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{{Short description|Type of mathematical space}} | {{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.]] | [[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]], | In [[mathematics]], especially [[general topology]] and [[analysis]], '''compactness''' is a property of a space that makes it behave in many ways like a [[finite set]].<ref>{{cite encyclopedia | ||
|last=Tao | |||
|first=Terence | |||
|author-link=Terence Tao | |||
|editor-last=Gowers | |||
|editor-first=Timothy | |||
|editor-link=Timothy Gowers | |||
|title=Compactness and compactification | |||
|encyclopedia=The Princeton Companion to Mathematics | |||
|publisher=Princeton University Press | |||
|year=2008 | |||
|pages=169–170 | |||
|isbn=978-0-691-11880-2 | |||
}}</ref> For instance, on a finite set every infinite sequence must take some value infinitely often, by the [[pigeonhole principle]]. For subsets of [[Euclidean space]], the analogous statement is [[sequential compactness]]: a set is compact if and only if every [[infinite sequence]] in the set has a [[subsequence]] that converges to a point of the set. Likewise, whereas every real-valued function on a finite set is bounded and attains its maximum and minimum, every [[continuous function|continuous]] real-valued function on a compact space has these properties. For compact subsets of Euclidean space, this is the [[extreme value theorem]]. | |||
Another basic property of finite sets is that every cover of a finite set by subsets has a finite subcover: one may choose, for each point of the finite set, a member of the cover containing it. The corresponding topological property is used to define compactness: a [[topological space]] is compact if every [[open cover]] has a finite subcover. In metric spaces this is equivalent to several other formulations, including [[sequential compactness]], though these equivalences can fail in more general topological spaces. Thus every sequence in the closed unit interval {{math|[0,1]}} has a convergent subsequence with limit in {{math|[0,1]}}, whereas this fails for spaces such as the open interval {{math|(0,1)}} and the [[real line]]. For subsets of Euclidean space, compactness is equivalent to being [[closed set|closed]] and [[bounded set|bounded]], by the [[Heine–Borel theorem]]. The property of compactness often allows local information to be combined into global conclusions. The term '''compact set''' may refer either to a compact topological space or, more commonly, to a subset of a topological space that is compact in the subspace topology. | |||
Compactness was formally introduced by [[Maurice Fréchet]] in 1906 | Compactness was formally introduced by [[Maurice Fréchet]] in 1906 in work generalizing the Bolzano–Weierstrass theorem from sets of points to spaces of functions. Later, [[Pavel Alexandrov]] and [[Pavel Urysohn]] developed the open-cover formulation that is now standard in topology. Compactness plays a central role throughout mathematics; for example, continuous real-valued functions on compact spaces attain maxima and minima, and major results such as the [[Arzelà–Ascoli theorem]] and the [[Peano existence theorem]] depend on compactness. | ||
== Historical development == | == Historical development == | ||
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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> | 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. | 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 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]]. | 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]]. | ||
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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 0, while the even-numbered ones get arbitrarily close to 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. | 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 0, while the even-numbered ones get arbitrarily close to 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. | 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. Likewise, a circle in the plane is compact (easily seen by its being closed and bounded). 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 == | == Definitions == | ||
Various definitions of compactness may apply, depending on the level of generality. | Various definitions of compactness may apply, depending on the level of generality. | ||
A subset of [[Euclidean space]] in particular is | A subset of [[Euclidean space]] in particular is compact if and only 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">{{cite web |title=Sequential compactness |series=MT 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 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). | 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). | ||
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=== Compactness of subsets === | === Compactness of subsets === | ||
A subset {{mvar|K}} of a topological space {{mvar|X | A subset {{mvar|K}} of a topological space {{mvar|X}} 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> | <math display="block">K \subseteq \bigcup_{S \in C} S\ ,</math> | ||
there is a | there is a finite subcollection {{mvar|F}} ⊆ {{mvar|C}} such that | ||
<math display="block">K \subseteq \bigcup_{S \in F} S\ .</math> | <math display="block">K \subseteq \bigcup_{S \in F} S\ .</math> | ||
Equivalently, {{mvar|K}} is compact as a subset of {{mvar|X}} if and only if the topological space {{mvar|K}} is compact in the [[subspace topology]]. In particular, if <math>K \subset Y \subset X</math>, with subset {{mvar|Y}} equipped with the subspace topology, then {{mvar|K}} is compact in {{mvar|Y}} if and only if {{mvar|K}} is compact in {{mvar|X}}. Furthermore, the compactness of {{mvar|K}} as a subset of a topological space {{mvar|X}} is independent of the embedding, provided that the subspace topology on {{mvar|K}} is the same. | |||
=== Characterization === | === Characterization === | ||
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# {{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 [[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 [[countably compact]]; that is, every countable open cover of {{mvar|X}} has a finite subcover. | ||
# {{math|(''X'', ''d'')}} is | # {{mvar|X}} is empty or {{math|(''X'', ''d'')}} is the image of a continuous function from the [[Cantor set]].<ref>{{harvnb|Willard|1970}} Theorem 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 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}}. | # 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}}. | ||
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An ordered space satisfying (any one of) these conditions is called a complete lattice. | 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 | 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 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 increasing sequence in {{mvar|X}} converges to a unique limit in {{mvar|X}}. | ||
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==== Characterization by continuous functions ==== | ==== Characterization by continuous functions ==== | ||
Let {{mvar|X}} be a | Let {{mvar|X}} be a [[completely regular space|completely regular]] Hausdorff space and {{math|C(''X'')}} the ring of real-valued continuous functions on {{mvar|X}}. | ||
For each {{math|''p'' ∈ ''X''}}, the evaluation map <math>\operatorname{ev}_p\colon C(X)\to \mathbb{R}</math> | For each {{math|''p'' ∈ ''X''}}, the evaluation map | ||
given by { | <math>\operatorname{ev}_p\colon C(X)\to \mathbb{R}</math> | ||
given by <math>\operatorname{ev}_p(f)=f(p)</math> | |||
is a [[ring homomorphism]]. | |||
Its [[kernel (algebra)|kernel]] | |||
<math>M_p=\ker(\operatorname{ev}_p)</math> | |||
is a [[maximal ideal]], since by the [[first isomorphism theorem]] | |||
<math display="block">C(X)/M_p \cong \mathbb{R}.</math> | |||
For a completely regular Hausdorff space {{mvar|X}}, {{mvar|X}} is [[pseudocompact space|pseudocompact]] if and only if every maximal ideal {{mvar|M}} of {{math|C(''X'')}} is ''real'', meaning that its residue field {{math|C(''X'')/''M''}} is isomorphic to <math>\mathbb{R}.</math> Moreover, {{mvar|X}} is [[realcompact space|realcompact]] if and only if every real maximal ideal is of the form {{math|''M''<sub>''p''</sub>}} for some {{math|''p'' ∈ ''X''}}. Consequently, {{mvar|X}} is compact if and only if every maximal ideal of {{math|C(''X'')}} is the kernel of an evaluation homomorphism.<ref>{{harvnb|Gillman|Jerison|1976|loc=§5.6}}</ref> | |||
If {{mvar|X}} is not pseudocompact, then {{math|C(''X'')}} has maximal ideals whose residue fields are proper ordered field extensions of <math>\mathbb{R}</math>, often called ''hyperreal'' fields. In the framework of [[non-standard analysis]], this corresponds to the following characterization of compactness:<ref>{{harvnb|Robinson|1996|loc=Theorem 4.1.13}}</ref> a topological space {{mvar|X}} is compact if and only if every point of the natural extension {{math|''*X''}} is [[infinitesimal|infinitely close]] to some point of {{mvar|X}} (that is, lies in the [[monad (non-standard analysis)|monad]] of a point of {{mvar|X}}). | |||
==== Hyperreal definition ==== | ==== Hyperreal definition ==== | ||
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* A closed subset of a compact space is compact.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.2.3}}</ref> | * A closed subset of a compact space is compact.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.2.3}}</ref> | ||
* | * The [[Union (set theory)|union]] of finitely many compact sets is compact. | ||
* | * The image of a compact space under a [[continuous function (topology)|continuous function]] 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) | * 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 | ** If {{mvar|X}} is not Hausdorff, then the intersection of two compact subsets may fail to be compact.{{efn| | ||
Let {{math|1=''X'' = {''a'', ''b''} ∪ <math>\mathbb{N}</math>}}, {{math|1=''U'' = {''a''} ∪ <math>\mathbb{N}</math>}}, and {{math|1=''V'' = {''b''} ∪ <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. | Let {{math|1=''X'' = {''a'', ''b''} ∪ <math>\mathbb{N}</math>}}, {{math|1=''U'' = {''a''} ∪ <math>\mathbb{N}</math>}}, and {{math|1=''V'' = {''b''} ∪ <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]].) | * 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]]) | * 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. | * A finite set endowed with any topology is compact. | ||
== Properties of compact spaces == | == Properties of compact spaces == | ||
* A compact subset of a [[Hausdorff space]] {{mvar|X}} is closed. | * A space in which every compact subset is closed is called a [[KC space]]. | ||
** If {{mvar|X}} is not Hausdorff then a compact subset of {{mvar|X}} may fail to be a closed subset of {{mvar|X}} | ** A compact subset of a [[Hausdorff space]] {{mvar|X}} is closed. | ||
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 a compact subset of {{mvar|X}} may fail to be a closed subset of {{mvar|X}}.{{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.}}{{efn|Any infinite proper subset of a space with the [[cofinite topology]] is compact but not closed.}} | ||
** If {{mvar|X}} is not Hausdorff then the closure of a compact set may fail to be compact | ** If {{mvar|X}} is not Hausdorff, then the closure of a compact set may fail to be compact.{{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. | 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. | ||
}} | }} | ||
** If {{mvar|X}} is not Hausdorff, then it can still be the case that every compact subset is closed.{{efn|For the [[cocountable topology]] on a space with [[uncountable set|uncountably]] many points, which is not Hausdorff, a subset is compact if and only if it is finite, and all the finite subsets are closed.}} | |||
* 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. | * 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''}}. | * 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 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]]. | * 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 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. | * If a subset of a metric space {{math|(''X'', ''d'')}} is compact, then it is {{mvar|d}}-bounded. | ||
=== Functions and compact spaces === | === Functions and compact spaces === | ||
Since a [[continuous function (topology)|continuous]] | Since the image of a compact space under a [[continuous function (topology)|continuous]] function 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. | (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. | ||
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A nonempty compact subset of the [[real number]]s has a greatest element and a least element. | A nonempty compact subset of the [[real number]]s has a greatest element and a least element. | ||
Let {{mvar|X}} be a [[total order| | Let {{mvar|X}} be a [[total order|totally 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> | 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 == | == Examples == | ||
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* 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. | * 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]]) | * 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. | * The [[Cantor set]] is compact. In fact, every non-empty compact metric space is a continuous image of the Cantor set. | ||
* Consider the set {{mvar|K}} of all functions {{math|''f'' : '''R''' → [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}}. | * Consider the set {{mvar|K}} of all functions {{math|''f'' : '''R''' → [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}}. The coarsest such topology, sometimes called the topology of [[pointwise convergence]], is the [[product topology]]. With this topology, {{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]]). | * 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'' : {{closed-closed|0, 1}} → {{closed-closed|0, 1}}}} satisfying the [[Lipschitz condition]] {{math|{{mabs|''f''(''x'') − ''f''(''y'')}} ≤ {{mabs|''x'' − ''y''}}}} for all {{math|''x'', ''y'' ∈ {{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. | * Consider the set {{mvar|K}} of all functions {{math|''f'' : {{closed-closed|0, 1}} → {{closed-closed|0, 1}}}} satisfying the [[Lipschitz condition]] {{math|{{mabs|''f''(''x'') − ''f''(''y'')}} ≤ {{mabs|''x'' − ''y''}}}} for all {{math|''x'', ''y'' ∈ {{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 [[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. | * 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. | * 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. [[Prokhorov'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 === | === Algebraic examples === | ||
* [[ | * Every [[semisimple Lie algebra|semisimple Lie group]] has a [[compact real form]], which is a compact [[topological group]]; an example is the [[orthogonal group]] of a positive-definite quadratic form. They also have non-compact real forms, such as the [[special linear group]] or the [[Lorentz group]]. | ||
* Since the [[p-adic numbers|{{mvar|p}}-adic integers]] are [[homeomorphic]] to the Cantor set, they form a compact set. | * 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]]. | * 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]]. | ||
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* [[Orthocompact space]] | * [[Orthocompact space]] | ||
* [[Paracompact space]] | * [[Paracompact space]] | ||
* [[Totally bounded space|Precompact set]] - also called ''[[totally bounded]]'' | |||
* [[Quasi-compact morphism]] | * [[Quasi-compact morphism]] | ||
* [[Relatively compact subspace]] | * [[Relatively compact subspace]] | ||
* [[Totally bounded]] | * [[Totally bounded]] | ||
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{{refbegin|colwidth=25em}} | {{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 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}}. | *{{cite book |last1=Arkhangel'skii |first1=A.V. |author-link=Alexander Arhangelskii |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. |editor2-link=Lev Pontryagin |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}}. | *{{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 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''). | ||
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[[Category:General topology]] | [[Category:General topology]] | ||
[[Category:Properties of topological spaces]] | [[Category:Properties of topological spaces]] | ||