelöf.
* The closed
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
is compact. This follows from the
Heine–Borel theorem. The open interval is not compact: the
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
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 ra ...
'' in the closed interval is not compact: the sets of rational numbers in the intervals