compact hyperkähler manifold

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## Idea

A [[compact topological space|compact]] (i.e. [[closed manifold|closed]]) [[hyperkähler manifold]].

## Examples

### Hilbert schemes on $K3$ and $\mathbb{T}^4$

The only known examples of [[compact hyperkähler manifolds]] are [[Hilbert schemes of points]] $X^{[n+1]}$ (for $n \in \mathbb{N}$) for $X$ either

1. a [[K3-surface]]

1. a [[4-torus]] (in which case the [[compact hyperkähler manifolds]] is really the [[fiber]] of $(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4$)

([Beauville 83](#Beauville83))
and two exceptional examples ([O'Grady 99](#OGrady99), [O'Grady 03](#OGrady03)
), see [Sawon 04, Sec. 5.3](#Sawon04).

### Coulomb- and Higgs-branches of $D=3$ $\mathcal{N} =4 $ SYM

Both the [[Coulomb branch]] and the [[Higgs branch]] of [[D=3 N=4 super Yang-Mills theories]] are [[hyperkähler manifolds]] ([Seiberg-Witten 96](D%3D3+N%3D4+super+Yang-Mills+theory#SeibergWitten96), see e.g. [dBHOO 96](D=3+N=4+super+Yang-Mills+theory#dBHOO96)).
In special cases they are [[compact hyperkähler manifolds]]
([Intriligator 99](#Intriligator99)).

## Properties

### Relation to Rozansky-Witten weight systems

In order for [[Rozansky-Witten weight systems]] to take values in the [[ground field]], hence to be actual [[weight systems]], the [[hyperkähler manifold]] has to be [[compact topological space|compact]] (i.e. [[closed manifold|closed]]).

## References

### General

* {#Beauville83} [[Arnaud Beauville]], _Variétés Kähleriennes dont la premiere classe de Chern est nulle_, Jour.
Diff. Geom. 18 (1983), 755–782 ([euclid.jdg/1214438181](https://projecteuclid.org/euclid.jdg/1214438181))

* {#OGrady99} [[Kieran O’Grady]], _Desingularized moduli spaces of sheaves on a K3_, J. Reine Angew. Math. 512 (1999), 49–117 ([arXiv:alg-geom/9708009](https://arxiv.org/abs/alg-geom/9708009), [arXiv:math/9805099](https://arxiv.org/abs/math/9805099))

* {#OGrady03} [[Kieran O’Grady]], _A new six dimensional irreducible symplectic variety_, J. Algebraic Geom. 12 (2003), 435-505 ([arXiv:math/0010187](https://arxiv.org/abs/math/0010187))

* [[Daniel Huybrechts]], _Compact Hyperkähler Manifolds_, In: Ellingsrud G., Ranestad K., Olson L., Strømme S.A. (eds.) _Calabi-Yau Manifolds and Related Geometries_, Universitext. Springer, Berlin, Heidelberg 2003 ([doi:10.1007/978-3-642-19004-9_3](https://doi.org/10.1007/978-3-642-19004-9_3),
[arXiv:alg-geom/9705025](https://arxiv.org/abs/alg-geom/9705025))

### Examples from Coulomb branches

On [[D=3 N=4 super Yang-Mills theories]] with [[compact hyperkähler manifold]] [[Coulomb branches]] obtained by [[KK-compactification]] of [[little string theories]]:

* {#Intriligator99} [[Kenneth Intriligator]], _Compactified Little String Theories and Compact Moduli Spaces of Vacua_, Phys. Rev. D61:106005, 2000 ([arXiv:hep-th/9909219](https://arxiv.org/abs/hep-th/9909219))

### In Rozansky-Witten theory

With an eye towards [[Rozansky-Witten theory]] ([[ground field]]-valued [[Rozansky-Witten weight systems]]):

* {#RobertsWillerton06} [[Justin Roberts]], [[Simon Willerton]], p. 17 of: _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

* {#Sawon04} [[Justin Sawon]], Section 5.3 of: _Rozansky-Witten invariants of hyperkähler manifolds_, Cambridge 2000 ([arXiv:math/0404360](https://arxiv.org/abs/math/0404360))

[[!redirects compact hyperkähler manifolds]]

[[!redirects compact hyperkaehler manifold]]
[[!redirects compact hyperkaehler manifolds]]


compact Lie algebra

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## Definition

A [[semisimple Lie algebra]] $\mathfrak{g}$ is _compact_ if its [[Killing form]]  $\langle -,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$  is a negative definite [[bilinear form]].

## Properties

+-- {: .num_prop}
###### Proposition

The [[Lie algebra]] of a [[compact space|compact]] [[semisimple Lie group]] is compact.

=--

A proof is spelled out for instance  in ([Woit, theorem 1](#Woit)).

## References

For instance

* Peter Woit, _Topics in Representation Theory: The Killing
Form, Reflections and Classification of Root
Systems_ ([pdf](http://www.math.columbia.edu/~woit/notes9.pdf))
 {#Woit}

[[!redirects compact Lie algebras]]


compact Lie group

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#Contents#
* table of contents
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## Definition

A real [[Lie group]] is **compact** if its underlying [[topological group]] is a [[compact topological group]].

## Properties

### Maximal tori

All [[maximal torus|maximal tori]] of a compact Lie group are conjugate by [[inner automorphisms]]. The [[dimension]] of a maximal torus $T$ of a compact Lie group is called the _[[rank of a Lie group|rank]]_ of $G$.

The [[normalizer]] $N(T)$ of a maximal torus $T$ determines $G$.

The [[Weyl group]] $W(G)=W(G,T)$ of $G$ with respect to a choice of a maximal torus $T$ is the [[group of automorphisms]] of $T$ which are restrictions of [[inner automorphisms]] of $G$.

The maximal torus is of [[finite number|finite]] [[index of a subgroup|index]] in its normalizer; the [[quotient group|quotient]] $N(T)/T$ is [[isomorphism|isomorphic]] to $W(G)$.

The [[cardinality]] of $W(G)$ for a compact [[connected topological space|connected]] $G$, equals the [[Euler characteristic]] of the [[homogeneous space]] $G/T$ ("[[flag variety]]").

See also at _[[relation between compact Lie groups and reductive algebraic groups]]_

### Abelian compact Lie groups
 {#AbelianCompactLieGroups}

\begin{proposition}
\label{MaximalToriOfConnectedCompactLieGroupsAreMaximalAbelianSubgroups}
  The [[maximal torus]] of a [[connected topological space|connected]] [[compact Lie group]] is also the maximal [[abelian group|abelian]] [[subgroup]].
\end{proposition}
(e.g. [Salamon 2021, Lem. 6.5](#Salamon21))

In particular:
\begin{proposition}
\label{ConnectedCompactLieGroupsAreTori}
All connected compact abelian Lie groups are [[tori]], up to [[isomorphism]].
\end{proposition}
([Adams 1982, Thm. 2.19, Cor. 2.20](#Adams82))

\begin{proposition}
**(Classification of compact abelian Lie groups)**
\linebreak
Assuming the [[axiom of choice]],
every [[abelian group|abelian]] [[compact Lie group]] is [[isomorphism|isomorphic]] to the [[direct product group]] of an [[n-torus]] with a [[finite group|finite]] [[abelian group]].
\end{proposition}
\begin{proof}
Write
$$
  G_{\mathrm{e}}
  \xhookrightarrow{\;\;}
  G
$$
for the [[subgroup]] which is the [[connected component]] of the [[neutral element]] in the given compact abelian Lie group $G$. By Prop. \ref{ConnectedCompactLieGroupsAreTori} this is a [[torus]]
$$
  G_{\mathrm{e}}
  \,\simeq\,
  T^n
$$
hence its [[underlying]] [[abelian group]] is a [[divisible group]] and therefore, by [this Prop.](injective+object#InjectiveAbelianGroupIsDivisibleGroup), an [[injective object]] in the [[category]] [[Ab]] of [[abelian group]]. This implies that the dashed [[extension]] in the following [[diagram]] in [[Ab]] exists:

\begin{tikzcd}
  G_{\mathrm{e}}
  \ar[r, -, shift left=1pt]
  \ar[r, -, shift right=1pt]
  \ar[d, hook, "i"{left}]
  &
  G_{\mathrm{e}}
  \\
  G
  \mathrlap{\,,}
  \ar[ur, dashed, "p"]
\end{tikzcd}
hence that $G$ [[retraction|retracts]] onto $G_0$.

While this is, a priori, a diagram in abelian [[discrete groups]] not it abelian Lie group, the fact that the dashed morphism $p \colon G \to G_{\mathrm{e}}$ [[restriction|restricts]] to the [[identity morphism]] on $G_{\mathrm{e}}$, together with the assumption that $G$ is a [[disjoint union]] of copies of this connected component and using the [[homomorphism]] property implies that $p$ is a [[continuous function|continuous]] homomorphism. But [[continuous homomorphisms of Lie groups are smooth]], so that $p$ is in fact smooth.

Therefore we have a [[split exact sequence]] of Lie groups
\begin{tikzcd}
  G_{\mathrm{e}}
  \ar[r, hook, "i"{below}]
  &
  G
  \ar[r, ->>]
  \ar[l, bend right=30, "p"{above}]
  &
  G/G_{\mathrm{e}}
  \,,
\end{tikzcd}
and hence an isomorphism
$$
  G
    \;\simeq\;
  G_{\mathrm{e}} \times G/G_{\mathrm{e}}
    \;\simeq\;
  T^n \times A
  \,.
$$
By the assumption that $G$ was compact abelian, $A \,\coloneqq\, G/G_{\mathrm{e}}$ is finite abelian.
\end{proof}

### Invariant metric
 {#InvariantMetric}

\begin{prop}\label{CompactLieGroupsAdmitBiinvariantMetrics}
**([[compact Lie groups]] admit [[bi-invariant Riemannian metrics]])**\linebreak
  Every compact Lie group admits a [[bi-invariant Riemannian metric]].
\end{prop}

([Milnor 76, Cor. 1.4](invariant+metric#Milnor76))

### Smooth actions

+-- {: .num_prop #SmoothActionOfCompactLieGroupOnSmoothManifoldIsProper}
###### Proposition

Let $X$ be a [[smooth manifold]] and let $G$ be a [[compact Lie group]].
Then every [[smooth function|smooth]] [[action]] of $G$ on $X$ is [[proper action|proper]].

=--

(e.g. [Lee 12, Corollary 21.6](#Lee12))

### Equivariant triangulation theorem
 {#EquivariantTriangulationTheorem}

The _[[equivariant triangulation theorem]]_ ([Illman 78](equivariant+triangulation+theorem#Illman78), [Illman 83](equivariant+triangulation+theorem#Illman83)) says that for $G$ a compact Lie group and $X$ a [[compact topological space|compact]] [[smooth manifold]] equipped with a [[smooth function|smooth]] $G$-[[action]], there exists a $G$-[[equivariant triangulation]] of $X$.

### Spaces of homomorphisms

* [[nearby homomorphisms from compact Lie groups are conjugate]]

## Examples

* A [[discrete group]] is a compact Lie group iff it is a [[finite group]].

* The [[classical Lie groups]] for *definite* [[inner product space|inner products]] are compact, such as the [[orthogonal groups]], the [[unitary groups]], the [[quaternionic unitary groups]], etc., but not the [[Lorentz group]] etc.

## Applications

### In equivariant homotopy theory
 {#InEquivariantHomotopyTheory}

Compact Lie groups make a somewhat unexpected appearance as [[equivariance groups]] in [[equivariant homotopy theory]], where the compact Lie condition on the [[equivariance group]] is needed in order for (the available proofs of) the [[equivariant Whitehead theorem]] to hold.

(Namely, the [[equivariant triangulation theorem]] [above](#EquivariantTriangulationTheorem) is used in these proofs to guaratee that [[Cartesian products]] of [[coset spaces]] $G/H$ are themselves [[G-CW-complexes]].)

### In gauge theory

In [[gauge theory]] ([[Yang-Mills theory]]/[[Chern-Simons theory]], ...) ...

## Related concepts

* [[locally compact topological group]]

* [[compact topological group]]

* [[maximal compact subgroup]]

* [[closed subgroup]]

* [[proper Lie groupoid]]

* [[p-compact group]]

* [[twisted ad-equivariant K-theory]]

## References

Textbook accounts:

* {#Bredon72} [[Glen Bredon]], Section 0.6 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* {#Adams82} [[Frank Adams]], *Lectures on Lie groups*, University of Chicago Press, 1982 ([ISBN:9780226005300](https://press.uchicago.edu/ucp/books/book/chicago/L/bo3614673.html), [gbooks](https://www.google.com/books/edition/Lectures_on_Lie_Groups/TC7d3ZcqjfsC?hl=en))

* [[Hans Duistermaat]], [[Johan A. C. Kolk]], Chapter 3 of: *Lie groups*, Springer (2000) [[doi:10.1007/978-3-642-56936-4](https://doi.org/10.1007/978-3-642-56936-4)]

In the broader context of [[smooth manifolds]]:

* {#Lee12} [[John Lee]], _Introduction to Smooth Manifolds_, Springer 2012 ([doi:10.1007/978-1-4419-9982-5](https://doi.org/10.1007/978-1-4419-9982-5), [Draft pdf of the 1st edition](https://lost-contact.mit.edu/afs/adrake.org/usr/rkh/Books/books/Introduction%20to%20Smooth%20Manifolds%20-%20J.%20Lee.pdf))

Dedicated lecture notes:

* {#Salamon21} [[Dietmar Salamon]], *Notes on compact Lie groups*, 2021 ([pdf](https://people.math.ethz.ch/~salamon/PREPRINTS/liegroup.pdf), [[Salamon_CompactLieGroups.pdf:file]])

Discussion in the context of [[representation theory]]:

* [[Tammo tom Dieck]], [[Theodor Bröcker]], *Representations of compact Lie groups*, Springer 1985 ([doi:10.1007/978-3-662-12918-0](https://link.springer.com/book/10.1007/978-3-662-12918-0))

For more see also the references at *[[equivariant homotopy theory]]*.

[[!redirects compact Lie groups]]


compact object

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# Contents
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## Idea

An [[object]] of a [[category]] is called *compact* if it is "finite" or "small" in some precise sense. There are however different formalizations of this idea. Here discussed is the notion, usually going by this term, where an object $X$  is called _compact_ if mapping out of it [[preserved colimit|preserves]] [[filtered colimits]].

This means that if any other object $A$ is given as the [[colimit]] of a "suitably increasing" family of objects $\{A_i\}$, then every morphism

$$
  X
  \longrightarrow
  A
   =
  \underset{\underset{i}{\longrightarrow}}{\lim} A_i
$$

out of the compact object $X$ into that colimit factors through one of the [[coprojections]] $A_i \to \underset{\underset{i}{\longrightarrow}}{lim A_i}$.

The notion of _[[small object]]_ is essentially the same, with a bit more flexibility on when the family $\{A_i\}$ is taken to be "suitably increasing". An important application of the above factorization property is accordingly named the _[[small object argument]]_.
On the other hand, there is also the notion of _[[finite object]]_ (in a [[topos]]) which, while closely related, is different. See also _[Subtleties and different meanings](#SubtletiesAndDifferentMeanings)_ below.

## Definition
{#FinitelyPresentableObject}

+-- {: .num_defn }
###### Definition

Let $C$ be a [[locally small category]] that admits [[filtered colimits]]. Then an [[object]] $X \in C$ is **compact**, or **finitely presented** or **finitely presentable**, or **of finite presentation**, if the [[corepresentable functor]]

$$
  Hom_C(X,-) \colon C \to Set
$$

[[preserved limit|preserves]] these [[filtered colimits]].  This means that for every [[filtered category]] $D$ and every functor $F : D \to C$, the canonical morphism

$$
  \underset{\to_d}{\lim} C(X,F(d)) \xrightarrow{\simeq}
  C(X, \underset{\to_d}{\lim} F(d))
$$

is an [[isomorphism]].

More generally, if $\kappa$ is a [[regular cardinal]], then an object $X$ such that $C(X,-)$ commutes with $\kappa$-[[filtered colimits]] is called **$\kappa$-compact**, or **$\kappa$-presented**, or **$\kappa$-presentable**.  An object which is $\kappa$-compact for some regular $\kappa$ is called a [[small object]].

=--

## Properties
  {#Properties}

\begin{proposition}
\label{SmoothColimitsOfCompactObjectsAreCompact}
**(smooth colimits of compact objects are compact)**
\linebreak
A $\kappa$-small [[colimit]] of $\kappa$-compact objects is again a $\kappa$-compact object.
\end{proposition}

\begin{proof}
Let $D$ be a $\kappa$-[[small category]] and $X : D \to C$ a [[diagram]] of $\kappa$-compact objects. Let $I$ be a $\kappa$-[[filtered category]] and $A : I \to C$ a $\kappa$-filtered diagram in $C$. Then

$$
  Hom
  \Big(
    \underset{\underset{d}{\longrightarrow}}{\lim}
    X_d
    ,\,
    \underset{\underset{i}{\longrightarrow}}{\lim}
    A_i
  \Big)
  \;\simeq\;
  \underset{\underset{d}{\longleftarrow}}{\lim}
  \;
   Hom
   \Big(
     X_d
     ,\,
    \underset{\underset{i}{\longrightarrow}}{\lim}
     A_i
   \Big)
$$

by [[hom-functor preserves limits|general properties of]] the [[hom functor]]. Now using that every $X_d$ is $\kappa$-compact and $I$ is $\kappa$-filtered this is

$$
  \cdots
  \;\simeq\;
  \underset{\underset{d}{\longleftarrow}}{\lim}
  \;
  \underset{\underset{i}{\longrightarrow}}{\lim}
  \;
  Hom\big(X_d, A_i\big)
  \,.
$$

Since this (co)limit is taken in [[Set]], the $\kappa$-small limit over $D$ [[limits commuting with colimits|commutes]] with the $\kappa$-filtered colimit

$$
  \cdots
  \;\simeq\;
  \underset{\underset{i}{\longrightarrow}}{\lim}
  \;
  \underset{\underset{d}{\longleftarrow}}{\lim}
  \,
  Hom\big(X_d, A_i\big)
  \,.
$$

Finally we make take the limit again to a colimit in the first argument

$$
  \cdots
  \;\simeq\;
  \underset{\underset{i}{\longrightarrow}}{\lim}
  \;
  Hom
  \big(
    \underset{\underset{d}{\longrightarrow}}{\lim}
    X_d
    ,\,
    A_i
  \big)
  \,,
$$

which yields the claim.
\end{proof}

+-- {: .num_prop}
###### Proposition
In a $\lambda$-[[accessible category]], if $\lambda$ is [[sharply smaller cardinal|sharply smaller]] than $\kappa$, then every $\kappa$-compact object is a [[retract]] of a $\kappa$-small $\lambda$-filtered colimit of $\lambda$-compact objects.
=--
+-- {: .proof}
###### Proof
See [Adamek-Rosicky, Remark 2.15](#AdamekRosicky94).  It is noted there that with the more technical proof of [Makkai-Pare, Proposition 2.3.11](#MakkaiPare89) the words "a retract of" can be omitted.
=--

If we weaken the hypothesis to $\lambda\le \kappa$, then we retain all of the result except for $\lambda$-filteredness of the colimit.

+-- {: .num_prop}
###### Proposition
In a locally $\lambda$-[[locally presentable category|presentable category]], if $\lambda\le \kappa$, then every $\kappa$-compact object is a [[retract]] of a $\kappa$-small colimit of $\lambda$-compact objects.
=--
+-- {: .proof}
###### Proof
See [Adamek-Rosicky, Remark 1.30](#AdamekRosicky94).  It is claimed there that the words "a retract of" can be omitted by reference to an argument in Makkai-Pare, but it seems unclear how this argument is intended to be used.  An alternative proof of this improvement is proposed [at this mathoverflow question](https://mathoverflow.net/q/325278).
=--

## Examples
 {#Examples}

* In $C = $ [[Set]] an object is compact precisely if it is a [[finite set]].
For this to hold constructively, [[filtered categories]] (appearing in the definition of _[[filtered colimit]]_) have to be understood as categories admitting cocones of every _Bishop-finite_ diagram.
(An object of Set is a [[finite set|Kuratowski-finite]] precisely if it is a [[finitely generated object]], or equivalently if it is [[compact space|compact]] when regarded as a [[discrete object|discrete]] topological space.)

* For $C$ a [[topos]], $X$ is compact if:
    * $C$ is a [[Grothendieck topos | sheaf topos]] on a site whose topology is generated by finite covering families and $X$ is a representable sheaf;

    * in particular if $C$ is a [[coherent topos]] and $X$ is a [[coherent object]].

However, there exist compact objects which are not coherent, c.f. the [[Elephant]], D3.3.12.

* In $C = $ [[Grp]] an object is compact precisely if it is [[finitely presented group|finitely presented]] as a group.

* More generally, if $C$ is any [[variety of algebras]], then an object is compact precisely if it is [[finitely presented algebra|finitely presented]] as an algebra.  A proof may be found in  [Adámek-Rosický 94, Corollary 3.13 ](#AdamekRosicky94).

* Let $X$ be a [[topological space]] and let $C = Op(X)$ be the [[category of open subsets]] of $X$. Then an [[open subset]] $U \in C$ is a compact object in $C$ precisely if it is a [[compact space|compact topological space]].  (It is *not* true that $X$ is a compact object of $Top$ iff it is a compact topological space; see below.)

* A [[finite-dimensional vector space]] is compact in [[Vect]], see [here](finite-dimensional+vector+space#CompactClosure).

## Subtleties and different meanings
 {#SubtletiesAndDifferentMeanings}

One has to be careful about the following variations of the theme of compactness.

(Some of these subtleties are resolved by noticing that there is a hierarchy of notions of compact objects that are secretly different but partly go by the same name. Some discussion of this is currently at _[[compact topos]]_, but more detailed discussion should eventually be somewhere...)

In the [[Elephant]], what Johnstone calls _compact objects_ are those objects such that the [[top]] element of the [[poset of subobjects]] $\operatorname{Sub}(C)$ is a [[compact element]]; he reserves the term _finitely-presented_ for the notion of compact on this page.

### Compactness in additive categories
  {#CompactnessInAdditiveCategories}

When $C$ is an [[additive category]] (often a [[triangulated category]]), an object $x$ in $C$ is called **compact** if for every set $S$ of objects of $C$ such that the coproduct $\coprod_{s\in S} s$ exists, the canonical map
$$
\coprod_{s\in S} C(x,s)\to C(x,\coprod_{s\in S}s)
$$
is an [[isomorphism]] of [[commutative monoid|commutative monoids]].

Here is an application of this concept to characterize which abelian categories are categories of modules of some ring:

+-- {: .un_theorem}
###### Theorem

Let $C$ be an abelian category.  If $C$ has all [[small category|small]] [[coproducts]] and has a compact [[projective object| projective]] [[generator]], then $C \simeq R Mod$ for some ring $R$.  In fact, in this situation we can take $R = C(x,x)^{op}$ where $x$ is any [[compact projective object|compact projective]] generator.   Conversely, if $C \simeq R Mod$, then $C$ has all small coproducts and $x = R$ is a compact projective generator.
=--

+-- {: .proof}
###### Proof

This theorem, minus the explicit description of $R$, can be found as Exercise F on page 103 of Peter Freyd's book [Abelian Categories](http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=132).
The first part of this theorem can also be found as Prop. 2.1.7. of Victor Ginzburg's [Lectures on noncommutative geometry](http://arxiv.org/PS_cache/math/pdf/0506/0506603v1.pdf#page=4).  Conversely, it is easy to see that $R$ is a compact projective generator of $R Mod$.
=--

+--{.query}
Zoran: While Ginzburg's reference is surely a worthy to look at, it would be better not to give false impression that this [[reconstruction theorem]] is due Ginzburg or at all new. It is rather a classical and well know fact probably from early 1960s, essentially small strengthening of a variant of a circle of abelian reconstruction theorems including the [Gabriel-Popescu theorem](http://myyn.org/m/article/gabriel-popescu-theorem-for-ab5-categories)(probably our variant could be read off from classical algera book by Faith for example, or Popescu's book on abelian categories, in any case it is well known in [[noncommutative algebraic geometry]]). In fact for this fact, if I think better, the reconstruction belongs usually to expositions which treat classical Morita theory for rings.
=--

A triangulated category is __compactly generated__ if it is generated (see [[generator]]) by a _set_ of compact objects.

The notion can be modified for categories [[enriched category|enriched]] over a [[closed monoidal category]] (compare to the notions of finite and/or rigid objects in various contexts).

Compact objects in the derived categories of quasicoherent sheaves over a scheme are called [[perfect complexes]]. Any [[compact object]] in the [[category of modules]] over a perfect ring is finitely generated as a module.

In non-additive contexts, the above definition is not right.  For instance, with this definition a [[topological space]] would be compact iff it is [[connected space|connected]].  In general one should expect to instead preserve filtered colimits, as above.

### Compact objects in $Top$
 {#CompactObjectsInTop}

Recall the above example of [[compact space|compact topological spaces]].  Notice that the statement which one might expect, that a [}topological space]] $X$ is [[compact space|compact]] if it is a compact object in [[Top]], is not quite right in general.

A [[counterexample]] is given for instance in [Hovey (1999), page 49](#Hovey99), which itself was corrected by Don Stanley (see the [errata](http://hopf.math.purdue.edu/Hovey/model-err.pdf) of that book). See also the blog discussion
[here](http://golem.ph.utexas.edu/category/2009/05/journal_club_geometric_infinit_3.html#c023790).

Namely, the two-element set with the [[indiscrete topology]] is a compact space $X$ for which
 \[
 Hom(X, -)
  M\colon
  Top \rightarrow Top
\]
doesn't preserve filtered colimits, in fact not even [[sequential colimit|colimits of sequences]] (functors out of the [[poset|ordered set]] of [[natural numbers]]).

For example, consider the sequence of spaces
 \[
 X_n=[n,\infty) \times \{0,1\}
\]
where the [[open sets]] are of the form
\[
 [n, \infty] \times \{0\} \cup [m,\infty) \times \{1\}
\]
(where $m \geq n$), plus the empty set. Define $X_n \rightarrow X_{n+1}$ so that it sends a pair $(k, \epsilon)$ to itself if $k \gt n$, and $(n,\epsilon)$ to $(n+1,\epsilon)$. This defines a functor
\[
F: \mathbb{N} \rightarrow Top
\]
The colimit $X_\infty$ of this sequence is the two-element set $\{0,1\}$ with the indiscrete topology. However, the identity map on this space does not factor through any of the canonical maps $X_n \rightarrow X_\infty$. It follows that
the comparison map
\[
 colim_n Hom(X_\infty, X_n) \rightarrow Hom(X_\infty, X_\infty)
\]
is not surjective, and therefore not an isomorphism.

+--{.query}
_[[Todd Trimble|Todd]]_ (posted from n-category cafe): I don't know if the story is any different for $X$ compact _Hausdorff_, but it could be worth considering.
=--

But with a bit of care on the assumptions, similar results do hold:

\begin{proposition} {#CompactSpace}
If $Y \in Top$ is a [[compact topological space]], then
$hom(Y,-)$ preserves colimits of functors mapping out of [[limit ordinals]], provided that the arrows of the cocone diagram,
 \[
 X_\alpha \rightarrow X_\beta,
\]
are [[closed map|closed]] inclusions of [[separation axiom|$T_1$-spaces]], and $X_\beta$ is the colimit of earlier $X_\alpha$ if $\beta$ is itself a limit ordinal.
\end{proposition}

This applies for example to the sequence of inclusions of $n$-skeleta in a [[CW-complex]]. By considering the cases $Y=S^k$ and $Y = S^k \times I$, this implies that the functor $\pi_k$ also preserves colimits of such sequences. Applications also occur in the theory of [[classifying space|classifying bundles]]. For example, the total space $EO(n)$ of the [[orthogonal group|$O(n)$]]-[[universal principal bundle]] can be constructed as the colimit $\underset{k \to \infty}{colim} \; V_n(\mathbb{R}^{n+k})$ of [[Stiefel manifolds]] of orthonormal $n$-frames in $\mathbb{R}^{n+k}$. To show $EO(n)$ is weakly contractible, it suffices, by this result, to show that for each $j$, the [[homotopy group]] $\pi_j(V_n(\mathbb{R}^{n+k}))$ vanishes for sufficiently large $k$.

\begin{proof}
Let $X$ denote the colimit of the diagram mapping out of the limit ordinal $\kappa$. We must show that every map $f \colon Y \to X$ factors through one of the universal cocone components $i_\alpha \colon X_\alpha \to X$, where $\alpha \lt \kappa$. Suppose not. Starting from any $\alpha \lt \kappa$, pick a point $x_0 \in f(Y)$ not in the image $i_\alpha(X_\alpha)$. There is a minimal $\alpha_1 \lt \kappa$ such that $x_0 \in i_{\alpha_1}(X_{\alpha_1})$. Proceeding inductively, choose $x_n \in f(Y)$ such that $x_n \notin i_{\alpha_n}(X_{\alpha_n})$, and let $\alpha_{n+1}$ be the minimal ordinal such that $x_n \in i_{\alpha_{n+1}}(X_{\alpha_{n+1}})$. Then let $\beta$ be the limit of the $\alpha_n$, and let $S$ be the set $\{x_n: n \in \mathbb{N}\}$, so that $S \subseteq i_\beta(X_\beta)$. Any subset $K$ of $S$ is closed, because closure of $K \subseteq i_\beta(X_\beta)$ is equivalent to closure of of each $K \cap i_{\alpha_n}(X_{\alpha_n})$ (since that is how the colimit topology on $X_\beta$ is defined), and all these intersections are finite and therefore closed by the $T_1$ condition. It follows that $f^{-1}(S)$ is an infinite discrete subspace of $Y$, but this contradicts compactness.
\end{proof}

This proof is essentially that given in [Hovey (1999), page 50](#Hovey99), where this result is needed towards the [[small object argument]] on the way to establishing the [[classical model structure on topological spaces]], see [this lemma](classical+model+structure+on+topological+spaces#CompactSubsetsAreSmallInCellComplexes).

## Related concepts

* [[compact element]] (in a [[(0,1)-category]])

* [[compact projective object]]

* [[compact topos]]

* [[compact object in an (∞,1)-category]]

* [[small object]], [[small object argument]]

* [[finitely generated object]]

* [[locally presentable category]], [[accessible category]]

* [[compactly generated (∞,1)-category]]

[[!include finite objects -- table]]

## References {#references}

Compact objects are discussed under the term "finitely presentable" or "finitely-presentable" objects for instance in

* {#AdamekRosicky94} [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally Presentable and Accessible Categories]]_, Cambridge University Press in the London Mathematical Society Lecture Note Series, number 189, (1994)

* {#MakkaiPare89} [[Michael Makkai]], [[Robert Paré]], _Accessible categories: The foundations of categorical model theory_ Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.

*  [[Masaki Kashiwara]], [[Pierre Schapira]], around Definition 6.3.3 of _[[Categories and Sheaves]]_;

* [[Peter Johnstone]], _[[Stone Spaces]]_, Definition VI.1.8;

* [[Peter Johnstone]], the _[[Elephant]]_, D2.3.1.

For the pages quoted in the context of the discussion of compact objects in [[Top]] see

* {#Hovey99} [[Mark Hovey]], _[[Model Categories]]_, Mathematical Surveys and Monographs, Volume 63, AMS (1999).  [ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s) [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063) [pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf), (errata: [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats-errata.pdf))
.

For the general definition with an eye towards the definition of [[compact object in an (infinity,1)-category]]  see section A.1.1 section 5.3.4 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects compact object]]
[[!redirects compact objects]]

[[!redirects finitely presentable object]]
[[!redirects finitely presentable objects]]

[[!redirects finitely-presentable object]]
[[!redirects finitely-presentable objects]]

[[!redirects presentable object]]
[[!redirects presentable objects]]


compact object - contents

**objects $d \in C$ such that $C(d,-)$ commutes with certain [[colimits]]**

* [[coproduct-preserving representable]]

* [[connected object]]

* [[compact object]]

  * [[compact element|compact object in a (0,1)-category]], [[compact object in an (∞,1)-category]]

  * [[finite object]]

  * [[small object]]

* [[tiny object]]

* [[infinitesimal object|atomic object]]

  * [[amazing right adjoint]]

* [[accessible category]]

  * [[presentable category]]

## Models

* [[compact topos]]

* [[compact topological space]]

## Relative version

* [[proper geometric morphism]]

* [[proper map]]

[Edit this sidebar](/nlab/edit/compact+object+-+contents)


compact object in an (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Compact objects
+--{: .hide}
[[!include compact object - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _compact object in an $(\infty,1)$-category_ is the analogue in [[(∞,1)-category theory]] of the notion of [[compact object]] in [[category theory]].

## Definition

+-- {: .num_defn}
###### Definition

Let $\kappa$ be a [[regular cardinal]] and $C$ an [[(∞,1)-category]]
with $\kappa$-[[filtered (∞,1)-colimits]].

Then an [[object]] $c \in C$ is called **$\kappa$-compact** if the [[(∞,1)-categorical hom space]] functor

$$
  C(c,-) : C \to \infty Grpd
$$

preserves $\kappa$-[[filtered (∞,1)-colimits]].

For $\omega$-compact we just say _compact_.

=--

This appears as ([[Higher Topos Theory|HTT, def. 5.3.4.5]]).

## Properties

### General

Let $\kappa$ be a [[regular cardinal]].

+-- {: .num_prop #StabilityUnderColimits}
###### Proposition

Let $C$ be an [[(∞,1)-category]] which admits small
$\kappa$-[[filtered (∞,1)-colimits]]. Then the full [[sub-(∞,1)-category]]
of $\kappa$-compact objects in closed under $\kappa$-small [[(∞,1)-colimits]]
in $C$.

=--

This is ([[Higher Topos Theory|HTT, cor. 5.3.4.15]]).

### Presentation in model categories

If the [[(∞,1)-category]] $\mathcal{C}$ is a [[locally presentable (∞,1)-category]], then it is the [[simplicial localization]] of a [[combinatorial model category]] $C$, and one may ask how the 1-categorical notion of [[compact object]] in $C$ relates to the $(\infty,1)$-categorical notion of compact in $\mathcal{C}$.

Since compactness is defined in terms of colimits, the question is closely related to the question which 1-categorical $\kappa$-[[filtered colimits]] in $C$ are already [[homotopy colimits]] (without having to [[derived functor|derive]] them first).

General statements seem not to be in the literature yet, but see [this MO discussion](http://mathoverflow.net/questions/95165/compact-objects-in-model-categories-and-infty-1-categories). For discussion of compactness in a [[model structure on simplicial sheaves]], see for instance ([Powell, section 4](#Powell)).

## Examples

*  In $C =$ [[(∞,1)Cat]], for uncountable $\kappa$, the $\kappa$-compact objects
   are precisely the $\kappa$-[[essentially small (∞,1)-categories]]. (See there
   for more details.)

*  In $C =$ [[∞Grpd]], for uncountable $\kappa$, the $\kappa$-compact objects are precisely the $\kappa$-[[essentially small ∞-groupoids]]. When $\kappa = \omega$, the compact objects in ∞Grpd are the [[retracts]] of the $\omega$-small ∞Grpds, i.e., the retracts of the [[finite homotopy types]] ([[finite CW-complexes]]). Not every such retract is equivalent to a $\omega$-small ∞-groupoid; the vanishing of [[Wall's finiteness obstruction]] is a necessary and sufficient condition for such an equivalence to exist.

## Related concepts

* [[compact object]]

* [[compact topos]]

* **compact object in an $(\infty,1)$-category**

* [[relatively k-compact morphism in an (infinity,1)-category]]

* [[compactly generated (∞,1)-category]]

## References

The general definition appears as definition 5.3.4.5  in

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

Compactness in presenting model categories of simplicial sheaves is discussed for instance in

section 4 of

* Geoffrey Powell, _The adjunction between $\mathcal{H}(k)$ and $DM^{eff}_-(k)$_ (2001) ([pdf](http://math.univ-angers.fr/~powell/documents/2001/adjoint.pdf))
  {#Powell}

[[!redirects compact object in an (∞,1)-category]]


compact operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

# Compact operators
* table of contents
{: toc}

## Definition

For a [[linear operator]] between [[Banach spaces]], compactness is a natural strengthening of [[bounded operator|continuity]] (boundedness). A linear operator is **compact** if it sends the [[bounded subspace|bounded subsets]] to [[relatively compact subset|relatively compact subsets]].

The [[quotient]] of the [[bounded operators]] by the compact operators is called the _[[Calkin algebra]]_.

## Properties

Since every relatively compact subspace (in a Banach space, or indeed in any [[metric space]]) is bounded, every compact operator is bounded. Instead of checking compactness on all bounded subsets it is sufficient to check it for a [[ball]] of one fixed radius: an operator is compact iff it sends the ball of unit radius to a relatively compact set.

See also [[compact self-adjoint operator]].

## Completely continuous operators

In the setup of [[Hilbert spaces]], instead of a compact operator, one sometimes says a completely continuous operator. However, in the full generality of Banach spaces, by a **completely continuous operator** one means slightly less: an operator that maps every [[weak convergence|weakly convergent]] sequence to a [[norm convergence|norm convergent]] sequence. For Hilbert spaces and, more generally, for [[reflexive Banach spaces]], the two notions are equivalent.

## References

* Wikipedia (English): [Compact operator](http://en.wikipedia.org/wiki/Compact_operator)

category: analysis
[[!redirects compact operator]]
[[!redirects compact operators]]
[[!redirects completely continuous operator]]
[[!redirects completely continuous operators]]


compact operators > history

< [[compact operator]]


compact ordered space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

Compact ordered spaces can be thought of as a refinement of the notion of [[compact Hausdorff space]] to [[order theory|ordered]] spaces. They share many properties of compact Hausdorff spaces, where equality is replaced by the order relation.
These spaces are a convenient choice for notions of [[convergence]] in an [[order theory|ordered]] setting.

In particular, just as [[compact Hausdorff spaces]] can be seen as [[algebra over a monad|algebras]] of the [[ultrafilter#ultrafilters_form_a_monad|ultrafilter monad]] on [[Set]], compact ordered spaces can be seen as algebras of the [[prime upper filter monad]] on [[Pos]], (see [below](#as_algebras)).

The concept was introduced by Nachbin (see [Nachbin '65](#nachbin)).

## Definition

A **compact ordered space** or **compact pospace** is a [[compact topological space]] $X$ equipped with a [[partial order]] which is [[closed subset|closed]] as a [[subset]] of $X\times X$.

## Properties

* A compact ordered space is always [[Hausdorff]], i.e. it is a [[compactum]]. To see this, note that since the [[relation]] is a closed subset of $X\times X$, so is the [[opposite relation]], and hence their intersection too, which is the [[diagonal]] of $X\times X$.

* Conversely, a [[compact Hausdorff space]] can be seen as a compact ordered space with the [[discrete order]].

* In a compact ordered space, the [[up-sets]] and [[down-sets]] $\uparrow\!\{x\}$ and $\downarrow\!\{x\}$ are always [[closed set|closed]] and [[compact]].

(...)

## Categories of compact ordered spaces

A canonical choice of [[morphisms]] between compact ordered spaces is [[continuous map|continuous]], [[monotone maps]], which form a category. This category is usually just called the **category of compact ordered spaces**, and denoted by __$CompOrd$__.

With the 2-cell structure given by the [[pointwise order]], __$CompOrd$__ becomes a [[locally posetal 2-category]].

(...)

## As algebras

For now, see [Flagg '96](#flagg).

(...)

## Connection with stably compact spaces

For now, see [Jung '04](#jung2).

(...)

## See also

* [[compact Hausdorff space]]
* [[stably compact space]]

## References

* {#nachbin} Leopoldo Nachbin, _Topology and Order_, Van Nostrand, 1965.

* {#flagg} Bob Flagg, _Algebraic theories of compact pospaces_, Topology and its Applications, 1996 ([doi:10.1016/S0166-8641(96)00117-4](https://doi.org/10.1016/S0166-8641%2896%2900117-4)).

* {#jung2} Achim Jung, _Stably compact spaces and the probabilistic powerspace construction_, ENTCS 87, 2004 ([doi:10.1016/j.entcs.2004.10.001](https://doi.org/10.1016/j.entcs.2004.10.001)).

[[!redirects compact ordered spaces]]
[[!redirects ordered compact space]]
[[!redirects ordered compact spaces]]
[[!redirects compact ordered topological space]]
[[!redirects compact ordered topological spaces]]
[[!redirects ordered compact topological space]]
[[!redirects ordered compact topological spaces]]
[[!redirects compact pospace]]
[[!redirects compact pospaces]]
[[!redirects CompOrd]]
[[!redirects category of compact ordered spaces]]
[[!redirects category of ordered compact spaces]]
[[!redirects category of compact ordered topological spaces]]
[[!redirects category of ordered compact topological spaces]]
[[!redirects category of compact ordered spaces]]
[[!redirects category of compact pospaces]]


compact projective object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

\tableofcontents

## Definition

An object $P$ of a [[category]] $C$ is a __compact projective object__
if its [[corepresentable functor]] $Hom(P,-)\colon C\to Set$ [[preserved colimit|preserves]] all [[small colimit|small]] [[sifted colimits]].

In the case of cocomplete [[Barr-exact categories]], it is equivalently an [[object]] that is

1. a [[compact object]]

   ($Hom(P,-)$ preserves all small [[filtered colimits]]),

1. a [[projective object]]

   ($Hom(P,-)$ preserves [[regular epimorphisms]], which follows from its preservation of [[coequalizers]]).

## Examples

In the category of [[algebra over an algebraic theory|algebras over]] an [[algebraic theory]],
compact projective objects are [[retracts]] of [[free construction|free]] algebras.

Conversely, if a [[locally small category]] has enough compact projective objects
(meaning that there is a set of compact projective objects that
generates it under small [[colimits]] and reflects isomorphisms),
then this category is equivalent to the category of algebras over an [[algebraic theory]].
Such a category is also known as a [[locally strongly finitely presentable category]]

## Related concepts

* [[sifted colimit]]

* [[algebraic theory]]

* [[locally strongly finitely presentable category]]

[[!redirects compact projective objects]]


compact self-adjoint operator

# Compact self-adjoint operators
* table of contents
{: toc}

A __compact self-adjoint operator__ is a [[linear operator]] that is both [[compact operator|compact]] and [[self-adjoint operator|self-adjoint]].

[[!redirects example of compact self-adjoint operators]]
[[!redirects example of compact self-adjoint opertors]]


compact space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Compact spaces
* table of contents
{: toc}

## Idea
 {#Idea}

A [[topological space]] (or more generally: a [[convergence space]]) is _compact_ if all [[sequences]] and more generally [[nets]] inside it [[convergence|converge]] as much as possible.

Compactness is a topological notion that was developed to abstract the key property of a [[subspace]] of a [[Euclidean space]] being "[[closed space|closed]] and [[bounded set|bounded]]": *every [[net]] must [[accumulation point|accumulate]] somewhere in the subspace*. (Roughly, the reason is that boundedness implies the net cannot escape the subspace, and the point to which it accumulates lies in the subspace by closure. This is the statement of the _[[Heine-Borel theorem]]_, see there for for more.)

Compactness provides an intrinsic way of formulating this property in the context of general topological spaces, without the need to view them as subspaces of an ambient space.  Still, it is common to work with _compact [[subsets]]_ of a given space.  These are those subsets which are compact spaces with the [[subspace topology]].

One often wishes to study [[compact Hausdorff spaces]] ("compacta") since these enjoy particularly useful properties. For instance they all arise as [[one-point compactifications]] (of [[locally compact Hausdorff spaces|locally compact]] [[Hausdorff spaces]], see [this remark](one-point+compactification#CompactHausdorffSpaceIsCompactificationOfComplementOfAnyPoint)).

As for most concepts related to [[topological spaces]], there is also a concept of compactness for _[[locales]]_. Observe that (using [[classical logic]]) every [[locally compact locale]] is aready [[spatial locale|spatial]] ([this prop.](locally+compact+locale#UsingClassicalLogicLocallyCompactLocaleIsSpatial)).
Instead of compact Hausdorff locales one usually considers compact [[regular space|regular]] locales, since regularity is easier to formulate and handle than Hausdorffness in locale theory (these are equivalent, since every locale is $T_0$ [[separation axiom|separated]] and hence $T_3$ if regular, while every Hausdorff space is $T_3$ if compact).

## Definitions/characterizations

There are many ways to say that a space $X$ is compact. The first is perhaps the most common:

+-- {: .num_defn #OpenCover}
###### Definition
**([[open cover]])**

Let $(X,\tau)$ be a [[topological space]]. Then an [[open cover]] is a [[set]] $\{U_i \subset X\}_{i \in I}$ of [[open subsets]] (i.e. $(U_i \subset X) \in \tau \subset P(X)$) such that their [[union]] is all of $X$

$$
  \underset{i \in I}{\cup} U_i = X
  \,.
$$

This is called a _[[finite open cover]]_ if $I$ is a (Kuratowski-)[[finite set]].

A _subcover_ of an open cover as above is a [[subset]] $J\subset I$ of the given open subsets, such that their union still exhausts $X$, i.e. $\underset{i \in J \subset I}{\cup} U_i = X$.

=--

+-- {: .num_defn #hb}
###### Definition
**(compact space)**

A [[topological space]] is called _compact_ if every [[open cover]] has a finite subcover (def. \ref{OpenCover}).

=--

For the purposes of exposition, this definition will be taken as the baseline definition. Throughout the remainder of this section we state a number of propositions of type "A space is compact (in the sense of Definition \ref{hb}) iff it satisfies property P", which can be read as saying that property P can be taken as an alternative definition of compactness (and may in fact be considered a more convenient or preferable definition over \ref{hb}, depending on author and context).

+-- {: .num_remark #DifferingTerminology}
###### Remark
**(differing terminology)**

Some authors use "compact" to mean "compact and [[Hausdorff topological space|Hausdorff]]" (a much [[nice topological space|nicer sort of space]], and forming a much [[nice category of spaces|nicer category of spaces]], see at _[[compact Hausdorff space]]_), and use the word "[[quasicompact]]" to refer to just "compact" as we are using it here. This custom seems to be prevalent among [[algebraic geometry|algebraic geometers]], for example, and particularly so within Francophone schools.

> But it is far from clear to me ([[Todd Trimble]]) that "quasicompact" is very well-established outside such circles (despite some arguments in favor of it), and using simply "compact" for the nicer concept therefore carries some risk of creating misunderstanding among mathematicians at large. My own habit at any rate is to say "compact Hausdorff" for the nicer concept, and I will continue using this on the $n$Lab until consensus is reached (if that happens).

Another term in usage is 'compactum' to mean a [[compact Hausdorff space]] (even when 'compact' is not used to imply Hausdorffness).

=--

The various reformulations of compactness fall into several families. Some are more or less tautological reformulations based on direct logical or set-theoretic manipulations of the open covering definition; we give these first. Others express compactness in terms of notions of convergence (nets, filters, ultrafilters). A third family expresses compactness of a space in terms of "stable" properties of maps sourced is that space; this is perhaps the most intrinsically categorical of the three families.

### Elementary reformulations

If [[excluded middle]] is assumed, then def. \ref{hb} has the following reformulations in terms of [[closed subsets]]:

+-- {: .num_prop #fip}
###### Proposition
**(compactness in terms of closed subsets)**

Let $(X,\tau)$ be a [[topological space]].
Assuming [[excluded middle]], then the following are equivalent:

1. $(X,\tau)$ is compact in the sense of def. \ref{hb}.

1. Let $\{C_i \subset X\}_{i \in I}$ be a set of [[closed subsets]] such that their [[intersection]] is [[empty set|empty]] $\underset{i \in I}{\cap} C_i = \emptyset$, then there is a [[finite set|finite]] [[subset]] $J \subset I$ such that the corresponding finite intersection is still empty: $\underset{i \in  J \subset i}{\cap} C_i = \emptyset$.

1. Let $\{C_i \subset X\}_{i \in I}$ be a set of [[closed subsets]] such that it enjoys the  _[[finite intersection property]]_, meaning that for every [[finite set|finite]] [[subset]] $J \subset I$ then the corresponding finite intersection is [[inhabited set|non-empty]] $\underset{i \in J \subset I}{\cap} C_i \neq \emptyset$. Then also the total intersection is [[inhabited set|inhabited]], $\underset{i \in I}{\cap} C_i \neq \emptyset$.

=--

+-- {: .proof}
###### Proof

The equivalence of the first two statements follows by [[de Morgan's law]] ([[complements]] interchange [[unions]] with [[intersections]]), the definition of [[closed subsets]] as the complements of [[open sets]], and (using [[excluded middle]]) that dually the complements of closed subsets are the open subsets:

Let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]]. Write $C_i \coloneqq X \backslash U_i$ for the corresponding closed complements. By [[de Morgan's law]] the condition that $\underset{i \in I}{\cup} U_i = X$ is equivalent to $\underset{i \in I}{\cap} C_i = \emptyset$. The second statement is that there is then a finite subset $J \subset I$ such that also $\underset{i \in J \subset I}{\cap} C_i = \emptyset$, and under forming complements again this is equivalently the first statement.

Then statement 3 is the [[contraposition]] of the second, and contrapositives are equivalent under [[excluded middle]].

=--

The closed-subset formulations of compactness appear frequently and are often more convenient. For example, [[compactness theorems]] in [[model theory]] draw on a connection between the finite intersection property and finite satisfiability of sets of axioms.

### Compactness for locales

In another direction, the definition (\ref{hb}) also works for [[locales]], since it refers only to the [[frame]] of open sets.  Here is an equivalent way to phrase it that is often convenient for locale theory.

+-- {: .num_prop #directed}
###### Proposition

$X$ is compact iff given any [[direction|directed]] collection of opens whose union is $X$ (a directed open cover), $X$ belongs to the collection.

=--

+-- {: .proof}
###### Proof
For the "only if" direction: if $\mathcal{U}= \{U_i\}_{i \in I}$ is a directed open cover, then by the open-cover definition of compactness, $X$ is the union of finitely many $U_i$. But by definition of directedness, any finite subfamily of $\mathcal{U}$ has an upper bound in $\mathcal{U}$; since the only upper bound of $X$ is $X$, it follows that $X$ belongs to $\mathcal{U}$.

For the "if" direction: given an open cover $\mathcal{U}$ of $X$, let $\mathcal{U}'$ be the family of open sets that are unions of finite subfamilies of $\mathcal{U}$. This $\mathcal{U}'$ is clearly directed, and an open cover of $X$ since $\mathcal{U}$ is. By hypothesis, $X$ belongs to $\mathcal{U}'$, so $X$ is a union of finitely many elements of $\mathcal{U}$. This shows $X$ is compact according to Definition \ref{hb}.
=--

As the union is a [[colimit]] in the [[category of open subsets]] $Op(X)$, we can also say

+-- {: .num_prop #object}
###### Proposition

$X$ is compact iff it is a [[compact object]] in $Op(X)$.

=--

+-- {: .proof}
###### Proof
To say $X$ is a compact object means that $\hom(X, -): Op(X) \to Set$ preserves [[filtered colimits]], or colimits of filtered diagrams. Here we may as well replace $Set$ by the full subcategory consisting of $0, 1$ (a $0$-element set or a $1$-element set) since the hom-sets for posets are of size $0$ or $1$. Further,
in a [[poset]] like $Op(X)$, a filtered diagram is the same as a nonempty directed diagram $D$, and $\hom(X, \bigcup_{d \in D} d)$ has size $1$ iff $X = \bigcup_{d \in D} d$. On the other hand, $\bigcup_{d \in D} \hom(X, d)$ has size $1$ iff $X = d$ for some $d$ in $D$. Thus that $\hom(X, -)$ preserves filtered colimits amounts to the same thing as saying $X$ belongs to any directed open cover $D$, which is the same as compactness by Proposition \ref{directed}.
=--

### Compactness via convergence

If the [[ultrafilter theorem]] (a weak form of the [[axiom of choice]]) is assumed, compactness may be characterized in terms of [[ultrafilter]] (or ultranet) [[convergence]]:

+-- {: .num_prop #ultrafilter}
###### Proposition

$X$ is compact iff every [[ultrafilter]] $\mathcal{U}$ (or ultranet $\nu$) on $X$ [[converges|converges]] to some point $x \in X$, meaning that $\mathcal{U}$ contains the filter of [[neighborhoods]] of $x$ (or that $\nu$ is eventually in any neighbourhood of $x$).

=--

In any case, if [[excluded middle]] is assumed, compactness can be characterized in terms of [[proper filter]] or equivalently (see at _[[eventuality filter]]_) of [[net]] [[convergence]] .

+-- {: .num_prop #refinement}
###### Proposition

Assuming [[excluded middle]], $X$ is compact iff every [[proper filter]]/[[net]] on $X$ has a [[convergence|convergent]] proper [[refinement]]/[[subnet]].

=--

This is equivalent to the characterization given in the [Idea-section](#Idea) above:

+-- {: .num_prop #clustering}
###### Proposition

Assuming [[excluded middle]], $X$ is compact iff every proper filter $\mathcal{U}$ (or net $\nu$) on $X$ has a cluster point $x$, meaning that every element of $\mathcal{U}$ meets (has [[inhabited set|inhabited]] intersection with) every neighbourhood of $x$ (or $\nu$ is frequently in every neighbourhood of $x$).

=--

While the usual definitions (\ref{hb}&\ref{fip}) are for [[topological spaces]], the convergence definitions (\ref{ultrafilter}--\ref{clustering}) make sense in any [[convergence space]]. However, in [[constructive mathematics]], that every [[topological space]] satisfying (\ref{hb}&\ref{fip}) also satisfies (\ref{refinement}&\ref{clustering}) as a [[convergence space]] is the same as [[excluded middle]], see [[compact spaces equivalently have converging subnets#Failure in constructive mathematics]] for more details.

### Compactness via completeness

+-- {: num_prop #completeness}
A [[uniform space]] $X$ is compact if and only if it is [[complete space|complete]] and [[totally bounded space|totally bounded]].  Moreover, a compact Hausdorff topological space has a unique compatible uniformity, which is complete and totally bounded.
=--

In [[constructive mathematics]], "complete and totally bounded" is sometimes taken as a substitute for open-cover compactness (to which it is no longer equivalent); see [[Bishop-compact space]].

### Compactness via stability properties

Frequently in [[category theory]], for example when we discuss [[internal logic]] in [[toposes]], we are interested in properties of maps that are stable under [[pullback]], and it turns out that compactness can be reformulated in terms of stability properties.

A first example concerns the property of a topological space $X$ that the unique map $!: X \to 1$ to the [[point space]] (the [[terminal object]] in [[Top]]) is a [[closed map]]. As a statement in ordinary [[point-set topology]], this is plainly a [[tautology]], trivially true for any space $X$. (Side remark: it is not at all a tautology in the more general setting of [[internal locales]] in [[toposes]]; the word "closed" is reserved for locales $X$ having that property. See also _[[closed morphism]]_.) However, even in ordinary point-set topology, $!: X \to 1$ is usually not *stably closed*. In more detail: the pullback of $!: X \to 1$ along a (or the) map $Y \to 1$ is the [[projection]] map

$$X \times Y \to Y$$

out of the [[product topological space]] and the issue is whether this map is closed. This leads us to the following proposition.

+-- {: .num_defn #projection}
###### Proposition
**([[closed-projection characterization of compactness]])**

A topological space $X$ is compact (def. \ref{hb}) precisely if for any topological space $Y$, the [[projection map]] $X \times Y \to Y$ out of their [[product topological space]] is [[closed map|closed]].

=--

Thus, compactness of $X$ is equivalent to $!: X \to 1$ being stably closed.
For a **proof**, see *[[closed-projection characterization of compactness]]*.

+-- {: .num_remark}
###### Remark

Contrary to possible appearance, the equivalence in prop. \ref{projection} does not require the [[axiom of choice]]; see [this MO question](http://mathoverflow.net/questions/42186/does-compact-iff-projections-are-closed-require-some-form-of-choice/42196) and answers, as well as [this page](/toddtrimble/published/Characterizations+of+compactness). See also the page [[compactness and stable closure]] (under construction).  This equivalence is also true for [[locales]], by way of proper maps; see below.

Moreover, the characterization of compact spaces via prop. \ref{projection} may be used to give an attractive proof of the [[Tychonoff theorem]], due to Clementino and Tholen. See [here](https://ncatlab.org/nlab/show/closed-projection%20characterization%20of%20compactness) for details.

=--

Of course, the notion of being stably closed applies to maps $p: X \to Y$ besides $!: X \to 1$. An analysis of this notion (that the pullback $f^\ast p: X \times_Y Z \to Z$ is a closed map for every map $f: Z \to Y$) leads to the correct notion of [[proper map]] in [[algebraic geometry]] and elsewhere.

Closely related to [[closed-projection characterization of compactness]], a characterisation of compactness in terms of logical [[quantification]] is featured in [[Paul Taylor]]'s [[Abstract Stone Duality]]:

+-- {: .num_prop #quantification}
###### Proposition

$X$ is compact iff for any space $Y$ and any open subset $U$ of $X \times Y$, the subset
$$ \forall_X U = \{ b : Y \;|\; \forall\; a: X,\; (a, b) \in U \} $$
is open in $Y$.

=--

To remove it from dependence on points, we can also write the definition like this:

+-- {: .num_defn #pointless}
###### Definition
$X$ is compact iff given any space $Y$ and any open $U$ in $X \times Y$, there exists an open $\forall_X U$ in $Y$ that satisfies the [[universal property]] of universal [[quantification]]:
$$ V \subseteq \forall_X U \;\Leftrightarrow\; X \times V \subseteq U $$
for every open $V$ in $Y$.
=--

A [[de Morgan duality|dual]] condition is satisfied by an [[overt space]].

## Properties

### Various

+-- {: .num_prop #UnionsAndIntersectionOfCompactSubspaces}
###### Proposition
**(unions and intersection of compact spaces)**

Let $(X,\tau)$ be a [[topological space]] and let

$$
  \{K_i \subset X\}_{i \in I}
$$

be a [[set]] of [[compact topological space|compact]] [[subspaces]].

1. If $I$ is a [[finite set]], then the [[union]] $\underset{i \in I}{\cup} K_i \subset X$ is itself a compact subspace;

1. If all $K_i \subset X$ are also [[closed subsets]] then their [[intersection]] $\underset{i \in I}{\cap} K_i \subset X$ is itself a compact subspace.

=--

+-- {: .proof}
###### Proof

Regarding the first statement:

Let $\{U_j \subset X\}_{j \in J}$ be an [[open cover]] of the union. Then this is also an open cover of each of the $K_i$, hence by their compactness there are finite subsets $J_i \subset J$ such that $\{U_{j_i} \subset X\}_{j_i \in J_i}$ is a finite cover of $K_i$. Accordingly then $\underset{i \in I}{\cup} \{ U_{j_i} \subset X \}_{j_i \in J_i}$ is a finite open cover of the union.

Regarding the second statement:

By the axioms on a [[topological space|topology]], the intersection of an arbitrary set of [[closed subsets]] is again closed. Hence the intersection of the closed compact subspaces is closed. But since [[subsets are closed in a closed subspace precisely if they are closed in the ambient space]] then for each $i \in I$ the intersection is a closed subspace of the compact space $K_i$. Since [[closed subspaces of compact spaces are compact]] it follows that the intersection is actually compact, too.

=--

+-- {: .num_prop #IntersectionCompactWithOpen}
###### Proposition
**(complements of compact with open subspaces is compact)**

Let $X$ be a [[topological space]]. Let

1. $K\subset X$ be a [[compact topological space|compact]] [[subspace]];

1. $U \subset X$ be an [[open subset]].

Then the [[complement]]

$$
  K \setminus U \subset Xcov
$$

is itself a compact subspace.

=--

+-- {: .proof #ProofIntersectionCompactWithOpen}
###### Proof

Let $\{V_i \subset K \setminus U\}$ be an [[open cover]] of the complement subspace. We need to show that this admits a finite subcover.

Observe that by definition of the [[subspace topologies]] on $K \setminus U$  and on $K$:

1. the $V_i \subset K \setminus U$ are still open when regarded as subsets $V_i \subset K$ of $K$;

1. also the intersection $K \cap U \subset K$ is an open subset of the compact subspace $K$.

This means that

$$
  \{
    V_i \subset K
  \}_{i \in I}
  \sqcup
   (U \cap K)
$$

is an [[open cover]] of $K$. Hence by the compactness of $K$, this has a finite subcover. If $U \cap K$ is retained in this subcover then it is of the form

$$
  \{ V_j \subset K\}_{j \subset J} \sqcup (U \cap K)
$$

otherwise it is of the form

$$
  \{ V_j \subset K\}_{j \subset J}
$$

with $J \subset I$ a [[finite set|finite]] [[subset]].

Since either of these is still a cover, also their restriction from $K$ to $K \setminus U$ is a cover of $K \setminus U$. But by assumption the $V_j$ are already restricted to $K \setminus U$, while the restriction of $(U \cap K)$ to $K \setminus U$ is empty. Therefore in either of the above cases we find that

$$
  \{ V_j \subset K \setminus U\}_{j \subset J}
$$

is a finite subcover of the original open cover of $K \setminus U$. Therefore $K \setminus U$ is compact.

=--

+-- {: .num_prop #ProductOfCompactSpacesIsCompact}
###### Proposition

Assuming the [[axiom of choice]], the category of compact spaces admits all small [[products]]. And, even without the axiom of choice, the category of compact locales admits all small products.

=--

\begin{proof}
The first statement follows from the [[Tychonoff theorem]]: every [[product topological space]] of compact spaces is itself again compact.
\end{proof}

+-- {: .num_remark}
###### Remark
However, the category of compact spaces does not admit [[equalizers]]. An explicit example is where one takes two maps $[0, 1] \to \{0, 1\}$ where the codomain is [[Sierpinski space]], where one of the maps $f$ has $f^{-1}(1) = [0, 1/2)$, and the other is the constant map at $1$. If an equalizer in $Comp$ existed, then $\hom(1, -): Comp \to Set$ would have to preserve it, so set-theoretically it would have to be $[0, 1/2)$. The topology on $[0, 1/2)$ would then have to be the same as or finer than the subspace topology in order for the equalizer map to be continuous. But if the subspace topology isn't compact, then no finer topology would make it compact either. (Here we are invoking the contrapositive of the proposition that if $(X, \tau)$ is compact and $\tau' \subseteq \tau$ is a coarser topology, then $(X, \tau')$ is also compact.)

On the other hand, the category of [[compactum|compact Hausdorff spaces]] does admit all [[limits]], and this fact does not require the full strength of the axiom of choice, but only a weaker choice principle called the [[ultrafilter principle]].
=--

+-- {: .num_prop }
###### Proposition

The direct [[image]] of a compact subspace under a [[continuous map]] is compact. A topological space becomes a [[bornological set]] by taking the bounded sets to be subsets contained in some compact subspace, and under this bornology, every continuous function is a bounded map.

If the spaces in question are $T_1$, then the sets with compact closure also constitute a bornology and continuous maps become bounded. In a non-Hausdorff situation these bornologies might differ because the closure of a compact set need not be compact.

=--

+-- {: .num_prop }
###### Proposition

A compact [[Hausdorff space]] must be [[normal space|normal]].  That is, the [[separation axioms]] $T_2$ through $T_4$ (when interpreted as an increasing sequence) are equivalent in the presence of compactness.

=--

+-- {: .num_prop }
###### Proposition

The [[Heine-Borel theorem]] asserts that a subspace $S \subset \mathbb{R}^n$ of a [[Cartesian space]] is compact precisely if it is [[closed subset|closed]] and [[bounded subset|bounded]].

=--

+-- {: .num_prop }
###### Proposition

We have:

1. [[compact subspaces of Hausdorff spaces are closed]].

1. [[closed subsets of compact spaces are compact]]

Hence:

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

=--

+-- {: .num_prop}
###### Proposition

[[sequentially compact metric spaces are equivalently compact metric spaces]]

[[countably compact metric spaces are equivalently compact metric spaces]]

[[compact spaces equivalently have converging subnet of every net]]

=--

+-- {: .num_prop}
###### Proposition

[[continuous images of compact spaces are compact]]

=--

+-- {: .num_prop}
###### Proposition

[[maps from compact spaces to Hausdorff spaces are closed and proper]]

=--

+-- {: .num_prop}
###### Proposition

[[continuous metric space valued function on compact metric space is uniformly continuous]]

=--

### Relation to compact objects in $Top$

One might expect that compact topological spaces are precisely the [[compact objects]] in [[Top]] in the abstract sense of [[category theory]], but this identification requires care, the naive version fails. See at _[compact object -- Compact objects in Top](compact+object#CompactObjectsInTop)_

## Examples

### General
 {#ExamplesGeneral}

+-- {: .num_prop }
###### Proposition

A [[discrete space]] is compact iff its underlying set is [[finite set|finite]].

In [[constructive mathematics]], a discrete space is compact iff its underlying set is [[Kuratowski-finite]].

=--

+-- {: .num_example #CompactClosedInterval}
###### Example
**([[closed intervals]] are [[compact topological space|compact]])**

For any $a \lt b \in \mathbb{R}$ the [[closed interval]]

$$
  [a,b] \subset \mathbb{R}
$$

regarded with its [[topological subspace|subspace topology]]
of [[Euclidean space]] with its [[metric topology]] is
a compact topological space.

=--

(See [[closed intervals are compact topological spaces]].)

In contrast:

+-- {: .num_example}
###### Nonexample

For all $n \in \mathbb{N}$, $n \gt 0$, the [[Euclidean space]] $\mathbb{R}^n$, regarded with its [[metric topology]], is _not_ compact.

=--

This is a special case of the [[Heine-Borel theorem]] (example \ref{HeineBorelTheorem} below), but for illustration it is instructive to consider the direct proof:

+-- {: .proof}
###### Proof

Pick any $\epsilon \in (0,1/2)$. Consider the open cover of $\mathbb{R}^n$ given by

$$
  \left\{
      U_n \coloneqq
     (n-\epsilon, n+1+\epsilon)
     \times
     \mathbb{R}^{n-1}
     \;\subset\;
     \mathbb{R}^{n+1}
  \right\}_{n \in \mathbb{Z}}
  \,.
$$

This is not a finite cover, and removing any one of its patches $U_n$, it ceases to be a cover, since the points of the form $(n + \epsilon, x_2, x_3, \cdots, x_n)$ are contained only in $U_n$ and in no other patch.

=--

+-- {: .num_example}
###### Example

By the [[Tychonoff theorem]], every [[product topological spaces]] of compact spaces is again compact.

=--

This implies the [[Heine-Borel theorem]], saying that:

+-- {: .num_example #HeineBorelTheorem}
###### Example
**([[Heine-Borel theorem]])**

Every [[bounded subset|bounded]] and [[closed subspace]] of a [[Euclidean space]] is compact.

=--

In particular:

+-- {: .num_example}
###### Nonexample

The [[open intervals]] $(a,b) \subset \mathbb{R}$ and the [[half-open intervals]] $[a,b)  \subset \mathbb{R}$ and $(a,b] \subset \mathbb{R}$ are not compact.

=--

+-- {: .num_example }
###### Example

The [[Mandelbrot set]], regarded as a [[subspace]] of the [[plane]], is a compact space (see [this prop.](/nlab/show/Mandelbrot+set#MndlbrtIsCompact)).

=--

+-- {: .num_example}
###### Example
Any set when equipped with the [[cofinite topology]] forms a compact space. This space is also $T_1$ (i.e., [[singletons]] are closed), but is not Hausdorff if the underlying set is [[infinite set|infinite]].
=--

### Compact spaces which are not sequentially compact
 {#CompactSpacesWhichAreNotSequentiallyCompact}

A famous example of a space that is compact, but not [[sequentially compact space|sequentially compact]] is the product space
$$
    \{0,1\}^{I} \coloneqq \{0, 1\}^{[0, 1]}
  \coloneqq \underset{[0,1]}{\prod} \{0,1\}
$$
with the [[product topology]].

This space is compact by the [[Tychonoff theorem]].

But it is not [[sequentially compact topological space|sequentially compact]]. We now discuss why. (We essentially follow [Steen-Seebach 70, item 105](#SteenSeebach70)).

Points of $\{0,1\}^{I}$ are functions $I \to \{0,1\}$, and the product topology is the topology of pointwise convergence.

Define a sequence $(a_n)$ in $I^{I}$ with $a_n(x)$ being the nth digit in the binary expansion of $x$ (we make the convention that binary expansions do not end in sequences of $1$s).  Let $a \coloneqq (a_{n_k})$ be a subsequence and define $p_a \in I$ by the binary expansion that has a $0$ in the $n_k$th position if $k$ is even and a $1$ if $k$ is odd (and, for definiteness and to ensure that this fits with our convention, define it to be $0$ elsewhere).  Then the projection of $(a_{n_k})$ at the $p_a$th coordinate is $1, 0, 1,0,...$ which is not convergent.  Hence $(a_{n_k})$ is not convergent.

(Basically that's the diagonal trick of [[Cantor's theorem]]).

However, as $\{0,1\}^I$ is compact, $a$ has a [[convergence|convergent]] [[subnet]].  An explicit construction of a convergent subset, given a [[cluster point]] $a$, is as follows.  A function $a \colon I \to \{0,1\}$ is a [[cluster point]] of $(a_n)$ if, for any $p_1, \dots, p_n \in I$ the set

$$
A(p_1,\dots,p_n) \coloneqq \{k \in \mathbb{R} : a_k(p_i) = a(p_i) \forall i\}
$$

is infinite.  We index our subnet by the family of finite subsets of $I$ and define the subnet function $\mathcal{F}(I) \to \mathbb{N}$ to be

$$
\{p_1,\dots,p_n\} \mapsto \min A(p_1,\dots,p_n)
$$

This is a convergent sub-[[net]].

## In synthetic topology

In [[synthetic topology]], where 'space' means simply 'set' (or [[type]], i.e. the basic objects of our foundational system), one natural notion of "compact space" is a [[covert set]], i.e. a set whose discrete topology is covert.  This includes the expected examples in various [[gros toposes]].

## Compact spaces and proper maps

A space $X$ is compact if and only if the unique map $X\to 1$ is [[proper map|proper]].  Thus, properness is a "relativized" version of compactness.

For topological spaces, this is either a definition of "proper map" (closed with compact fibers) or follows from the above characterization of compactness in terms of projections being closed maps (if proper maps are defined to be those that are universally closed).  For locales, it follows from the definition of proper map (a closed map such that $f_*$ preserves directed joins) and the fact that compact locales are automatically [[covert space|covert]] (see [[covert space]] for a proof).

## History

Compact spaces were introduced (under the name of “bicompact spaces”) by [[Paul Alexandroff]] and [[Paul Urysohn]] around 1924, see the 92nd volume of Mathematische Annalen, especially [AU](#AlexandroffUrysohn1924).

## Related concepts

* [[Lebesgue number lemma]]

* [[sequentially compact topological space]], [[countably compact topological space]], [[paracompact topological space]], [[locally compact topological space]], [[strongly compact topological space]], [[compactly generated topological space]]

* [[compactification]]

  * [[one-point compactification]]

  * [[Stone-Cech compactification]]

* [[compact-open topology]]

* [[closed manifold]]

* [[compact topos]]

* [[compactly supported cohomology]]

* [[complete algebraic variety]]

## References

Introduced and studied by [[Paul Alexandroff]] and [[Paul Urysohn]] in

* {#AlexandroffUrysohn1924} [[Paul Alexandroff]], [[Paul Urysohn]], _Zur Theorie der topologischen Räume_, [EuDML](https://eudml.org/doc/159069).

For general references see the [list](topology#References) at *[[topology]]*.

Examples of compact spaces that are not sequentially compact are in

* {#SteenSeebach70} [[Lynn Arthur Steen]], [[J. Arthur Seebach]], _[[Counterexamples in Topology]]_, Springer 1970/1978 ([doi:10.1007/978-1-4612-6290-9](https://link.springer.com/book/10.1007/978-1-4612-6290-9))

On compact [[metric spaces]]:

* Roland Speicher, _Compact metric spaces_ ([pdf](http://www.mast.queensu.ca/~speicher/Chapter10.pdf))

* Alan Sokal, _Compactness of metric spaces_ ([pdf](http://www.ucl.ac.uk/~ucahad0/3103_handout_2.pdf))

For [[proper base change theorem]] e.g.

* {#Milne} [[James Milne]], section 17 of _[[Lectures on Étale Cohomology]]_

[[!redirects compact space]]
[[!redirects compact spaces]]
[[!redirects compact topological space]]
[[!redirects compact topological spaces]]
[[!redirects compact convergence space]]
[[!redirects compact convergence spaces]]
[[!redirects compact locale]]
[[!redirects compact locales]]

[[!redirects compact manifold]]
[[!redirects compact manifolds]]

[[!redirects compact subspace]]
[[!redirects compact subspaces]]
[[!redirects compact subset]]
[[!redirects compact subsets]]
[[!redirects compact set]]
[[!redirects compact sets]]

[[!redirects compact]]

[[!redirects compactness]]


compact spaces equivalently have converging subnets

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[classical mathematics]], for [[metric spaces]] with their [[metric topology]] it is true that [[sequentially compact metric spaces are equivalently compact metric spaces]]. The analogous statement fails for more general [[topological spaces]]: for them, being [[sequentially compact topological space|sequentially compact]] in general neither implies nor is implied by being [[compact topological space|compact]] (see the counter-examples [here](sequentially+compact+topological+space#Examples)).

But the failure of this equivalence is not due to a deficit in the concept of [[convergence]] but in the concept of [[sequences]] and sub-sequences. If the latter are generalized to [[nets]] and [[sub-nets]] (beware that the definition of sub-nets is slightly non-obvious), then the analogue of the statement remains true in [[classical mathematics]]: A topological space is [[compact topological space|compact]] precisely if every [[net]] in it has a [[sub-net]] that [[converges]].

In [[constructive mathematics]], on the other hand, this property implies the [[law of excluded middle]], so is thus equivalent to the [[law of excluded middle]].

## Statement

+-- {: .num_prop #CompactSpacesEquivalentlyHaveConvergetSubnets}
###### Proposition
**(compact spaces are equivalently those for which every net has a converging subnet)**

Assuming [[excluded middle]] and the [[axiom of choice]], then:

A [[topological space]] $(X,\tau)$ is [[compact topological space|compact]] precisely if every [[net]] in $X$ has a [[sub-net]] that [[convergence|converges]].

=--

We break this up into lemmas \ref{InACompactSpaceEveryNetHasAConvergentSubnet} and \ref{IfEveryNetHasConvergentSubnetThenSpaceIsCompact}:

+-- {: .num_example #InACompactSpaceEveryNetHasAConvergentSubnet}
###### Lemma
**(in a compact space, every net has a convergent subnet)**

Let $(X,\tau)$ be a [[compact topological space]]. Then every net in $X$ has a convergent subnet.

=--

+-- {: .proof}
###### Proof

Let $\nu \colon A \to X$ be a net. We need to show that there is a subnet which converges.

For $a \in A$ consider the [[topological closures]] $Cl(S_a)$ of the sets $S_a$ of elements of the net beyond some fixed index:

$$
  S_a
    \;\coloneqq\;
  \left\{
    \nu_b  \in X \;\vert\; b \geq a
  \right\}
  \subset X
  \,.
$$

Observe that the set $\{S_a \subset X\}_{a \in A}$ and hence also the set $\{Cl(S_a) \subset X\}_{a \in A}$ has the [[finite intersection property]], by the fact that $A$ is a [[directed set]]. Therefore [this prop. ](finite+intersection+property#CompactnessInTermsOfFiniteIntersectionProperty) implies from the assumption of $X$ being compact that the intersection of _all_ the $Cl(S_a)$ is non-empty, hence that there is an element

$$
  x \in \underset{a \in A}{\cap}  Cl(S_a)
  \,.
$$

In particular every [[neighbourhood]] $U_x$ of $x$ intersects each of the $Cl(S_a)$, and hence also each of the $S_a$. By definition of the $S_a$, this means that for every $a \in A$ there exists $b \geq a$ such that $\nu_b \in U_x$, hence that $x$ is a [[cluster point]] of the net.

We will now produce a [[sub-net]]

$$
  \array{
     B && \overset{f}{\longrightarrow} && A
     \\
     & \searrow && \swarrow_{\nu}
     \\
     && X
  }
$$

that converges to this cluster point. To this end, we first need to build the domain directed set $B$. Take it to be the sub-directed set of the Cartesian product directed set of $A$ with the directed neighbourhood set $Nbhd_X(x)$ of $x$

$$
  B \subset A_{\leq} \times Nbhd_X(x)_{\supset}
$$

on those pairs such that the element of the net indexed by the first component is is contained in the second component:

$$
  B
    \;\coloneqq\;
  \left\{
     (a,U_x)  \,\vert \, \nu_a \in U_X
  \right\}
  \,.
$$

It is clear $B$ is a [[preordered set]]. We need to check that it is indeed directed, in that every pair of elements $(a_1, U_1)$, $(a_2, U_2)$ has a common upper bound $(a_{bd}, U_{bd})$. Now since $A$ itself is directed, there is an upper bound $a_3 \geq a_1, a_2$, and since $x$ is a cluster point of the net there is moreover an $a_{bd} \geq a_3 \geq a_1, a_3$ such that $\nu_{a_{bd}} \in U_1 \cap U_2$. Hence with $U_{bd} \coloneqq U_1 \cap U_2$ we have obtained the required pair.

Next take the function $f$ to be given by

$$
  \array{
     B &\overset{f}{\longrightarrow}& A
     \\
     (a, U) &\overset{\phantom{AAA}}{\mapsto}& a
  }
  \,.
$$

This is clearly order preserving, and it is [[cofinal]] since it is even a surjection. Hence we have defined a subnet $\nu \circ f$.

It now remains to see that $\nu \circ f$ converges to $x$, hence that for every open neighbourhood $U_x$ of $x$ we may find $(a,U)$ such that for all $(b,V)$ with $a \leq b$ and $U \supset V$ then $\nu(f(b,V)) = \nu(b) \in U_x$.  Now by the nature of $x$ there exists some $a$ with $\nu_a \in U_x$, and hence if we take $U \coloneqq U_x$ then nature of $B$ implies that with $(b, V) \geq (a,U_x)$ then $b \in V \subset U_x$.

=--

+-- {: .num_example #IfEveryNetHasConvergentSubnetThenSpaceIsCompact}
###### Lemma

Assuming [[excluded middle]], then:

Let $(X,\tau)$ be a [[topological space]]. If every [[net]] in $X$ has a [[subnet]] that [[convergence|converges]], then $(X,\tau)$ is a [[compact topological space]].

=--

+-- {: .proof}
###### Proof

By [[excluded middle]] we may equivalently prove the [[contrapositive]]: If $(X,\tau)$ is not compact, then not every net in $X$ has a convergent subnet.

Hence assume that $(X,\tau)$ is not compact. We need to produce a net without a convergent subnet.

Again by [[excluded middle]], then by [this prop.](finite+intersection+property#CompactnessInTermsOfFiniteIntersectionProperty) $(X,\tau)$ not being compact means equivalently that there exists a set $\{C_i \subset X\}_{i \in I}$ of [[closed subsets]] satisfying the [[finite intersection property]], but such that their intersection is empty: $\underset{i \in I}{\cap} C_i = \emptyset$.

Consider then $P_{fin}(I)$, the set of [[finite set|finite]] [[subsets]] of $I$. By the assumption that $\{C_i \subset X\}_{i \in I}$ satisfies the [[finite intersection property]], we may [[axiom of choice|choose]] for each $J \in P_{fin}(I)$ an element

$$
  x_J \in \underset{i \in J \subset I}{\cap} C_i
  \,.
$$

Now $P_{fin}(X)$ regarded as a [[preordered set]] under inclusion of subsets is clearly a [[directed set]], with an upper bound of two finite subsets given by their [[union]]. Therefore we have defined a net

$$
  \array{
     P_{fin}(X)_{\subset} &\overset{\nu}{\longrightarrow}& X
     \\
     J &\overset{\phantom{AAA}}{\mapsto}& x_J
  }
  \,.
$$

We will show that this net has no converging subnet.

Assume on the contrary that there were a subnet

$$
  \array{
     B && \overset{f}{\longrightarrow} && P_{fin}(X)
     \\
     & \searrow && \swarrow_{\nu}
     \\
     && X
  }
$$

which converges to some $x \in X$.

By the assumption that $\underset{i \in I}{\cap} C_i = \emptyset$, there would exist an $i_x \in I$ such that $x \neq C_{i_x}$, and because $C_i$ is a [[closed subset]], there would exist even an [[open neighbourhood]] $U_x$ of $x$ such that $U_x \cap C_{i_x} = \emptyset$. This would imply that $x_J \neq U_x$ for all $J \supset \{i_x\}$.

Now since the function $f$ defining the subset is cofinal, there would exist $b_1 \in B$ such that $\{i_x\} \subset f(b_1)$. Moreover, by the assumption that the subnet converges, there would also be $b_2 \in B$ such that $\nu_{b_2 \leq \bullet} \in U_x$. Since $B$ is directed, there would then be an upper bound $b \geq b_1, b_2$ of these two elements. This hence satisfies both $\nu_{f(e)} \in U_x$ as well as $\{i_x\} \subset f(b_1) \subset f(b)$. But the latter of these two means that $\nu_{f(b)}$ is not in $U_x$, which is a contradiction to the former. Thus we have a [[proof by contradiction]].

=--

## Failure in constructive mathematics

In [[constructive mathematics]], this statement is equivalent to [[excluded middle]].

+--{: .num_prop #TwoElementSetWithDiscreteTopologyIsCompact}
###### Proposition
The set of two elements $\{0, 1\}$ with its [[discrete topology]] is a [[compact space]].
=--

+-- {: .proof}
###### Proof
Let $U$ be an [[open cover]] of $\{0, 1\}$. Then there exists $i_0$ such that $0 \in U_{i_0}$ and $i_1$ such that $1 \in U_{i_1}$. Then $U_{i_0}, U_{i_1}$ form a [[finite]] [[subobject|sub]][[cover]] of $\{0, 1\}$: the [[universal property]] of the set of two elements implies that if $x \in \{0, 1\}$, then $x = 0$ or $x = 1$, which means that if $x = 0$, then $x \in U_{i_0}$, and if $x = 1$, then $x \in U_{i_1}$.
=--

+--{: .num_prop #InhabitedSubsetsOfTwoElementSetAreDirectedPosets}
###### Proposition
Every [[inhabited set|inhabited]] [[subset]] of $\{0, 1\}$ is a [[directed set|directed]] [[poset]].
=--

+-- {: .proof}
###### Proof
Let $P$ be an inhabited subset of $\{0, 1\}$. Then $P$ is a [[directed poset]]: it is inhabited, it has a [[partial order]] inherited from the partial order defined on $\{0, 1\}$ by $0 \leq 1$, $0 \leq 0$, $1 \leq 1$, and there is an upper bound function $\beta:P \times P \to P$ defined for elements $a \in P$ and $b \in P$ as $\beta(a, b) = b$ if $a = 0$, $\beta(a, b) = a$ if $a = 1$, $\beta(a, b) = a$ if $b = 0$, and $\beta(a, b) = b$ if $b = 1$: this means that $a \leq \beta(a, b)$ and $b \leq \beta(a, b)$.
=--

+--{: .num_lem #SubsetInclusionsToTwoElementSetAreNets}
###### Lemma
Every subset inclusion $i:P \hookrightarrow \{0, 1\}$ of an inhabited subset $P \subseteq \{0, 1\}$ is a net in $\{0, 1\}$.
=--

+-- {: .proof}
###### Proof
Since $P$ is a directed poset, it follows that any [[function]] $i:P \to \{0, 1\}$ is a net in $\{0, 1\}$. Thus the subset inclusion $i:P \hookrightarrow \{0, 1\}$ is a net.
=--

+-- {: .num_prop #TwoElementSetHasConvergetSubnetsImpliesExcludedMiddle}
###### Proposition
Given the set of two elements $\{0, 1\}$ with its [[discrete topology]], if every net in $\{0, 1\}$ has a converging subnet, then the [[law of excluded middle]] is true.
=--

+-- {: .proof}
###### Proof
Let $Q$ be a [[directed set]] (i.e. a directed [[preordered set]]). Assume that every [[net]] in $\{0, 1\}$ has a converging subnet, where $\{0, 1\}$ has the discrete topology. Because the subset inclusion $i \colon P \to \{0, 1\}$ for any subset $P \subseteq \{0, 1\}$ is a net, there is a [[cofinal]] function $f:Q \to P$ from $Q$ to $P$ such that $i \circ f$ is a net which converges to an element $x \in \{0, 1\}$, which by the universal property of $\{0, 1\}$ satisfies either $x = 0$ or $x = 1$.

Because $\{0, 1\}$ has the discrete topology, $\{1\}$ is an open of $\{0, 1\}$. If $x = 1$, then there is an element $q \in Q$ such that $f(q) \in \{1\}$, but $f(q) \in P$, so $1 \in P$.

If $x = 0$, then suppose that $1 \in P$. Because $f$ is [[cofinal]], there is an element $q \in Q$ such that $f(q) \geq 1$, and by the definition of the partial order in $\{0, 1\}$, $f(q) = 1$, which means that for any other element $r \in Q$ such that $r \geq q$, $f(r) \geq f(q)$, $f(r) \geq 1$, and $f(r) = 1$, which contradicts the fact that $f$ converges to $0$. Thus, $1 \notin P$.

Thus, assuming that $\{0, 1\}$ comes with its [[discrete topology]], if every net in $\{0, 1\}$ has a converging subnet, then for every inhabited subset $P$ of $\{0, 1\}$, either $1 \in P$ or $1 \notin P$. For the empty subset, $1 \notin \emptyset$, so for every subset $P$ of $\{0, 1\}$, either $1 \in P$ or $1 \notin P$.

Given any [[proposition]] $p$, let us define the subset of $\{0, 1\}$ as

$$
  P_p \;\coloneqq\;
  \Big\{
    x \in \{0, 1\}
  \Big\vert
    (x = 0) \vee \big((x = 1) \wedge p\big)
  \Big\}
$$

It follows that if $1 \in P_p$, then $p$, and if $1 \notin P_p$, then $\not p$. Thus, the [[law of excluded middle]] is true for all [[propositions]] $p$.
=--

## Related concepts

* [[sequentially compact metric spaces are equivalently compact metric spaces]]

* [[countably compact metric spaces are equivalently compact metric spaces]]

* [[sequentially compact metric spaces are totally bounded]]

* [[Lebesgue number lemma]]

[[!redirects compact spaces equivalently have converging subnet of every net]]


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[[!include topology - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_lemma #SeparationByNeighbourhoodsOfPointsFromCompactSubsetsInHausdorffSpaces}
###### Lemma
**(compact subspaces in Hausdorff spaces are separated by neighbourhoods from points)**

Let

1. $(X,\tau)$ be a [[Hausdorff topological space]];

1. $Y \subset X$ a [[compact topological space|compact]] [[subspace]].

Then for every $x \in X \backslash Y$ there exists

1. an [[open neighbourhood]] $U_x \supset \{x\}$;

1. an open neighbourhood $U_Y \supset Y$

such that

* they are still disjoint: $U_x \cap U_Y = \emptyset$.

=--

As a direct consequence [[compact subspaces of Hausdorff spaces are closed]].

+-- {: .proof}
###### Proof

By the assumption that $(X,\tau)$ is Hausdorff, we find for every point $y \in Y$ disjoint open neighbourhoods $U_{x,y} \supset \{x\}$ and $U_y \supset \{y\}$. By the nature of the [[subspace topology]] of $Y$, the restriction of all the $U_y$ to $Y$ is an [[open cover]] of $Y$:

$$
  \left\{
    (U_y \cap Y) \subset Y
  \right\}_{y \in Y}
  \,.
$$

Now by the assumption that $Y$ is compact, there exists a finite subcover, hence a [[finite set]] $S \subset Y$ such that

$$
  \left\{
    (U_y \cap Y) \subset Y
  \right\}_{y \in S \subset Y}
$$

is still a cover.

But the finite intersection

$$
  U_x \coloneqq \underset{s \in S \subset Y}{\cap} U_{x,s}
$$

of the corresponding open neighbourhoods of $x$ is still open, and by construction it is disjoint from all the $U_{s}$, hence also from their union

$$
  U_Y \coloneqq \underset{s \in S \subset Y}{\cup}  U_s
  \,.
$$

Therefore $U_x$ and $U_Y$ are two open subsets as required.

=--


compact subspaces of Hausdorff spaces are closed

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=--
=--
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#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop #CompactSubspacesOfHausdorffSpacesAreClosed}
###### Proposition

A [[compact subspace]] $K$ of a [[Hausdorff topological space]] $X$ is a [[closed subspace]].

=--

+-- {: .proof}
###### Proof

Let $x \in X \backslash K$ be any point of $X$ not contained in $K$. We need to show that there exists an [[open neighbourhood]] of $x$ in $X$ which does not [[intersection|intersect]] $K$.

By assumption that $X$ is Hausdorff, there exist for each $y \in K$ disjoint open neighbourhoods $y \subset U_y \subset X$ and $x \subset V_y \subset X$. Clearly the [[union]] of all the $U_y$ is an [[open cover]] of $K$

$$
  K \subset \underset{y \in K}{\cup} U_y
  \,.
$$

Hence by assumption that $K$ is compact, there exists a [[finite set|finite]] [[subset]] $S \subset K$ of points in $K$ such that the $U_s$ for $s \in S$ still cover $K$:

$$
  K \subset \underset{y \in S \subset K}{\cup} U_y
  \,.
$$

Since $S$ is finite, the intersection

$$
  U_x \coloneqq  \underset{y \in S \subset K}{\cap} V_y
$$

is still open, and by construction it is disjoint from all $U_y$ for $y \in S$, hence in particular disjoint from $K$, and it contains $x$. Hence $U_x$ is an open neighbourhood of $x$ as required.

=--

## Remarks

+-- {: .num_defn #NecessityOfHausdorffAssumption}
###### Remark

The statement of prop. \ref{CompactSubspacesOfHausdorffSpacesAreClosed} does not hold with the Hausdorff condition replaced by weaker [[separation axioms|separation assumptions]].

To see this, consider a [[dense subspace]] of a topological space. For instance the [[affine scheme]] [[Spec(Z)]] is $T_1$ but the conclusion of the proposition does not hold for the [[generic point]].

=--

## Related entries

* [[closed subsets of compact spaces are compact]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[continuous images of compact spaces are compact]]

* [[a CW-complex is a Hausdorff space]]

* [[Hausdorff implies sober]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

## References

* ProofWiki, _[Compact Subspace of Hausdorff Space is Closed](https://proofwiki.org/wiki/Compact_Subspace_of_Hausdorff_Space_is_Closed)_


compact support

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+-- {: .hide}
[[!include topology - contents]]
=--
=--
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# Contents
* table of contents
{: toc}

## Definition

A [[function]] $f\colon X \to V$ on a [[topological space]] with values in a [[vector space]] $V$ (or really any [[pointed set]] with the basepoint called $0$) has __compact support__ (or is __compactly supported__) if the [[closure]] of its [[support]], the set of points where it is non-zero, is a [[compact space|compact subset]].  That is, the subset $\overline{f^{-1}(V \setminus \{0\})}$ is a [[compact subspace|compact subset]] of $X$.

Typically, $X$ is [[Hausdorff space|Hausdorff]], $f$ is a [[continuous function]], and $V$ is a Hausdorff [[topological vector space]] (or at least a pointed topological space whose basepoint is closed), so that $f^{-1}(V \setminus \{0\})$ is an [[open subspace]] of $X$, yet any compact subspace of $X$ must be [[closed subspace|closed]]; this is why we take the closure.

If we work with [[locales]] instead of topological spaces, then a closed point $0$ in $V$ still has an open subspace of $V$ as its formal dual, and we use this in the place of $V \setminus \{0\}$.

## Related concepts

* [[bump function]]

* [[compactly supported distribution]]

* [[integrable function]], [[square-integrable function]]

* [[bounded function]]

* __compactly supported function__

* [[measurable function]]

* [[rapidly decreasing function]]

[[!redirects compact support]]
[[!redirects compact supports]]

[[!redirects compactly supported continuous function]]
[[!redirects compactly supported continuous functions]]

[[!redirects compactly supported function]]
[[!redirects compactly supported functions]]


compact symplectic group

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+-- {: .hide}
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=--
=--
=--

# The compact symplectic groups
* table of contents
{: toc}

## Definition

The __compact symplectic group__ (in $n$ dimensions) is the [[simply-connected]], maximal [[compact]] real [[Lie group]] $Sp(n)$ sitting inside the complex [[symplectic group]] $Sp(2n,\mathbb{C})$. It can also be seen as the [[orthogonal group of an inner product space|orthogonal group]] of $\mathbb{H}^n$. It may also be defined as the [[intersection]] of $Sp(2n,\mathbb{C})$ with the [[unitary group]] $\mathrm{U}(2n) = \mathrm{U}(2n,\mathbb{C})$ within the [[general linear group]] $GL(2n,\mathbb{C})$.  It should *not* be confused with the real [[symplectic group]] $Sp(2n,\mathbb{R})$ (which is not simply-connected).

The compact symplectic group is also called the _[[quaternionic unitary group]]_, and see there for more.

## Examples

* The group $Sp(1)$ is isomorphic to the [[special unitary group]] $SU(2)$ and also to the [[spin group]] $Spin(3)$. Its underlying [[smooth manifold]] is the 3-[[sphere]].

* $Sp(2)$ is isomorphic to [[Spin(5)]] (see there for a proof).


compact topological group

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+-- {: .hide}
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=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[topological group]] is called **compact** if its underlying [[topological space]] is a [[compact topological space]].

## Related concepts

* [[locally compact topological group]]

* [[maximal compact subgroup]]

* [[proper topological groupoid]]

* [[compact Lie group]]

## References

On space of [[subgroups]] of compact topological groups:

* S. Fisher, P. Gartside, *On the space of subgroups of a compact group I*, Topology and its Applications Volume 156, Issue 5, 15 February 2009, Pages 862-871 ([doi:10.1016/j.topol.2008.11.005](https://doi.org/10.1016/j.topol.2008.11.005))

[[!redirects compact topological groups]]

[[!redirects compact group]]
[[!redirects compact groups]]


compact topos > history

see _[[proper geometric morphism]]_