View source for Compact space

{{redirect|Compactness}}
{{Short description|Type of mathematical space}}
[[File:Compact.svg|thumb|upright=1.6| Per the compactness criteria for Euclidean space as stated in the [[Heine–Borel theorem]], 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 (but bounded). The interval {{math|''B'' {{=}} [0, 1]}} is compact because it is both closed and bounded.]]
In [[mathematics]], specifically [[general topology]], '''compactness''' is a property that seeks to generalize the notion of a [[closed set|closed]] and [[bounded set|bounded]] subset of [[Euclidean space]].<ref>{{cite encyclopedia |title=Compactness |department=mathematics |encyclopedia=[[Encyclopaedia Britannica]] |lang=en |url=https://www.britannica.com/science/compactness |access-date=2019-11-25 |via=britannica.com}}</ref> The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all [[limit (mathematics)|''limiting values'']] of points. For example, the open [[interval (mathematics)|interval]] (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of [[rational number]]s <math>\mathbb{Q}</math> is not compact, because it has infinitely many "punctures" corresponding to the [[irrational number]]s, and the space of [[real number]]s <math>\mathbb{R}</math> is not compact either, because it excludes the two limiting values <math>+\infty</math> and <math>-\infty</math>. However, the [[extended real numbers|''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]], but may not be [[logical equivalence|equivalent]] in other [[topological space]]s.

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.<ref>{{cite book |last=Engelking |first=Ryszard |year=1977 |title=General Topology |publisher=PWN |place=Warsaw, PL |pages=266 |oclc= |url= }}</ref> 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 in the sequence {{nobr| {{sfrac|1|2}}, {{sfrac|4|5}}, {{sfrac|1|3}}, {{sfrac|5|6}}, {{sfrac|1|4}}, {{sfrac|6|7}}, ...}} accumulate to 0 (while others accumulate to 1). 
Since neither 0 nor 1 are members of the open unit interval {{math|(0, 1)}}, those same sets of points would not accumulate to any point of it, 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 <math>\mathbb{R}^1</math> (the real number line), the sequence of points {{nobr| 0,  1,  2,  3, ...}} 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 [[function space|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 [[Sequentially compact space|sequential compactness]] and [[limit point compact]]ness, were developed in general [[metric space]]s.<ref name=":0">{{cite web |title=Sequential compactness |series=MT&nbsp;4522 course lectures |volume=L22 |website=www-groups.mcs.st-andrews.ac.uk |url=http://www-groups.mcs.st-andrews.ac.uk/~john/MT4522/Lectures/L22.html |access-date=2019-11-25}}</ref> 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]]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]] – 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 [[#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 of a sequence|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|1990|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 space|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|1990|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]] – 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. {{lang|fr|italic=no|"Generalisation d'un theorem de Weierstrass"}}. {{lang|fr|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 {{mvar|X}} is a [[Hausdorff space]], and we have a point {{math|''x'' ∈ ''X''}} and a finite subset {{mvar|A}} of {{mvar|X}} not containing {{mvar|x}}. Then we can [[separated sets|separate]] {{mvar|x}} and {{mvar|A}} by [[neighborhood (topology)|neighborhoods]]: for each {{math|''a'' ∈ ''A''}}, let {{math|''U''(''x'')}} and {{math|''V''(''a'')}} be disjoint neighborhoods containing {{mvar|x}} and {{mvar|a}}, respectively. Then the intersection of all the {{math|''U''(''x'')}} and the union of all the {{math|''V''(''a'')}} are the required neighborhoods of {{mvar|x}} and {{mvar|A}}.

Note that if {{mvar|A}} is [[Infinity|infinite]], the proof fails, because the intersection of arbitrarily many neighborhoods of {{mvar|x}} might not be a neighborhood of {{mvar|x}}. 
The proof can be "rescued", however, if {{mvar|A}} is compact: we simply take a finite subcover of the cover {{math|{''V''(''a'') : ''a'' ∈ A}}} of {{mvar|A}}, then intersect the corresponding finitely many {{math|''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 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]] {{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]] among these points in that interval. For instance, the odd-numbered terms of the sequence {{nowrap|1, {{sfrac|1|2}}, {{sfrac|1|3}}, {{sfrac|3|4}}, {{sfrac|1|5}}, {{sfrac|5|6}}, {{sfrac|1|7}}, {{sfrac|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 of a sequence|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 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 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]] 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.<ref name=":0"/>

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 (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]] – in a [[neighbourhood (mathematics)|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]] {{mvar|X}} is called ''compact'' if every [[open cover]] of {{mvar|X}} has a [[finite set|finite]] [[subcover]].<ref>{{cite web |title=Compact Space |last=Weisstein |first=Eric W. |website=Wolfram MathWorld |lang=en |url=http://mathworld.wolfram.com/CompactSpace.html |access-date=2019-11-25}}</ref> That is, {{mvar|X}} is compact if for every collection {{mvar|C}} of open subsets<ref>Here, "collection" means "[[set (mathematics)|set]]" but is used because "collection of open subsets" is less awkward than "set of open subsets". Similarly, "subcollection" means "subset".</ref> of {{mvar|X}} such that

<math display="block">X = \bigcup_{S \in C}S\ ,</math>

there is a '''finite''' subcollection {{mvar|F}} ⊆ {{mvar|C}} such that

<math display="block">X = \bigcup_{S \in F} S\ .</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 display="block">K \subseteq \bigcup_{S \in C} S\ ,</math>

there is a '''finite''' subcollection {{mvar|F}} ⊆ {{mvar|C}} such that

<math display="block">K \subseteq \bigcup_{S \in F} S\ .</math>

Because compactness is a [[topological property]], the compactness of a subset depends only on the subspace topology induced on it.  It follows that, 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}}.

=== Characterization ===

If {{mvar|X}} is a topological space then the following are equivalent:
# {{mvar|X}} is compact; i.e., every [[open cover]] of {{mvar|X}} has a finite [[subcover]].
# {{mvar|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]]).
# {{mvar|X}} is [[Lindelöf space|Lindelöf]] and [[countably compact]].{{sfn | Howes | 1995 | pp=xxvi-xxviii}}
# Any collection of closed subsets of {{mvar|X}} with the [[finite intersection property]] has nonempty intersection.
# Every [[net (mathematics)|net]] on {{mvar|X}} has a convergent subnet (see the article on [[net (mathematics)|nets]] for a proof).
# Every [[Filters in topology|filter]] on {{mvar|X}} has a convergent refinement.
# Every net on {{mvar|X}} has a cluster point.
# Every filter on {{mvar|X}} has a cluster point.
# Every [[ultrafilter (set theory)|ultrafilter]] on {{mvar|X}} converges to at least one point.
# Every infinite subset of {{mvar|X}} has a [[complete accumulation point]].<ref>{{harvnb|Kelley|1955|p=163}}</ref>
# For every topological space {{mvar|Y}}, the projection <math>X \times Y \to Y</math> is a [[closed mapping]]<ref name="Bourbaki">{{harvnb|Bourbaki|2007|loc=§ 10.2. Theorem&nbsp;1, Corollary&nbsp;1.}}</ref> (see [[proper map]]).
# Every open cover linearly ordered by subset inclusion contains {{mvar|X}}.{{sfn|Mack|1967}}

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).<ref name="BourbakiDefinition">{{harvnb|Bourbaki|2007|loc=§&nbsp;9.1. Definition&nbsp;1.}}</ref>
 
==== Euclidean space ====

For any [[subset]] {{mvar|A}} of [[Euclidean space]], {{mvar|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 {{mvar|n}}-ball.

==== Metric spaces ====

For any metric space {{math|(''X'', ''d'')}}, the following are equivalent (assuming [[countable choice]]):
# {{math|(''X'', ''d'')}} is compact.
# {{math|(''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&nbsp;5.3.7}}</ref>
# {{math|(''X'', ''d'')}} is sequentially compact; that is, every [[sequence]] in {{mvar|X}} has a convergent subsequence whose limit is in {{mvar|X}} (this is also equivalent to compactness for [[first-countable]] [[uniform space]]s).
# {{math|(''X'', ''d'')}} is [[limit point compact]] (also called weakly countably compact); that is, every infinite subset of {{mvar|X}} has at least one [[Limit point of a set|limit point]] in {{mvar|X}}.
# {{math|(''X'', ''d'')}} is [[countably compact]]; that is, every countable open cover of {{mvar|X}} has a finite subcover.
# {{math|(''X'', ''d'')}} is an image of a continuous function from the [[Cantor set]].<ref>{{harvnb|Willard|1970}} Theorem&nbsp;30.7.</ref>
# Every decreasing nested sequence of nonempty closed subsets {{math|''S''<sub>1</sub> ⊇ ''S''<sub>2</sub> ⊇ ...}} in {{math|(''X'', ''d'')}} has a nonempty intersection.
# Every increasing nested sequence of proper open subsets {{math|''S''<sub>1</sub> ⊆ ''S''<sub>2</sub> ⊆ ...}} in {{math|(''X'', ''d'')}} fails to cover {{mvar|X}}.

A compact metric space {{math|(''X'', ''d'')}} also satisfies the following properties:
# [[Lebesgue's number lemma]]: For every open cover of {{mvar|X}}, there exists a number {{nowrap|''δ'' > 0}} such that every subset of {{mvar|X}} of diameter < {{mvar|δ}} is contained in some member of the cover.
# {{math|(''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.
# {{mvar|X}} is closed and bounded (as a subset of any metric space whose restricted metric is {{mvar|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.

==== Ordered spaces ====

For an ordered space {{math|(''X'', <)}} (i.e. a totally ordered set equipped with the order topology), the following are equivalent:
# {{math|(''X'', <)}} is compact.
# Every subset of {{mvar|X}} has a supremum (i.e. a least upper bound) in {{mvar|X}}.
# Every subset of {{mvar|X}} has an infimum (i.e. a greatest lower bound) in {{mvar|X}}.
# Every nonempty closed subset of {{mvar|X}} 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 {{math|(''X'', <)}}, and (assuming [[countable choice]]) are true whenever {{math|(''X'', <)}} is compact. (The converse in general fails if {{math|(''X'', <)}} is not also metrizable.):
# Every sequence in {{math|(''X'', <)}} has a subsequence that converges in {{math|(''X'', <)}}.
# Every monotone increasing sequence in {{mvar|X}} converges to a unique limit in {{mvar|X}}.
# Every monotone decreasing sequence in {{mvar|X}} converges to a unique limit in {{mvar|X}}.
# Every decreasing nested sequence of nonempty closed subsets {{mvar|S}}<sub>1</sub> ⊇ {{mvar|S}}<sub>2</sub> ⊇ ... in {{math|(''X'', <)}} has a nonempty intersection.
# Every increasing nested sequence of proper open subsets {{mvar|S}}<sub>1</sub> ⊆ {{mvar|S}}<sub>2</sub> ⊆ ... in {{math|(''X'', <)}} fails to cover {{mvar|X}}.

==== Characterization by continuous functions ====

Let {{mvar|X}} be a topological space and {{math|C(''X'')}} the ring of real continuous functions on {{mvar|X}}. 
For each {{math|''p'' ∈ ''X''}}, the evaluation map <math>\operatorname{ev}_p\colon C(X)\to \mathbb{R}</math>
given by {{math|1=ev<sub>''p''</sub>(''f'') = ''f''(''p'')}} is a ring homomorphism. 
The [[kernel (algebra)|kernel]] of {{math|ev<sub>''p''</sub>}} is a [[maximal ideal]], since the [[residue field]] {{nowrap|{{math|C(''X'')/ker ev<sub>''p''</sub>}}}} is the field of real numbers, by the [[first isomorphism theorem]]. A topological space {{mvar|X}} is [[pseudocompact space|pseudocompact]] if and only if every maximal ideal in {{math|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 {{mvar|m}} in {{math|C(''X'')}} such that the residue field {{math|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|1996|loc=Theorem&nbsp;4.1.13}}</ref> a topological space {{mvar|X}} is compact if and only if every point {{mvar|x}} of the natural extension {{math|''*X''}} is [[infinitesimal|infinitely close]] to a point {{math|''x''<sub>0</sub>}} of {{mvar|X}} (more precisely, {{mvar|x}} is contained in the [[monad (non-standard analysis)|monad]] of {{math|''x''<sub>0</sub>}}).

==== Hyperreal definition ====

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

== Sufficient conditions ==

* A closed subset of a compact space is compact.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.2.3}}</ref>
* A finite [[Union (set theory)|union]] of compact sets is compact.
* A [[continuous function (topology)|continuous]] image of a compact space is compact.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.2.2}}</ref>
* The intersection of any non-empty collection of compact subsets of a Hausdorff space is compact (and closed);
** If {{mvar|X}} is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).{{efn|
Let {{math|1=''X'' = {''a'', ''b''} &cup; <math>\mathbb{N}</math>}}, {{math|1=''U'' = {''a''} &cup; <math>\mathbb{N}</math>}}, and {{math|1=''V'' = {''b''} &cup; <math>\mathbb{N}</math>}}. Endow {{math|X}} with the topology generated by the following basic open sets: every subset of <math>\mathbb{N}</math> is open; the only open sets containing {{mvar|a}} are {{mvar|X}} and {{mvar|U}}; and the only open sets containing {{mvar|b}} are {{mvar|X}} and {{mvar|V}}. Then {{mvar|U}} and {{mvar|V}} are both compact subsets but their intersection, which is <math>\mathbb{N}</math>, is not compact. Note that both {{mvar|U}} and {{mvar|V}} are compact open subsets, neither one of which is closed.
}}
* The [[product topology|product]] of any collection of compact spaces is compact. (This is [[Tychonoff's theorem]], which is equivalent to the [[axiom of choice]].)
* In a [[metrizable space]], a subset is compact if and only if it is [[sequentially compact]] (assuming [[axiom of countable choice|countable choice]])
* A finite set endowed with any topology is compact.

== Properties of compact spaces ==

* A compact subset of a [[Hausdorff space]] {{mvar|X}} is closed. 
** If {{mvar|X}} is not Hausdorff then a compact subset of {{mvar|X}} may fail to be a closed subset of {{mvar|X}} (see footnote for example).{{efn|
Let {{math|1=''X'' = {''a'', ''b''}<!---->}} and endow {{mvar|X}} with the topology {{math|{''X'', ∅, {''a''}<!---->}<!---->}}. Then {{math|{''a''}<!---->}} is a compact set but it is not closed.
}}
** If {{mvar|X}} is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).{{efn|
Let {{mvar|X}} be the set of non-negative integers. We endow {{mvar|X}} with the [[particular point topology]] by defining a subset {{math|''U'' ⊆ ''X''}} to be open if and only if {{math|0 ∈ ''U''}}. Then {{math|1=''S'' := {0}<!---->}} is compact, the closure of {{mvar|S}} is all of {{mvar|X}}, but {{mvar|X}} is not compact since the collection of open subsets {{math|{<!---->{0, ''x''} : ''x'' ∈ ''X''}<!---->}} does not have a finite subcover.
}}
* In any [[topological vector space]] (TVS), a compact subset is [[complete space|complete]]. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are ''not'' closed.
* If {{mvar|A}} and {{mvar|B}} are disjoint compact subsets of a Hausdorff space {{mvar|X}}, then there exist disjoint open sets {{mvar|U}} and {{mvar|V}} in {{mvar|X}} such that {{math|''A'' ⊆ ''U''}} and {{math|''B'' ⊆ ''V''}}.
* A continuous bijection from a compact space into a Hausdorff space is a [[homeomorphism]].
* A compact Hausdorff space is [[Normal space|normal]] and [[Regular space|regular]].
* If a space {{mvar|X}} is compact and Hausdorff, then no finer topology on {{mvar|X}} is compact and no coarser topology on {{mvar|X}} is Hausdorff.
* If a subset of a metric space {{math|(''X'', ''d'')}} is compact then it is {{mvar|d}}-bounded.

=== Functions and compact spaces ===

Since a [[continuous function (topology)|continuous]] image of a compact space is compact, the [[extreme value theorem]] holds for such spaces: 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.

=== Compactifications ===
{{Main|Compactification (mathematics)}}

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

=== Ordered compact spaces ===

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

Let {{mvar|X}} be a [[total order|simply ordered]] set endowed with the [[order topology]]. 
Then {{mvar|X}} is compact if and only if {{mvar|X}} is a [[complete lattice]] (i.e. all subsets have suprema and infima).<ref>{{harvnb|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 <math>\mathbb{R}</math> is homeomorphic to the circle {{math|'''S'''<sup>1</sup>}}; the one-point compactification of <math>\mathbb{R}^2</math> is homeomorphic to the sphere {{math|'''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 <math>\mathbb{R}</math> 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]] {{math|{{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 display="inline">\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 display="inline">\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;<math>\mathbb{R}</math>.
* The set <math>\mathbb{R}</math> 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 {{math|{{open-open|{{mvar|n}} − 1, {{mvar|n}} + 1}} }}, where {{mvar|n}} takes all integer values in {{math|'''Z'''}}, cover <math>\mathbb{R}</math> 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 and thus would ''not'' cover the extended real line.  In fact, the set has the [[homeomorphism]] to [−1,&nbsp;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 {{mvar|K}} of all functions {{math|''f'' : <math>\mathbb{R}</math> → [0, 1]}} from the real number line to the closed unit interval, and define a topology on {{mvar|K}} so that a sequence <math>\{f_n\}</math> in {{mvar|K}} converges towards {{math|''f'' ∈ ''K''}} if and only if <math>\{f_n(x)\}</math> converges towards {{math|''f''(''x'')}} for all real numbers {{mvar|x}}. There is only one such topology; it is called the topology of [[pointwise convergence]] or the [[product topology]]. Then {{mvar|K}} is a compact topological space; this follows from the [[Tychonoff theorem]].
* A subset of the Banach space of real-valued continuous functions on a compact Hausdorff space is relatively compact if and only if it is equicontinuous and pointwise bounded ([[Arzelà–Ascoli theorem]]).
* Consider the set {{mvar|K}} of all functions {{math|''f''&nbsp;: {{closed-closed|0, 1}}&nbsp;→ {{closed-closed|0, 1}}}} satisfying the [[Lipschitz condition]] {{math|{{mabs|''f''(''x'')&nbsp;−&nbsp;''f''(''y'')}}&nbsp;≤ {{mabs|''x''&nbsp;−&nbsp;''y''}}}} for all {{math|''x'',&nbsp;''y''&nbsp;∈&nbsp;{{closed-closed|0,1}}}}.  Consider on {{mvar|K}} 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 the Arzelà–Ascoli theorem the space {{mvar|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 <math>\mathbb{C}</math>. Conversely, any compact subset of <math>\mathbb{C}</math> 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 <math>\mathbb{C}</math> as spectrum.
* The space of Borel [[probability measure]]s on a compact Hausdorff space is compact for the [[vague topology]], by the Alaoglu theorem.
* A collection of probability measures on the Borel sets of Euclidean space is called ''tight'' if, for any positive epsilon, there exists a compact subset containing all but at most epsilon of the mass of each of the measures.  Helly's theorem then asserts that a collection of probability measures is relatively compact for the vague topology if and only if it is tight.

=== Algebraic examples ===

* [[Topological group]]s such as an [[orthogonal group]] are compact, while groups such as a [[general linear group]] are not.
* Since the [[p-adic numbers|{{mvar|p}}-adic integers]] are [[homeomorphic]] to the Cantor set, they form a compact set.
* Any [[global field]] ''K'' is a discrete additive subgroup of its [[adele ring]], and the quotient space is compact.  This was used in [[John Tate (mathematician)|John Tate]]'s [[Tate's thesis|thesis]] to allow [[harmonic analysis]] to be used in [[number theory]].
* 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]] 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 ==
{{Div col|colwidth=16em}}
* [[Compactly generated space]]
* [[Compactness theorem]]
* [[Eberlein compactum]]
* [[Exhaustion by compact sets]]
* [[Lindelöf space]]
* [[Metacompact space]]
* [[Noetherian topological space]]
* [[Orthocompact space]]
* [[Paracompact space]]
* [[Quasi-compact morphism]]
* [[Totally bounded space|Precompact set]] - also called ''[[totally bounded]]''
* [[Relatively compact subspace]]
* [[Totally bounded]]
{{Div col end}}

== Notes==
{{notelist}}

== References ==
{{reflist|25em}}

== Bibliography ==
{{refbegin|colwidth=25em}}
*{{cite journal |last1=Alexandrov |first1=Pavel |author-link1=Pavel Alexandrov |last2=Urysohn |first2=Pavel |author-link2=Pavel Urysohn |year=1929 |title=Mémoire sur les espaces topologiques compacts |journal=Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Mathematical Sciences |volume=14}} 
*{{cite book |last1=Arkhangel'skii |first1=A.V. |last2=Fedorchuk |first2=V.V. |year=1990 |contribution=The basic concepts and constructions of general topology |editor1=Arkhangel'skii, A.V. |editor2=Pontrjagin, L.S. |title=General Topology I |publisher=Springer |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}}.
*{{cite book |first=Bernard |last=Bolzano |author-link=Bernard Bolzano |year=1817 |title=Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege |url=https://books.google.com/books?id=EoW4AAAAIAAJ&q=%22Rein%20analytischer%20Beweis%20des%20Lehrsatzes%22&pg=PA2-IA3 |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'').
*{{cite journal |last=Borel |first=Émile |author-link=Émile Borel |year=1895 |title=Sur quelques points de la théorie des fonctions |journal=[[Annales Scientifiques de l'École Normale Supérieure]] |series= 3 |volume=12 |pages=9–55 |jfm=26.0429.03 |doi=10.24033/asens.406 |doi-access=free }}
*{{cite book |last1=Bourbaki |first1=Nicolas |title=Topologie générale. Chapitres 1 à 4 |date=2007 |publisher=Springer |location=Berlin |doi=10.1007/978-3-540-33982-3 |isbn=978-3-540-33982-3 |url=https://link.springer.com/book/10.1007/978-3-540-33982-3}}
*{{cite book | last=Boyer | first=Carl B. | author-link=Carl Benjamin Boyer | year=1959 | title=The history of the calculus and its conceptual development | publisher=Dover Publications | location=New York | mr=0124178}}
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* {{cite journal |first=Cesare |last=Arzelà |author-link=Cesare Arzelà |year=1895 |title=Sulle funzioni di linee |journal=Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. |volume=5 |issue=5 |pages=55–74}}
* {{cite journal |last=Arzelà |first=Cesare |author-link=Cesare Arzelà |year=1882–1883 |title=Un'osservazione intorno alle serie di funzioni |journal=Rend. Dell' Accad. R. Delle Sci. dell'Istituto di Bologna |pages=142–159}}
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*{{cite journal |last=Fréchet | first=Maurice |author-link=Maurice Fréchet |year=1906 |title=Sur quelques points du calcul fonctionnel |journal=[[Rendiconti del Circolo Matematico di Palermo]] |volume=22 |issue=1 |pages=1–72 |doi=10.1007/BF03018603 |hdl=10338.dmlcz/100655 |s2cid=123251660 |url=https://zenodo.org/record/1428464 }}
*{{cite book | last1=Gillman|first1=Leonard |last2=Jerison |first2=Meyer |year=1976 |title=Rings of continuous functions |publisher=Springer-Verlag}}
* {{Howes Modern Analysis and Topology 1995}} <!-- {{sfn | Howes | 1995 | p=}} -->
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*{{cite book | last1=Kline | first1=Morris | author1-link=Morris Kline |orig-year=1972 |year=1990 | title=Mathematical thought from ancient to modern times | publisher=Oxford University Press | edition=3rd | isbn=978-0-19-506136-9 | url-access=registration | url=https://archive.org/details/mathematicalthou00klin }} 
*{{cite book |first=Henri |last=Lebesgue |year=1904 |title=Leçons sur l'intégration et la recherche des fonctions primitives |publisher=Gauthier-Villars |url=https://books.google.com/books?id=VfUKAAAAYAAJ&q=%22Lebesgue%22%20%22Le%C3%A7ons%20sur%20l'int%C3%A9gration%20et%20la%20recherche%20des%20fonctions%20...%22&pg=PA1}} 
*{{cite journal
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*{{cite book | last1=Robinson | first1=Abraham | author1-link=Abraham Robinson | year=1996 | title=Non-standard analysis | publisher=Princeton University Press | isbn=978-0-691-04490-3 |mr=0205854}} 
* {{cite journal |first1=C.T. |last1= Scarborough |first2= A.H. |last2= Stone |year=1966 |title= Products of nearly compact spaces |journal= Transactions of the American Mathematical Society |volume= 124 |issue=1 |pages= 131–147 |doi=10.2307/1994440 |jstor=1994440 |url=https://www.ams.org/tran/1966-124-01/S0002-9947-1966-0203679-7/S0002-9947-1966-0203679-7.pdf |archive-url=https://web.archive.org/web/20170816203049/http://www.ams.org/tran/1966-124-01/S0002-9947-1966-0203679-7/S0002-9947-1966-0203679-7.pdf |archive-date=2017-08-16 |url-status=live |doi-access=free }}.
* {{cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | year=1995 | title=Counterexamples in Topology | orig-year=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover Publications reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | title-link=Counterexamples in Topology }}
* {{cite book |last=Willard |first=Stephen |year=1970 |title=General Topology |publisher=Dover publications |isbn=0-486-43479-6}}
{{refend}}

== External links ==

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

{{Topology}}

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