View source for Compact space

{{short description|Topological notions of all points being "close"}}
{{redirect|Compactness}}
[[File:Compact.svg|thumb|upright=1.6|The interval {{math|''A'' {{=}} (−∞, −2]}} is not compact because it is not bounded. The interval {{math|''C'' {{=}} (2, 4)}} is not compact because it is not closed. The interval {{math|''B'' {{=}} [0, 1]}} is compact because it is both closed and bounded.]]
In [[mathematics]], and more specifically in [[general topology]], '''compactness''' is a property that generalizes the notion of a subset of [[Euclidean space]] being [[closed set|closed]] (that is, containing all its [[limit point]]s) and [[bounded set|bounded]] (that is, 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|''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]] {{math|[0, 1]}} some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers {{nowrap|1/2, 4/5, 1/3, 5/6, 1/4, 6/7, &hellip;}} accumulate to 0 (others accumulate to 1). The same set of points would not accumulate to any point of the ''open'' unit interval {{math|(0, 1)}}; 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 {{nowrap|0, 1, 2, 3, …}} has no subsequence that converges to any real number.

Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces are encountered in [[mathematical analysis]], where the property of compactness of some topological spaces arises in the hypotheses or in the conclusions of many fundamental theorems, such as the [[Bolzano–Weierstrass theorem]], the [[extreme value theorem]], the [[Arzelà–Ascoli theorem]], and the [[Peano existence theorem]]. Another example is the definition of [[distribution (mathematics)|distributions]] which uses the space of [[smooth function]]s that are zero outside some (unspecified) compact space.

Various equivalent notions of compactness, including [[Sequentially compact space|sequential compactness]] and [[limit point compact]]ness, can be developed in general [[metric space]]s. 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 (topology)|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 [[local property|locally]]&mdash;that is, in a neighborhood of each point&mdash;into corresponding statements that hold throughout the space, and many [[#Theorems|theorems]] are of this character.

The term '''compact set''' is sometimes a synonym for compact space, but usually refers to a [[#Compactness of subsets|compact subspace]] of a topological space.

== 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]] ([[#CITEREFBolzano1817|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–Weierstrass theorem|Bolzano's theorem]], and its method of proof, would not emerge until almost 50 years later when it was rediscovered by [[Karl Weierstrass]].<ref>{{harvnb|Kline|1972|pp=952–953}}; {{harvnb|Boyer|Merzbach|1991|p=561}}</ref>

In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for [[function spaces|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]].<ref>{{harvnb|Kline|1972|loc=Chapter 46, §2}}</ref>  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 [[uniform convergence|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]]&mdash;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 René Fréchet|Maurice Fréchet]] who, in [[#CITEREFFréchet1906|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<ref>Frechet, M. 1904. Generalisation d'un theorem de Weierstrass. Analyse Mathematique.</ref> 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 [[linear continuum|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]] ([[#CITEREFBorel1895|1895]]), and it was generalized to arbitrary collections of intervals by [[Pierre Cousin (mathematician)|Pierre Cousin]] (1895) and [[Henri Lebesgue]] ([[#CITEREFLebesgue1904|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 property|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 {{harvtxt|Lebesgue|1904}}, who also exploited it in the development of the [[Lebesgue integral|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]].  {{harvtxt|Alexandrov|Urysohn|1929}} 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.
<!--
One of the main reasons for studying compact spaces is because they are in some ways very similar to [[finite set]]s: there are many results which are easy to show for finite sets, whose proofs carry over with minimal change to compact spaces. Here is an example:

* Suppose ''X'' is a [[Hausdorff space]], and we have a point ''x'' in ''X'' and a finite subset ''A'' of ''X'' not containing ''x''. Then we can [[separated sets|separate]] ''x'' and ''A'' by [[neighborhood (topology)|neighborhoods]]: for each ''a'' in ''A'', let ''U''(''x'') and ''V''(''a'') be disjoint neighborhoods containing ''x'' and ''a'', respectively. Then the intersection of all the ''U''(''x'') and the union of all the ''V''(''a'') are the required neighborhoods of ''x'' and ''A''.

Note that if ''A'' is [[Infinity|infinite]], the proof fails, because the intersection of arbitrarily many neighborhoods of ''x'' might not be a neighborhood of ''x''. The proof can be "rescued", however, if ''A'' is compact: we simply take a finite subcover of the cover {{math|{''V''(''a'') : ''a'' ∈ A}}} of ''A'', then intersect the corresponding finitely many ''U''(''x''). In this way, we see that in a Hausdorff space, any point can be separated by neighborhoods from any compact set not containing it. In fact, repeating the argument shows that any two disjoint compact sets in a Hausdorff space can be separated by neighborhoods&nbsp;– note that this is precisely what we get if we replace "point" (i.e. [[singleton set]]) with "compact set" in the Hausdorff [[separation axiom]]. Many of the arguments and results involving compact spaces follow such a pattern.
-->

==Basic examples==
Any [[finite topological space|finite space]] is trivially compact. A non-trivial example of a compact space is the (closed) [[unit interval]] {{closed-closed|0,1}} 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 {{nowrap|1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, ...}} get arbitrarily close to&nbsp;0, while the even-numbered ones get arbitrarily close to&nbsp;1.  The given example sequence shows the importance of including the [[boundary (topology)|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 set|bounded]], since in the interval {{closed-open|0,∞}} one could choose the sequence of points {{nowrap|0, 1, 2, 3, ...}}, of which no sub-sequence ultimately gets arbitrarily close to any given real number.

In two dimensions, closed [[Disk (mathematics)|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 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]] in particular is called compact if it is [[closed set|closed]] and [[bounded set|bounded]].  This implies, by the [[Bolzano–Weierstrass theorem]], that any infinite [[sequence (mathematics)|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]]s.

In contrast, the different notions of compactness are not equivalent in general [[topological space]]s, and the most useful notion of compactness&mdash;originally called ''bicompactness''&mdash;is defined using [[cover (topology)|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 [[local property|locally]]&mdash;in a [[neighbourhood (mathematics)|neighbourhood]] of each point of the space&mdash;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]] {{mvar|X}} is called ''compact'' if each of its [[open cover]]s has a [[finite set|finite]] [[subcover]]. That is, {{mvar|X}} is compact if for every collection {{mvar|C}} of open subsets of {{mvar|X}} such that

:<math>X = \bigcup_{x \in C}x</math>,

there is a '''finite''' subset {{mvar|F}} of {{mvar|C}} such that

:<math>X = \bigcup_{x \in F}x.</math>

Some branches of mathematics such as [[algebraic geometry]], typically influenced by the French school of [[Nicolas Bourbaki|Bourbaki]], use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both [[Hausdorff space|Hausdorff]] and ''quasi-compact''.  A compact set is sometimes referred to as a ''compactum'', plural ''compacta''.

===Compactness of subsets===
A subset {{mvar|K}} of a topological space {{mvar|X}} is said to be compact if it is compact as a subspace (in the [[subspace topology]]).  That is, {{mvar|K}} is compact if for every arbitrary collection {{mvar|C}} of open subsets of {{mvar|X}} such that

:<math>K \subseteq \bigcup_{c \in C} c</math>,

there is a '''finite''' subset {{mvar|F}} of {{mvar|C}} such that

:<math>K \subseteq \bigcup_{c \in F} c</math>.

Compactness is a "topological" property. That is, if <math>K \subset Z \subset Y</math>, with subset {{mvar|Z}} equipped with the subspace topology, then {{mvar|K}} is compact in {{mvar|Z}} if and only if {{mvar|K}} is compact in {{mvar|Y}}.

===Equivalent definitions===
Assuming the [[axiom of choice]], the following are equivalent:
# A topological space ''X''  is compact.
# Every [[open cover]] of ''X'' has a finite [[subcover]].
# ''X'' 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]])
# Any collection of closed subsets of ''X'' with the [[finite intersection property]] has nonempty intersection.
# Every [[Net (mathematics)|net]] on ''X'' has a convergent subnet (see the article on [[Net (mathematics)|nets]] for a proof).
# Every [[mathematical filter|filter]] on ''X'' has a convergent refinement.
# Every net on ''X'' has a cluster point.
# Every filter on ''X'' has a cluster point.
# Every [[ultrafilter]] on ''X'' converges to at least one point.
# Every infinite subset of ''X'' has a [[complete accumulation point]].<ref>{{harv|Kelley|1955|p=163}}</ref>

====Euclidean space====
For any [[subset]] ''A'' of [[Euclidean space]] '''R'''<sup>''n''</sup>, ''A'' is compact if and only if it is [[closed set|closed]] and [[bounded set|bounded]]; 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 (mathematics)|interval]] or closed ''n''-ball.

====Metric spaces====
For any metric space (''X'', ''d''), the following are equivalent:
# (''X'', ''d'') is compact.
# (''X'', ''d'') is [[completeness (topology)|complete]] and [[totally bounded]] (this is also equivalent to compactness for [[uniform space]]s).<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.3.7}}</ref>
# (''X'', ''d'') is sequentially compact; that is, every [[sequence]] in ''X'' has a convergent subsequence whose limit is in ''X'' (this is also equivalent to compactness for [[first-countable]] [[uniform space]]s).
# (''X'', ''d'') is limit point compact (also called countably compact); that is, every infinite subset of ''X'' has at least one [[limit point]] in ''X''.
# (''X'', ''d'') is an image of a continuous function from the [[Cantor set]].<ref>{{harvnb|Willard|1970}} Theorem 30.7.</ref>

A compact metric space (''X'', ''d'') also satisfies the following properties:
# [[Lebesgue's number lemma]]: For every open cover of ''X'', there exists a number {{nowrap|''δ'' > 0}} such that every subset of ''X'' of diameter < ''δ'' is contained in some member of the cover.
# (''X'', ''d'') is [[second-countable space|second-countable]], [[Separable space|separable]] and [[Lindelöf space|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.
# ''X'' is closed and bounded (as a subset of any metric space whose restricted metric is ''d''). 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 [[Singleton (mathematics)|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 ''X'' be a topological space and C(''X'') the ring of real continuous functions on ''X''.  For each ''p''&isin;''X'', the evaluation map <math>\operatorname{ev}_p\colon C(X)\to \mathbf{R}</math>
given by ev<sub>''p''</sub>(''f'')=''f''(''p'') is a ring homomorphism.  The [[kernel (algebra)|kernel]] of ev<sub>''p''</sub> is a [[maximal ideal]], since the [[residue field]] {{nowrap|C(''X'')/ker ev<sub>''p''</sub>}} is the field of real numbers, by the [[first isomorphism theorem]].  A topological space ''X'' is [[pseudocompact space|pseudocompact]] if and only if every maximal ideal in C(''X'') 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.<ref>{{harvnb|Gillman|Jerison|1976|loc=§5.6}}</ref>  There are pseudocompact spaces that are not compact, though.

In general, for non-pseudocompact spaces there are always maximal ideals ''m'' in C(''X'') such that the residue field C(''X'')/''m'' is a ([[non-archimedean field|non-archimedean]]) [[hyperreal field]].  The framework of [[non-standard analysis]] allows for the following alternative characterization of compactness:<ref>{{harvnb|Robinson||loc=Theorem 4.1.13}}</ref> a topological space ''X'' is compact if and only if every point ''x'' of the natural extension ''*X'' is [[infinitesimal|infinitely close]] to a point ''x''<sub>0</sub> of ''X'' (more precisely, ''x'' is contained in the [[monad (non-standard analysis)|monad]] of ''x''<sub>0</sub>).

====Hyperreal definition====
A space ''X'' is compact if its [[hyperreal number|hyperreal extension]] ''*X'' (constructed, for example, by the [[ultrapower construction]]) has the property that every point of ''*X'' is infinitely close to some point of ''X''⊂''*X''. For example, an open real interval {{nowrap|''X'' {{=}} (0, 1)}} is not compact because its hyperreal extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of ''X''.

== Properties of compact spaces ==

===Functions and compact spaces===
A [[continuous function (topology)|continuous]] image of a compact space is compact.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.2.2}}; See also {{PlanetMath|urlname=Compactnessispreservedunderacontinuousmap|title=Compactness is preserved under a continuous map}}</ref>
This implies the [[extreme value theorem]]: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Corollary 5.2.1}}</ref> (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.

===Compact spaces and set operations===
A closed subset of a compact space is compact,<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.2.3}}; {{PlanetMath|urlname=closedsetinacompactspaceiscompact|title=Closed set in a compact space is compact}}; {{PlanetMath|urlname=closedsubsetsofacompactsetarecompact|title=Closed subsets of a compact set are compact}}</ref> and a finite [[Union (set theory)|union]] of compact sets is compact.

The [[product topology|product]] of any collection of compact spaces is compact. (This is [[Tychonoff's theorem]], which is equivalent to the [[axiom of choice]].)

Every topological space ''X'' is an open [[dense topological subspace|dense subspace]] of a compact space having at most one point more than ''X'', by the [[Compactification (mathematics)|Alexandroff one-point compactification]].  By the same construction, every [[locally compact]] Hausdorff space ''X'' is an open dense subspace of a compact Hausdorff space having at most one point more than ''X''.

===Ordered compact spaces===
A nonempty compact subset of the [[real number]]s has a greatest element and a least element.

Let ''X'' be a [[total order|simply ordered]] set endowed with the [[order topology]]. Then ''X'' is compact if and only if ''X'' is a [[complete lattice]] (i.e. all subsets have suprema and infima).<ref>{{harv|Steen|Seebach|1995|p=67}}</ref>

==Examples==
* Any [[finite topological space]], including the [[empty set]], is compact.  More generally, any space with a [[finite topology]] (only finitely many open sets) is compact; this includes in particular the [[trivial topology]].
* Any space carrying the [[cofinite topology]] is compact.
* Any [[locally compact]] Hausdorff space can be turned into a compact space by adding a single point to it, by means of [[Alexandroff one-point compactification]]. The one-point compactification of '''R''' is homeomorphic to the circle '''S'''<sup>1</sup>; the one-point compactification of '''R'''<sup>2</sup> is homeomorphic to the sphere '''S'''<sup>2</sup>. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
* The [[right order topology]] or [[left order topology]] on any bounded [[totally ordered set]] is compact. In particular, [[Sierpiński space]] is compact.
* No [[discrete space]] with an infinite number of points is compact. The collection of all [[Singleton (mathematics)|singletons]] of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.
* In '''R''' 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 space|Lindelöf]].
* The closed [[unit interval]] {{closed-closed|0,1}} is compact. This follows from the [[Heine–Borel theorem]]. The open interval {{open-open|0,1}} is not compact: the [[open cover]] <math>\left ( \frac{1}{n}, 1 - \frac{1}{n} \right )</math> for {{math|1={{mvar|n}} = 3, 4, … }} does not have a finite subcover. Similarly, the set of ''[[rational numbers]]'' in the closed interval {{closed-closed|0,1}} is not compact: the sets of rational numbers in the intervals <math>\left[0,\frac{1}{\pi}-\frac{1}{n}\right] \ \text{and} \ \left[\frac{1}{\pi}+\frac{1}{n},1\right]</math> cover all the rationals in [0,&nbsp;1] for {{math|1={{mvar|n}} = 4, 5, ... }} but this cover does not have a finite subcover. Here, the sets are open in the subspace topology even though they are not open as subsets of&nbsp;'''R'''.
* The set '''R''' of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals {{open-open|{{mvar|n}}−1, {{mvar|n}}+1}} , where {{mvar|n}} takes all integer values in '''Z''', cover '''R''' but there is no finite subcover.
* On the other hand, the [[extended real number line]] carrying the analogous topology ''is'' compact; note that the cover described above would never reach the points at infinity.  In fact, the set has the [[homeomorphism]] to [-1,1] of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homeomorphisms preserve covers, the Heine-Borel property can be inferred.
* For every [[natural number]] {{mvar|n}}, the [[n-sphere|{{mvar|n}}-sphere]] is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional [[normed vector space]] is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its [[closed unit ball]] is compact.
* On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. ([[Alaoglu's theorem]])
* The [[Cantor set]] is compact. In fact, every compact metric space is a continuous image of the Cantor set.
* Consider the set ''K'' of all functions ''f'' : '''R''' → [0,1] from the real number line to the closed unit interval, and define a topology on ''K'' so that a sequence <math>\{f_n\}</math> in ''K'' converges towards <math>f\in K</math> if and only if <math>\{f_n(x)\}</math> converges towards ''f''(''x'') for all real numbers ''x''. There is only one such topology; it is called the topology of [[pointwise convergence]] or the [[product topology]]. Then ''K'' is a compact topological space; this follows from the [[Tychonoff theorem]].
* Consider the set ''K'' of all functions ''f''&nbsp;: {{closed-closed|0,1}}&nbsp;→ {{closed-closed|0,1}} satisfying the [[Lipschitz condition]] |''f''(''x'')&nbsp;−&nbsp;''f''(''y'')|&nbsp;≤ |''x''&nbsp;−&nbsp;''y''| for all ''x'',&nbsp;''y''&nbsp;∈&nbsp;{{closed-closed|0,1}}.  Consider on ''K''&thinsp; the metric induced by the [[uniform convergence|uniform distance]] <math>d(f,g)=\sup_{x \in [0, 1]} |f(x)-g(x)|.</math> Then by [[Arzelà–Ascoli theorem]] the space ''K'' is compact.
* The [[spectrum of an operator|spectrum]] of any [[bounded linear operator]] on a [[Banach space]] is a nonempty compact subset of the [[complex number]]s '''C'''. Conversely, any compact subset of '''C''' arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space [[sequence spaces#ℓp spaces|<math>\ell^2</math>]] may have any compact nonempty subset of '''C''' as spectrum.

=== Algebraic examples ===
* [[Compact group]]s such as an [[orthogonal group]] are compact, while groups such as a [[general linear group]] are not.
* Since the [[p-adic numbers|''p''-adic integers]] are [[homeomorphic]] to the Cantor set, they form a compact set.
* The [[spectrum of a ring|spectrum]] of any [[commutative ring]] with the [[Zariski topology]] (that is, the set of all prime ideals) is compact, but never [[Hausdorff space|Hausdorff]] (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact [[scheme (mathematics)|schemes]], "quasi" referring to the non-Hausdorff nature of the topology.
* The [[spectrum of a boolean algebra|spectrum of a Boolean algebra]] is compact, a fact which is part of the [[Stone representation theorem]]. [[Stone space]]s, compact [[totally disconnected space|totally disconnected]] Hausdorff spaces, form the abstract framework in which these spectra are studied.  Such spaces are also useful in the study of [[profinite group]]s.
* The [[structure space]] of a commutative unital [[Banach algebra]] is a compact Hausdorff space.
* The [[Hilbert cube]] is compact, again a consequence of Tychonoff's theorem.
* A [[profinite group]] (e.g., Galois group) is compact.

== See also ==
* [[Compactly generated space]]
* [[Eberlein compactum]]
* [[Exhaustion by compact sets]]
* [[Lindelöf space]]
* [[Metacompact space]]
* [[Noetherian topological space]]
* [[Orthocompact space]]
* [[Paracompact space]]

==Notes==
{{reflist}}

==References==
*{{citation |last1=Alexandrov |first1=Pavel |authorlink1=Pavel Alexandrov |last2=Urysohn |first2=Pavel |authorlink2=Pavel Urysohn |title=Mémoire sur les espaces topologiques compacts |journal=Koninklijke Nederlandse Akademie van Wetenschappen Te Amsterdam, Proceedings of the Section of Mathematical Sciences |volume=14 |year=1929}}.
*{{citation |last1=Arkhangel'skii |first1=A.V. |last2=Fedorchuk |first2=V.V. |contribution=The basic concepts and constructions of general topology |editor1=Arkhangel'skii, A.V. |editor2=Pontrjagin, L.S. |title=General topology I |publisher=Springer |year=1990 |isbn=978-0-387-18178-3 |series=Encyclopedia of the Mathematical Sciences |volume=17}}.
*{{springer|id=C/c023530|title=Compact space|first=A.V.|last=Arkhangel'skii}}.
*{{citation |first=Bernard |last=Bolzano |authorlink=Bernard Bolzano |title=Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege |year=1817 |url=https://books.google.com/?id=EoW4AAAAIAAJ&dq=%22Rein%20analytischer%20Beweis%20des%20Lehrsatzes%22&pg=PA2-IA3#v=onepage&q= |publisher=Wilhelm Engelmann}} (''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'').
*{{citation |last=Borel |first=Émile |authorlink=Émile Borel |title=Sur quelques points de la théorie des fonctions |journal=[[Annales Scientifiques de l'École Normale Supérieure]]|series= 3 |volume=12 |year=1895 |pages=9–55 |jfm=26.0429.03 |doi=10.24033/asens.406 }}
*{{Citation | last1=Boyer | first1=Carl B. | author1-link=Carl Benjamin Boyer | title=The history of the calculus and its conceptual development | publisher=Dover Publications | location=New York | mr=0124178 | year=1959}}.
* {{citation |first=Cesare |last=Arzelà |authorlink=Cesare Arzelà |title=Sulle funzioni di linee |journal=Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. |volume=5 |issue=5 |pages=55–74 |year=1895}}.
* {{citation |first=Cesare |last=Arzelà |authorlink=Cesare Arzelà |title=Un'osservazione intorno alle serie di funzioni |journal=Rend. Dell' Accad. R. Delle Sci. Dell'Istituto di Bologna |pages=142–159 |year=1882–1883}}.
* {{citation |first=G. |last=Ascoli |authorlink=Giulio Ascoli |title=Le curve limiti di una varietà data di curve |journal=Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat. |volume=18 |issue=3 |pages=521–586 |year=1883–1884}}.
*{{citation | first=Maurice |last=Fréchet |authorlink=Maurice Fréchet |title=Sur quelques points du calcul fonctionnel |year=1906 |journal= [[Rendiconti del Circolo Matematico di Palermo]] |volume=22 |doi=10.1007/BF03018603 |pages=1–72 |issue=1|hdl=10338.dmlcz/100655 |url=https://zenodo.org/record/1428464/files/article.pdf }}.
*{{citation | last1=Gillman|first1=Leonard|last2=Jerison|first2=Meyer|title=Rings of continuous functions|publisher=Springer-Verlag|year=1976}}.
*{{citation | last=Kelley |first=John |title=General topology |publisher=Springer-Verlag |year=1955 |series=Graduate Texts in Mathematics |volume=27}}.
*{{Citation | last1=Kline | first1=Morris | author1-link=Morris Kline | title=Mathematical thought from ancient to modern times | year=1972 | publisher=Oxford University Press | edition=3rd | isbn=978-0-19-506136-9 | publication-date=1990 | url-access=registration | url=https://archive.org/details/mathematicalthou00klin }}.
*{{citation |first=Henri |last=Lebesgue |title=Leçons sur l'intégration et la recherche des fonctions primitives |url=https://books.google.com/?id=VfUKAAAAYAAJ&dq=%22Lebesgue%22%20%22Le%C3%A7ons%20sur%20l'int%C3%A9gration%20et%20la%20recherche%20des%20fonctions%20...%22&pg=PA1#v=onepage&q= |year=1904 |publisher=Gauthier-Villars}}.
*{{Citation | last1=Robinson | first1=Abraham | author1-link=Abraham Robinson | title=Non-standard analysis | publisher=Princeton University Press | isbn=978-0-691-04490-3 |mr=0205854 | year=1996}}.
*{{citation |first=C.T. |last= Scarborough |first2= A.H. |last2= Stone |title= Products of nearly compact spaces |journal= Transactions of the American Mathematical Society |volume= 124 |year=1966 |pages= 131–147 |doi=10.2307/1994440 |issue=1 |publisher=Transactions of the American Mathematical Society, Vol. 124, No. 1 |jstor=1994440|url=http://www.ams.org/tran/1966-124-01/S0002-9947-1966-0203679-7/S0002-9947-1966-0203679-7.pdf }}.
* {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | origyear=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover Publications reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995| title-link=Counterexamples in Topology }}
*{{Citation|last=Willard|first=Stephen|title=General Topology|publisher=Dover publications|year=1970|isbn=0-486-43479-6}}

==External links==
* {{PlanetMath|urlname=countablycompact|title=Countably compact}}
* {{cite arXiv |last=Sundström |first=Manya Raman |eprint=1006.4131v1 |title=A pedagogical history of compactness |class=math.HO |year=2010 |version=}}
----
{{PlanetMath attribution|urlname=examplesofcompactspaces|title=Examples of compact spaces}}

{{Topology}}

{{DEFAULTSORT:Compact Space}}
[[Category:Compactness (mathematics)]]
[[Category:General topology]]
[[Category:Properties of topological spaces]]
[[Category:Topology]]