, more specifically in general topology
, compactness is a property that generalizes the notion of a subset of Euclidean space
(i.e., containing all its limit point
s) and bounded
(i.e., having all its points lie within some fixed distance of each other).
Examples include a closed interval
, a rectangle
, or a finite set of points. This notion is defined for more general topological space
s than Euclidean space in various ways.
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
, 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. Euclidean space itself is not compact since it is not bounded.
In particular, the sequence of points , which is not bounded, has no subsequence that converges to any real number.
The concept of a compact space was formally introduced by Maurice Fréchet
in 1906 to generalize the Bolzano–Weierstrass theorem to spaces of functions
, rather than geometrical points. Applications of compactness to classical analysis, such as the Arzelà–Ascoli theorem
and the Peano existence theorem
are of this kind. Following the initial introduction of the concept, various equivalent notions of compactness, including sequential compactness
and limit point compact
ness, were developed in general metric space
In general topological spaces, however, different notions of compactness are not necessarily equivalent. The most useful notion, which is the standard definition of the unqualified term ''compactness'', is phrased in terms of the existence of finite families of open set
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
and Pavel Urysohn
in 1929, exhibits compact spaces as generalizations of finite set
s. 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 often refers to a compact subspace
of a topological space as well.
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
) 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
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
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
s, as investigated by David Hilbert
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
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
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
), and it was generalized to arbitrary collections of intervals by Pierre Cousin
(1895) and Henri Lebesgue
). 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
, under the direction of Pavel Alexandrov
and Pavel Urysohn
, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a topological space
. 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.
Any finite space
is trivially compact.
A non-trivial example of a compact space is the (closed) unit interval
of real number
s. If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point
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 point
s must be in the space itself — an open (or half-open) interval of the real numbers is not compact.
It is also crucial that the interval be bounded
, since in the interval , one could choose the sequence of points , of which no sub-sequence ultimately gets arbitrarily close to any given real number.
In two dimensions, closed disks
are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary.
However, an open disk is not compact, because a sequence of points can tend to the boundary—without getting arbitrarily close to any point in the interior.
Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point ''within'' the space.
Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.
Various definitions of compactness may apply, depending on the level of generality.
A subset of Euclidean space
in particular is called compact if it is closed
This implies, by the Bolzano–Weierstrass theorem
, that any infinite sequence
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 compact
ness, can be developed in general metric space
In contrast, the different notions of compactness are not equivalent in general topological space
s, and the most useful notion of compactness—originally called ''bicompactness''—is defined using cover
s consisting of open set
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
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
is called ''compact'' if each of its open cover
s has a finite subcover
. That is, is compact if for every collection of open subsets of such that
there is a finite subset of such that
Some branches of mathematics such as algebraic geometry
, typically influenced by the French school of Bourbaki
, use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both Hausdorff
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 subset of 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 .
If is a topological space then the following are equivalent:
# is compact.
# Every open cover
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 any subset
of Euclidean space
, is compact if and only if it is closed
; this is the Heine–Borel theorem
As a Euclidean space
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.
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
# is sequentially compact; that is, every sequence
in has a convergent subsequence whose limit is in (this is also equivalent to compactness for first-countable uniform space
# is limit point compact (also called countably compact); that is, every infinite subset of has at least one limit point
# is an image of a continuous function from the Cantor set
A compact metric space also satisfies the following properties:
# Lebesgue's number lemma
: For every open cover of , there exists a number such that every subset of of diameter < ''δ'' is contained in some member of the cover.
# is second-countable
– 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
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.
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.
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 space
s, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism. There are pseudocompact spaces that are not compact, though.
In general, for non-pseudocompact spaces there are always maximal ideals in such that the residue field is a (non-Archimedean
) hyperreal field
The framework of non-standard analysis
allows for the following alternative characterization of compactness: a topological space is compact if and only if every point of the natural extension is infinitely close
to a point of (more precisely, is contained in the monad
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 .
* 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 collection of compact subsets of a Hausdorff space is compact (and closed);
** If is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).
[Let , , and . Endow with the topology generated by the following basic open sets: every subset of is open; the only open sets containing are and ; and the only open sets containing are and . Then and are both compact subsets but their intersection, which is , is not compact. Note that both and are compact open subsets, neither one of which is closed.]
* The product
of any collection of compact spaces is compact. (This is Tychonoff's theorem
, which is equivalent to the axiom of choice
* In a metrizable space
, a subset is compact if and only if it is sequentially compact
(assuming countable choice
* A finite set endowed with any topology is compact.
Properties of compact spaces
* A compact subset of a Hausdorff space
** If is not Hausdorff then a compact subset of may fail to be a closed subset of (see footnote for example).
[Let and endow with the topology . Then is a compact set but it is not closed.]
** If is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).
[Let be the set of non-negative integers. We endow with the particular point topology by defining a subset to be open if and only if . Then is compact, the closure of is all of , but is not compact since the collection of open subsets does not have a finite subcover.]
* In any topological vector space
(TVS), a compact subset is complete
. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are ''not'' closed.
* If and are disjoint compact subsets of a Hausdorff space , then there exist disjoint open set and in such that and .
* A continuous bijection from a compact space into a Hausdorff space is a homeomorphism
* A compact Hausdorff space is normal
* 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
: 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
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
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
(i.e. all subsets have suprema and infima).
* Any finite topological space
, including the empty set
, is compact. More generally, any space with a finite topology
(only finitely many open sets) is compact; this includes in particular the trivial topology
* Any space carrying the cofinite topology
* Any locally compact
Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification
. The one-point compactification of is homeomorphic to the circle ; the one-point compactification of is homeomorphic to the sphere . Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
* The right order topology
or left order topology
on any bounded totally ordered set
is compact. In particular, Sierpiński space
* No discrete space
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
is compact. This follows from the Heine–Borel theorem
. The open interval is not compact: the open cover
for does not have a finite subcover. Similarly, the set of ''rational numbers
'' in the closed interval is not compact: the sets of rational numbers in the intervals