Editing Compact space (section)

== 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}}.