closed monoidal category

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### Context
#### Monoidal categories
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[[!include monoidal categories - contents]]
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# Contents
* table of contents
{: toc}

## Idea

A *closed monoidal category* is a [[monoidal category]] $\big( \mathcal{C}, \, \mathbb{1},\, \otimes \big)$ that is also a [[closed category]], in a compatible way, i.e. such that for each [[object]] $Y \,\in\, \mathcal{C}$

1. the [[functor]]

   $$
     (-) \otimes Y  \;\colon\; C \longrightarrow C
   $$

   of forming the [[tensor product]] with $Y$,

1. has a [[right adjoint]] [[functor]]

   $$
     [Y,-] \;\colon\; C \longrightarrow C
   $$

   forming the [[internal-hom]] out of $Y$

in that for all [[triples]] of [[objects]] $X,\, Y,\, Z $ there is a [[natural isomorphism|natural]] [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) of the following form:

$$
  Hom_{\mathcal{C}}(X \otimes Y,\, Z)
    \;\simeq\;
  Hom_{\mathcal{C}}\big(X, [Y,Z] \big)
  \,.
$$

The archetypical example is the [[cartesian closed category]] of [[Sets]], where the [[internal hom]] is given by forming [[function sets]], so that the above [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) expresses the fact that a [[function]] $(x,y) \mapsto f(x,y)$ of two variables is equivalently a function of a single variable with values in functions of the other variable: $x \mapsto \big(y \mapsto f(x,y) \big)$. In [[formal logic]] this is known as "[[currying]]".

More abstractly, from the point of view of [[2-category]] theory, closed monoidal categories may be regarded as a special case of the notion of [[closed pseudomonoids]] in a [[monoidal bicategory]].

## Definitions

### Symmetric closed monoidal category

A [[symmetric monoidal category|symmetric]] [[monoidal category]] $C$ is **closed** (cf. *[[symmetric monoidal closed category]]*) if for all [[objects]] $b \in C_0$ the [[tensor product]] [[functor]] $ b \otimes - \colon C \to C$ has a [[right adjoint]] functor $[b,-] \colon C \to C$.

$$
  \underset{b \in C}{\forall}
  \;\;
  C
  \underoverset
    {\underset{[b,-]}{\longrightarrow}}
    {\overset{b \otimes (-)}{\longleftarrow}}
    {\;\;\;\;\;\bot\;\;\;\;\;}
  C
  \,.
$$

This [means](adjoint+functor#InTermsOfHomIsomorphism) that for all $a,b,c \in C_0$ we have a [[natural bijection]]

$$
  Hom_C(a \otimes b, c)
   \;\simeq\;
  Hom_C(a, [b,c])
  \,,
$$

[[natural transformation|natural]] in all [[variables|arguments]].

The object $[b,c]$ is called the **[[internal hom]]** of $b$ and $c$. This is commonly also denoted by lower case $hom(b,c)$ (and then sometimes underlined).

### Cartesian closed monoidal category

If the monoidal structure of $C$ is [[cartesian monoidal category|cartesian]] (and so in particular [[symmetric monoidal category|symmetric monoidal]]), then $C$ is called **[[cartesian closed]]**.  In this case the internal hom is often called an **[[exponential object]]** and written $c^b$.

### Left-, right- and bi-closed monoidal category

If $C$ is [[monoidal category|monoidal]] but not necessarily [[symmetric monoidal category|symmetric]] or even [[braided monoidal category|braided]], then left and right [[tensor product]] $-\otimes b$ and $b\otimes -$ may be inequivalent functors, and either one or both may have an adjoint.  The terminology here is less standard, but many people use **left closed**, **right closed**, and **biclosed** monoidal category to indicate, respectively, that the left tensor product, the right tensor product functors or both have [[right adjoints]].  (Other authors simply say **closed** instead of biclosed.)  So in particular a symmetric closed monoidal category is automatically biclosed.

The analogue of exponential objects for monoidal categories are [[residual|left and right residuals]].

## Properties

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

For $(\mathcal{C}, \otimes, 1)$ a closed monoidal category with [[internal hom]] denoted $[-,-]$, then not only are there [[natural bijections]]

$$
   Hom_{\mathcal{C}}(X \otimes Y, Z)
  \simeq
   Hom_{\mathcal{C}}(X, [Y,Z])
$$

but these isomorphisms themselves "internalize" to isomorphisms in $\mathcal{C}$ of the form

$$
  [X \otimes Y, Z]
    \simeq
  [X,[Y,Z]]
  \,.
$$

=--

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

By the external natural bijections there is for every $A \in \mathcal{C}$ a composite natural bijection

$$
  Hom_{\mathcal{C}}(A, [X \otimes Y, Z])
   \simeq
  Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z)
   \simeq
  Hom_{\mathcal{C}}((A \otimes X) \otimes Y, Z)
    \simeq
  Hom_{\mathcal{C}}(A \otimes X, [Y,Z])
    \simeq
  Hom_{\mathcal{C}}(A,[X,[Y,Z]])
  \,.
$$

Since this holds for every $A \in \mathcal{C}$, the [[Yoneda lemma]] (namely the [[fully faithful functor|fully faithfulness]] of the [[Yoneda embedding]]) implies that there is already an isomorphism

$$
  [X \otimes Y, Z] \simeq [X,[Y,Z]]
  \,.
$$

=--

## Examples
 {#Examples}

\begin{example}\label{ExampleSets}
The tautological example is the category [[Set]] of sets with its [[Cartesian product]] (a [[cartesian closed monoidal category]]): the collection of functions between any two sets is itself a set -- the [[function set]].
\end{example}

\begin{example}\label{ExamplePresheaves}
In generalization of Exp. \ref{ExampleSets} and as a special case of Exp. \ref{FunctorCategory}, every [[category of presheaves]], i.e. every [[functor category]] of the form $Func(\mathcal{C}^{op}, Set)$, is a [[cartesian closed monoidal category]], details are spelled out at *[[closed monoidal structure on presheaves]]*.
\end{example}

\begin{example}
In further generalization of Exp \ref{ExamplePresheaves}, any [[topos]] is a [[cartesian closed monoidal category]].
\end{example}

\begin{example}\label{AbelianGroups}
The category [[Ab]] of [[abelian groups]] with its [[tensor product of abelian groups]] is (non-cartesian!) closed: for any two abelian groups $A, B$ the set of homomorphisms $A \to B$ carries (pointwise defined) abelian group structure.
\end{example}

\begin{example}\label{ExampleModules}
In generalization of Exp. \ref{AbelianGroups} (which is the case $R =$ [[integers|$\mathbb{Z}$]]), for $R$ a [[commutative ring]], the category $R$[[Mod]] of $R$-[[modules]] with its usual [[tensor product of modules]] is (non-cartesian!) closed monoidal. This is one of the first hom-tensor adjunctions that appeared in [[algebra]].
\end{example}

\begin{example}
In specialization of Exp. \ref{ExampleModules} to the case that $R = \mathbb{K}$ is a [[field]], the category $\mathbb{K}$-[[Vect]] of $\mathbb{K}$-[[vector spaces]] and $\mathbb{K}$-[[linear maps]] between them is (non-cartesian!) closed monoidal with respect to the usual [[tensor product of vector spaces]]. Here the [[internal hom]] assigns vector spaces of [[linear maps]] (with the vector operations given argument-wise):

  $$
    V,\,W \,\in\, \mathbb{K}Vect
    \;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;
    [V,W]
    \;\coloneqq\;
    \left(
      \mathbb{K}Vec(V,W)
      ,\,
      \begin{array}{l}
       k \cdot f \,\colon\, v \mapsto k \cdot f(v)
       \\
       f + g \,\colon\, v \mapsto f(v) + g(v)
      \end{array}
    \right)
  $$
\end{example}

\begin{example}
A [[homotopical algebra|higher homotopical]] variant of Exp. \ref{ExampleModules}: For $R$ a [[commutative ring]] the [[category of chain complexes]] over $R$ is closed monoidal, via the [[tensor product of chain complexes]] and the [[internal hom of chain complexes]], see there for more.
\end{example}

\begin{example}
Given a closed monoidal category $C$ with [[equalizers]] and [[coequalizers]] and a [[commutative monad]] $T$, the category of [[algebra over a monad|$T$-algebras]] inherits a closed monoidal structure. See at *[[tensor product of algebras over a commutative monad]]*.
\end{example}

\begin{example}\label{FunctorCategory}
Let $C$ be a [[complete category|complete]] closed monoidal category and $I$ any [[small category]].  Then the [[functor category]] $[I,C]$ is closed monoidal with the pointwise tensor product, $(F\otimes G)(x) = F(x) \otimes G(x)$.
\end{example}
\begin{proof}
Since $C$ is complete, the category $[I,C]$ is [[comonadic functor|comonadic]] over $C^{ob I}$; the comonad is defined by right [[Kan extension]] along the inclusion $ob I \hookrightarrow I$.  Now for any $F\in [I,C]$, consider the following square:
$$
  \array{
    [I,C]
    & \overset{F\otimes - }{\longrightarrow} &
    [I,C]
    \\
    \big\downarrow && \big\downarrow
    \\
    C^{ob I}
    & \underset{F_0 \otimes -}{\longrightarrow} &
    C^{ob I}
  }
$$
This [[commuting diagram|commutes]] because the tensor product in $[I,C]$ is pointwise (here $F_0$ means the family of objects $F(x)$ in $C^{ob I}$).
Since $C$ is closed, $F_0 \otimes -$ has a [[right adjoint]].
Since the vertical functors are [[comonadic functor|comonadic]], the (dual of the) [[adjoint lifting theorem]] implies that $F\otimes -$ has a right adjoint as well.
\end{proof}

\begin{example}
A [[discrete category|discrete]] monoidal category (i.e., a [[monoid]]) is left closed iff it is right closed iff every object has an inverse, i.e. iff the monoid (aka [[semigroup]]) is in fact a [[group]].
\end{example}

\begin{example}
The [[delooping]] $\mathbf{B}M$ of a [[commutative monoid]] $M$ is a closed monoidal (in fact, [[compact closed]]) category with one object, with both the tensor and internal hom defined (on morphisms) using the multiplication operation $f \otimes g = [f,g] = f g$.  Conversely, any closed monoidal category with one object must be isomorphic to one constructed from a commutative monoid. (See [Eilenberg and Kelly (1965)](#EilenbergKelly65), IV.3, p.553.)
\end{example}

* Certain [[nice category of spaces|nice categories]] of [[topological spaces]] are cartesian closed: for any two nice enough topological spaces $X$, $Y$ the set of continuous maps $X \to Y$ can be equipped with a topology to become a nice topological space itself.

* Certain nice categories of _[[pointed object|pointed/based]]_  topological spaces are closed symmetric monoidal.  The monoidal structure is the [[smash product]] and the internal-hom is the set of basepoint-preserving maps with topology induced from the space of unbased ones.

* The category [[Cat]] is cartesian closed: the internal-hom is the [[functor category]] of functors and natural transformations. There is exactly one other closed monoidal structure on [[Cat]], given by the [[funny tensor product]].

* The category $2 Cat$ of [[strict 2-category|strict 2-categories]] and strict 2-functors is closed symmetric monoidal under the [[Gray tensor product]].  The internal-hom is the 2-category of strict 2-functors, _pseudo_ natural transformations, and modifications.

* The category of strict $\omega$-[[strict omega-category|categories]] is also biclosed monoidal, under the [[Crans-Gray tensor product]].

* If $M$ is a monoidal category and $Set^{M^{op}}$ is endowed with the tensor product given by the induced [[Day convolution]] product, then the [[category of presheaves]] $Set^{M^{op}}$ is biclosed monoidal.

* The category of [[species]], with the monoidal structure given by substitution product of species, is closed monoidal (each functor $- \circ G$ admits a right adjoint) but not biclosed monoidal.

* The category of [[modules]] over any [[Hopf monoid]] in a closed monoidal category, or more generally algebras for any [[Hopf monad]], is again a closed monoidal category.  In particular, the category of modules over any [[group object]] in a [[cartesian closed category]] is (cartesian) closed monoidal. For more on this phenomenon see at _[[Tannaka duality]]_.

* The category of [[locally convex topological vector spaces]] with the [[inductive tensor product]] and internal hom the space of continuous linear maps with the topology of pointwise convergence is symmetric closed monoidal.

## Related concepts

* [[monoidal category]], [[monoidal (∞,1)-category]]

* [[symmetric monoidal category]], [[symmetric monoidal (∞,1)-category]]

* **closed monoidal category** ,  [[closed monoidal (∞,1)-category]]

  * [[doubly closed monoidal category]]

  * [[closed monoidal functor]]

  * [[internal hom]]

  * [[indexed closed monoidal category]]

  * [[cartesian closed category]], [[exponential object]]

* [[string diagram]], [[Kelly-Mac Lane diagram]], [[linguistics|natural language syntax]]

## References

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  §III of: *Closed Categories*, in: *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

Textbook accounts:

* {#MacLane97} [[Saunders MacLane]], VII.7 in: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5**, Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Borceux94} [[Francis Borceux]], vol 2 section 6.1 of: _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]

Original articles studying monoidal biclosed categories are

* {#Lambek68} [[Joachim Lambek]], _Deductive systems and categories_, Mathematical Systems Theory 2 (1968), 287-318.

* {#Lambek69} [[Joachim Lambek]], _Deductive systems and categories II_, Lecture Notes in Math. 86, Springer-Verlag (1969), 76-122.

For more historical development see at _[linear type theory -- History of linear categorical semantics](linear+type+theory#HistoryCategoricalSemantics)_.

In [[enriched category theory]] the enriching category is taken to be closed monoidal. Accordingly the standard textbook on enriched category theory

* [[Max Kelly]], _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press 1982, 245 pp. ([ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029));

  republished as:

  Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 ([tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))

has a chapter on just closed monoidal categories.

[[!redirects closed monoidal category]]
[[!redirects closed monoidal categories]]
[[!redirects monoidal closed category]]
[[!redirects monoidal closed categories]]
[[!redirects closed monoidal structure]]
[[!redirects closed monoidal structures]]

[[!redirects biclosed monoidal category]]
[[!redirects biclosed monoidal categories]]
[[!redirects monoidal biclosed category]]
[[!redirects monoidal biclosed categories]]
[[!redirects left closed monoidal category]]
[[!redirects left closed monoidal categories]]
[[!redirects right closed monoidal category]]
[[!redirects right closed monoidal categories]]

[[!redirects hom-tensor adjunction]]
[[!redirects hom-tensor adjunctions]]
[[!redirects hom tensor adjunction]]
[[!redirects hom tensor adjunctions]]


closed monoidal category module

A  _closed monoidal category module_ $C$ is a [[monoidal category module]] over a [[closed monoidal category|closed]] [[monoidal category|symmetric monoidal]] [[category]] $V$ such that the (left) action

$$
  \otimes : V \times C \to C
$$

extends to a [[two-variable adjunction]].  Such closed modules can be identified with $V$-[[enriched category|enriched categories]] having [[power]]s and [[copower]]s.

#Refererences#

This is in chapter 4 of

* M. Hovey, _Model Categories_, volume 63 of _Mathematical Surveys and Monographs_, AMS, 1999.


closed monoidal deformation retract

A _closed monoidal deformation retract_ on a [[closed monoidal category|closed]] [[monoidal category|(symmetric) monoidal]] [[homotopical category|homotopical]] category $(V, \otimes, [-,-])$ is a [[deformation retract of a homotopical category]] $(V_Q, V_Q, V_R)$ for the [[two-variable adjunction]]

$$
  (\otimes , [-,-], [-,-]) : V \times V \to V
$$

such that a bunch of conditions are satisfied.

#Remarks#

Such closed monoidal deformation retracts are used to characterize [[closed monoidal homotopical category|closed monoidal homotopical categories]].

#References#

This appears as [definition 15.3, p. 44](http://arxiv.org/PS_cache/math/pdf/0610/0610194v1.pdf#page=44) in Shulman: [Homotopy limits and colimits in enriched homotopy theory](http://arxiv.org/abs/math.AT/0610194v1)


closed monoidal homotopical category

#Idea#

In the context of [[homotopy coherent category theory]] one wishes to nicely pair [[enriched category theory]] with [[homotopy theory]] for the case that the category $V$ being enriched over is a [[model category]] or at least a [[homotopical category]].

The idea is that a _closed (symmetric) monoidal homotopical category_ is a [[closed monoidal category]] which is also  [[homotopical category|homotopical]] such that its [[homotopy category]] $Ho_V$ is still itself [[closed monoidal category|closed monoidal]].

A category enriched over such $V$ may be an [[enriched homotopical category]].

#Definition#

A _closed (symmetric) monoidal homotopical category_ $V_0$ is a [[closed monoidal category|closed]], [[monoidal category|symmetric monoidal]] [[homotopical category|homotopical]] category which is furthermore equipped with a [[closed monoidal deformation retract]].

#Examples#

The most obvious example is:

* Any (symmetric) [[monoidal model category]] is a closed (symmetric) monoidal homotopical category.

In this case, the cofibrant and fibrant replacements serve as a closed monoidal deformation retract.   Closed monoidal homotopical categories that are not monoidal model categories seem surprisingly hard to come by, given how much stronger the axioms of a monoidal model category appear.  One other example that can probably be extracted from the theory of homotopy tensor products in Shulman ([below](http://arxiv.org/abs/math.AT/0610194)) is:

* If $V$ is a closed symmetric monoidal homotopical category (such as a monoidal model category) and $C$ is any small category, then the functor category $V^C$, with its [[Day tensor product]] and levelwise homotopical structure, is a closed symmetric monoidal homotopical category.  The closed monoidal deformation retract is provided by the deformations in $V$ combined with the [[bar construction]].

#Properties#

+-- {: .un_prop}
###### Proposition
The [[homotopy category]] $Ho_V$ of a closed (symmetric)
monoidal homotopical category $V$ is itself
closed (symmetric) monoidal.
=--

#References#

The definition appears as [definition 15.3, p. 44](http://arxiv.org/PS_cache/math/pdf/0610/0610194v1.pdf#page=44) in Shulman: [Homotopy limits and colimits in enriched homotopy theory](http://arxiv.org/abs/math.AT/0610194v1),
the proposition as [proposition 15.4, p. 45](http://arxiv.org/PS_cache/math/pdf/0610/0610194v1.pdf#page=45).


closed monoidal structure on presheaves

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

As every [[topos]], a  [[category of presheaves]] is  [[cartesian monoidal category|cartesian]] [[closed monoidal category|closed monoidal]].

## Definition

+-- {: .num_prop #CartesianClosedMonoidalnessOfCategoriesOfPresheaves}
###### Proposition
**([[cartesian closed category|cartesian closure]] of [[categories of presheaves]])**

Let $\mathcal{C}$ be a [[small category]] and write $[\mathcal{C}^{op}, Set]$ for its [[category of presheaves]].

This is

1. a [[cartesian monoidal category]], whose [[Cartesian product]] is given objectwise in $\mathcal{C}$ by the [[Cartesian product]] in [[Set]]:

   for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$, their [[Cartesian product]] $\mathbf{X} \times \mathbf{Y}$ exists and is given by

   $$
     \mathbf{X} \times \mathbf{Y}
     \;\;\colon\;\;\phantom{A}
     \array{
       c_1 &\mapsto& \mathbf{X}(c_1) \times \mathbf{Y}(c_1)
       \\
       {}^{\mathllap{f}}\big\downarrow
         &&
       \big\uparrow^{ \mathrlap{ \mathbf{X}(f) \times \mathbf{Y}(f) } }
       \\
       c_2 &\mapsto& \mathbf{X}(c_2) \times \mathbf{Y}(c_2)
     }
   $$

1. a [[cartesian closed category]], whose [[internal hom]] is given for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ by

   $$
     [\mathbf{X}, \mathbf{Y}]
     \;\;\colon\;\;\phantom{A}
     \array{
       c_1 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_1) \times \mathbf{X}, \mathbf{Y} )
       \\
       {}^{ \mathllap{ f } }\big\downarrow
         &&
       \big\uparrow^{ \mathrlap{ Hom_{[\mathcal{C}^{op}, Set]}( y(f) \times \mathbf{X}, \mathbf{Y} ) } }
       \\
       c_2 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_2) \times \mathbf{X}, \mathbf{Y} )
     }
   $$

   Here $y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]$ denotes the [[Yoneda embedding]] and $Hom_{[\mathcal{C}^{op}, Set]}(-,-)$ is the [[hom-functor]] on the [[category of presheaves]].

=--

(e.g. [MacLane-Moerdijk, section I.6, pages 46-47](#MacLaneMoerdijk)).

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

The first statement is a special case of the general fact that [[limits of presheaves are computed objectwise]].

For the second statement, first assume that $[\mathbf{X}, \mathbf{Y}]$ does exist. Then by the [[adjoint functor|hom-adjunction isomorphism]]  we have for any other presheaf $\mathbf{Z}$ a [[natural isomorphism]] of the form

\[
  \label{InternalHomIsoInPresheaves}
  Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z}, [\mathbf{X},\mathbf{Y}])
    \;\simeq\;
  Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z} \times \mathbf{X}, \mathbf{Y})
  \,.
\]

This holds in particular for $\mathbf{Z} = y(c)$ a [[representable functor]] (i.e. in the [[essential image]] of the [[Yoneda embedding]]) and so the [[Yoneda lemma]] implies that if it exists, then $[\mathbf{X}, \mathbf{Y}]$ must have the claimed form:

$$
  \begin{aligned}
    [\mathbf{X}, \mathbf{Y}](c)
    & \simeq
    Hom_{[\mathcal{C}^{op}, Set]}( y(c), [\mathbf{X}, \mathbf{Y}] )
    \\
    & \simeq Hom_{ [\mathcal{C}^{op}, Set] }( y(c) \times \mathbf{X}, \mathbf{Y} )
    \,.
  \end{aligned}
$$

Hence it remains to show that this formula does make (eq:InternalHomIsoInPresheaves) hold generally.

For this we use the equivalent definition of [[adjoint functors]] in terms of the [[adjunction counit]] providing a system of [[universal arrows]].

Define a would-be [[adjunction counit]], which here is called an _[[evaluation]]_ morphism, by

$$
  \array{
    \mathbf{X} \times [\mathbf{X} , \mathbf{Y}]
    &\overset{ev}{\longrightarrow}&
    \mathbf{Y}
    \\
    \mathbf{X}(c) \times Hom_{[\mathcal{C}^{op}, Set]}(y(c) \times \mathbf{X}, \mathbf{Y})
    &\overset{ev_c}{\longrightarrow}&
    \mathbf{Y}(c)
    \\
    (x, \phi) &\mapsto& \phi_c( id_c, x )
  }
$$

Then it remains to show that for every morphism of presheaves of the form
$ \mathbf{X} \times \mathbf{A} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{Y} $ there is a _unique_  morphism
$\widetilde f \;\colon\; \mathbf{A} \longrightarrow [\mathbf{X}, \mathbf{Y}]$ such that

\[
  \label{UniversalArrowConditionForEvaluationMapInPresheaves}
  \array{
    \mathbf{X} \times \mathbf{A}
    && \overset{ \mathbf{X} \times \widetilde f  }{\longrightarrow} &&
    \mathbf{X} \times [\mathbf{X}, \mathbf{Y}]
    \\
    & {}_{\mathllap{ \mathrlap{f} }}\searrow
      &&
    \swarrow_{ \mathrlap{  ev } }
    \\
    && \mathbf{Y}
  }
\]

The [[commuting diagram|commutativity]] of this diagram means in components at $c \in \mathcal{C}$ that, that for all $x \in \mathbf{X}(c)$ and $a \in \mathbf{A}(c)$ we have

$$
  \begin{aligned}
    ev_c( x, \widetilde f_c(a) )
    & \coloneqq
    (\widetilde f_c(a))_c( id_c, x )
    \\
    & = f_c( x, a )
  \end{aligned}
$$

Hence this fixes the component $\widetilde f_c(a)_c$ when its first argument is the [[identity morphism]] $id_c$. But let $g \;\colon\; d \to c$ be any morphism and chase $(id_c, x )$ through the naturality diagram for $\widetilde f_c(a)$:

$$
  \array{
    Hom_{\mathcal{C}}(c,c) \times \mathbf{X}(c)
    &\overset{ (\widetilde f_c(a))_c }{\longrightarrow}&
    \mathbf{Y}(c)
    \\
    {}^{\mathllap{ g^\ast }}\big\downarrow
     &&
    \big\downarrow^{\mathrlap{ g^\ast }}
    \\
    Hom_{\mathcal{C}}(d,c) \times \mathbf{X}(d)
    &\overset{ (\widetilde f_c(a))_d }{\longrightarrow}&
    \mathbf{Y}(d)
  }
  \phantom{AAAA}
  \array{
    \{ (id_c, x ) \} &\longrightarrow& \{ f_c( x, a ) \}
    \\
    \big\downarrow && \big\downarrow
    \\
    \{ (g, g^\ast(x)) \}  &\longrightarrow& \{ f_d( g^\ast(x), g^\ast(a) ) \}
  }
$$

This shows that $(\widetilde f_c(a))_d$ is fixed to be given by

\[
  \label{ComponentFormulaForEvaluationMapInPresheaves}
  (\widetilde f_c(a))_d( g, x' )
  \;=\;
  f_d( x', g^\ast(a) )
\]

at least on those pairs $(g,x')$ such that $x'$ is in the image of $g^\ast$.

But, finally, $(\widetilde f_c(a))_d$ is also natural in $c$

$$
  \array{
    \mathbf{A}(c)
    &\overset{ \widetilde f_c }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](c)
    \\
    {}^{\mathllap{g^\ast}}\big\downarrow && \big\downarrow^{\mathrlap{g^\ast}}
    \\
    \mathbf{A}(d)
    &\overset{ \widetilde f_d }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](d)
  }
$$

which implies that (eq:ComponentFormulaForEvaluationMapInPresheaves) must hold generally. Hence naturality implies that (eq:UniversalArrowConditionForEvaluationMapInPresheaves) indeed has a unique solution.

=--

## Definition in terms of homs of direct images ##

Often another, equivalent, expression is used to express the internal hom of presheaves:

Let $X$ be a [[site|pre-site]] with underlying [[category]] $S_X$. Recall from the discussion at [[site]] that just means that we have a category $S_X$ on which we consider [[presheaf|presheaves]] $F \in PSh(S_X) := [S_X^{op}, Set]$, but that it suggests that

* to each object $U \in PSh(X)$ and in particular to each $U \in S_X \hookrightarrow PSh(X)$ there is naturally associated the pre-site $U$ with underlying category the [[comma category]] $S_U = (Y/Y(U))$;

* that the canonical [[stuff, structure, property|forgetful]] [[functor]] $j^t_{U \to X} :  S_U \to S_X$, which can be thought of as a [[site|morphism of pre-sites]] $j_{U \to X} : X \to U$ induces the [[direct image]] functor $(j_{U \to X})_* : PSh(X) \to PSh(U)$ which we write $F \mapsto F|_U$.

Then in these terms the above **internal hom**  for presheaves

$$
  hom : PSh(X)^{op} \times PSh(X) \to PSh(X)
$$

is expressed for all $F,G \in PSh(X)$ by

$$
  hom(F,G) = U \mapsto Hom_{PSh(U)}(F|_U, G|_U)
  \,.
$$

## Relation of the two definitions ##

To see the equivalence of the two  definitions, notice

* that by the [[Yoneda lemma]] we have that $S_U$
is simply the [[over category]]
$S_U = S_X/U$;
* by the general properties of [[functors and comma categories]] there is an equivalence $PSh(S_X/U) \simeq PSh(S_X)/y(U)$;
* which identifies the functor $(-)|_U : PSh(S_X) \to PSh(S_U)$ with the functor
$((-)\times y(U) \stackrel{p_2}{\to} y(U)) : PSh(S_X) \to PSh(S_X)/y(U)$;
* and that $Hom_{PSh(S_X)/y(U)}(y(U) \times F, y(U) \times G) \simeq Hom_{PSh(S_X)}(y(U) \times F, G)$.

## Presheaves over a monoidal category

It is worth noting that in the case where $X$ is itself a [[monoidal category]] $(X, \otimes, I)$, $Psh(X)$ is equipped with another (bi)closed monoidal structure given by the [[Day convolution]] product and its componentwise right adjoints.  Let $F$ and $G$ be two presheaves over $X$.  Their tensor product $F \star G$ can be defined by the following [[coend]] formula:

$$F\star G = U \mapsto \int^{U_1,U_2\in X} Hom_X(U, U_1\otimes U_2) \times F(U_1) \times G(U_2)$$

Then we can define two right adjoints

$$F\star - \dashv F \backslash -
\qquad -\star G \dashv - / G$$

by the following [[end]] formulas:

$$F \backslash H = V \mapsto \int_{U\in X} F(U) \to H(U\otimes V)$$
$$H / G = U \mapsto \int_{V\in X} G(V) \to H(U\otimes V)$$

In the case where the monoidal structure on $X$ is cartesian, the induced closed monoidal structure on $Psh(X)$ coincides with the cartesian closed structure described in the previous sections.

## Related concepts

* The cartesian closure restricts from presheaves to [[categories of sheaves]] (e.g. [MacLane-Moerdijk, section III.6, p. 136-138](#MacLaneMoerdijk))

## References

The first definition is discussed for instance in section I.6 of

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_

The second definition is discussed for instance in section 17.1 of

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_

[[!redirects ccc structure on presheaves]]
[[!redirects cartesian closed structure on presheaves]]
[[!redirects cartesian closed monoidal structure on presheaves]]

[[!redirects cartesian closure of categories of presheaves]]
[[!redirects categories of presheaves are cartesian closed]]


closed monoidal structure on sheaves

#Definition#

Let $X$ be a [[site]] with underlying category $S_X$. Write $PSh(X) = [S^{op}, Set]$ for the [[presheaf]] category of $X$ and $Sh(X)$ for the corresponding [[subcategory]] [[category of sheaves|of sheaves]].

The [[closed monoidal structure on presheaves]] restricts under the inclusion $Sh(X) \hookrightarrow PSh(X)$ to a closed monoidal structure on sheaves.

#Properties#

* For $f : X \to Y$ a left axact [[site|morphism of sites]], there is for $F \in Sh(X)$, $F \in Sh(Y)$ a natural [[isomorphism]]

$$
  f_* hom(f^{-1}G, F) \simeq hom(G, f_* F)
  \,.
$$

Here $f_*$ is the [[direct image]] and $f^{-1}$ the [[inverse image]] operation.


closed morphism

> This entry is about the general concept. For the concept in [[topology]] see at _[[closed map]]_.

+-- {: .rightHandSide}
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[[!include topology - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### Between topological spaces

A [[continuous function]]  $f \colon X \longrightarrow Y$ between [[topological spaces]]  is called a __[[closed map]]__ if the [[image]] of every [[closed subset]] in $X$ is also closed in $Y$.

Recall that $f$ is a __[[continuous map]]__ if the [[preimage]] of every [[closed subspace|closed set]] in $Y$ is closed in $X$.  For defining closed maps typically one restricts attention to closed [[continuous maps]], although it also makes sense to speak of closed functions that are not continuous.

### Between locales

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

A map $f : X \to Y$ of [[locales]] is _closed_ iff for any $u \in O(X)$ and $v \in O(Y)$ the [[reciprocity]] relation
$$ f_*(u \vee f^*v) = f_*(u) \vee v $$
holds. ("$\geq$" is trivially always satisfied.) The map $f_* : O(X) \to O(Y)$ is the monotone [[right adjoint]] to $f^* : O(Y) \to O(X)$, explicitly given by
$$ f_*(u) = \sup\{ v \in O(Y) \,|\, f^*(v) \leq u \}. $$

=--

### Between toposes

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

A [[geometric morphism]] $f \;\colon\; \mathcal{F} \longrightarrow \mathcal{E}$ of [[toposes]] is _closed_ iff for any [[object]] $A \in \mathcal{E}$, the induced [[locale]] [[homomorphism]]

$$
  \mathrm{Sub}_{\mathcal{F}/f^*A}(1) \to \mathrm{Sub}_{\mathcal{E}}(1)
$$

between the spaces of [[subobjects]] of the corresponding [[terminal objects]] is closed in the sense of def. \ref{ClosedMapOfLocales}.

=--

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

Let $A$ be an object of a topos $\mathcal{E}$. Then the canonical [[etale geometric morphism]] $\mathcal{E}/A \to A$ is closed iff $A$ fulfills the following condition, formulated in the [[internal language]]:
$$ \mathcal{E} \models \forall U \subseteq A{:} \forall p \in \Omega{:} \quad A \subseteq (U \cup \{ x \in A \,|\, p \}) \quad\Rightarrow\quad (A \subseteq U) \vee p. $$
Note that this condition is satisfied for any $A$ whatsoever if the [[internal language]] of $\mathcal{E}$ is [[Boolean topos|boolean]].

=--

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

Let $f : A \to B$ be a morphism in a topos $\mathcal{E}$. Then the induced geometric morphism $\mathcal{E}/A \to \mathcal{E}/B$ is closed iff, in the internal language of $\mathcal{E}$, the fibers of $f$ fulfill the condition displayed at the previous example.
=--

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

A [[geometric morphism]] $f : \mathcal{F} \to \mathcal{E}$ is closed iff, in the [[internal language]] of $\mathcal{E}$, the unique locale map $f_* \Omega_{\mathcal{F}} \to \mathrm{pt}$ into the one-point space (given by the frame $\Omega_{\mathcal{E}}$) is closed.
=--

## Related concepts

* [[open morphism]], [[open geometric morphism]]

* [[proper morphism]]

* [[étale morphism]]

* [[separated morphism]]

* [[covert space]]

## References

Closed maps of locales and toposes are discussed in section C3.2 of

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_.

[[!redirects closed morphisms]]

[[!redirects closed geometric morphism]]
[[!redirects closed geometric morphisms]]


closed natural transformation

#Contents#
* table of contents
{:toc}

## Idea

A *closed natural transformation* is the appropriate sort of [[natural transformation]] relating [[closed functors]] between [[closed categories]].

## Definition

Let $F,G\colon C\to D$ be closed functors.  A **closed natural transformation** $\alpha\colon F\to G$ is simply an ordinary natural transformation such that:

* The following diagram commutes:
  $$\array{I_D & \overset{F^0}{\to} & F(I_C)\\
   & _{G^0}\searrow & \downarrow^{\alpha}\\
  &  & G(I_C)}$$

* The following diagram commutes for any $X,Y$:
  $$\array{F([X,Y]) & \overset{\hat{F}}{\to} & [F(X),F(Y)]\\
  && \downarrow^{[1,\alpha]}\\
  ^{\alpha }\downarrow && [F(X),G(Y)]\\
  && \uparrow^{[\alpha,1]}\\
  G([X,Y])& \underset{\hat{G}}{\to} & [G(X),G(Y)]}$$

## Examples

* Just as [[closed functors]] between [[closed monoidal categories]] are in bijective correspondence with (lax) [[monoidal functors]], closed natural transformations between such are in bijective corrrespondence with monoidal transformations.

* The same is true for transformations between closed unital [[multicategories]], regarded as closed categories.

## References

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  §I.4 of: *Closed Categories*, in: *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

[[!redirects closed natural transformations]]


closed point

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

#Content#
* table of contents
{:toc}

## Definition

In a [[topological space]] $(X,\tau)$ a [[point]] ([[element]]) $x \in X$ is called a _closed point_ if the [[singleton set]] $\{x\} \subset X$ is a [[closed subset]] of $X$.

## Related concepts

* [[irreducible closed subset]]

* [[open point]]

* [[interior point]]

## Examples
 {#Examples}

* A [[topological space]] is a [[T1-topological space|T1-space]] precisely if all its points are closed points.

* In the [[Zariski topology]] on an [[algebraic variety]] $Spec(R)$, the closed points correspond to the _[[maximal ideals]]_ in $R$ ([this Prop.](Zariski+topology#MaximalIdealsAreClosedPoints)).

* In particular the [[prime numbers]] correspond to the closed points in [[Spec(Z)]] ([this Example](Zariski+topology#SpecZ)).

[[!redirects closed points]]


closed subgroup

+-- {: .rightHandSide}
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#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A common condition on [[subgroups]] of a [[topological group]] is that they be [[closed subspaces|closed]].

This is typically required in [[equivariant differential topology]]/[[equivariant homotopy theory]], for instance in the definition of the [[orbit category]].

## Definition

\begin{defn}\label{ClosedSubgroup}
**(closed subgroup)**\linebreak

A [[topological group|topological]] [[subgroup]] $H \subset G$ of a [[topological group]] $G$ is called a _closed subgroup_ if as a [[topological subspace]] it is a [[closed subspace]].

\end{defn}

+-- {: .num_lemma #OpenSubgroupOfTopologicalGroupIsClosed}
###### Lemma
**(open subgroups of topological groups are closed)**

Every [[open subset|open]] [[subgroup]] $H \subset G$ of a topological group is [[closed subset|closed]], hence a [[closed subgroup]].

=--

(e.g [Arhangel'skii-Tkachenko 08, theorem 1.3.5](#ArhangelskiiTkachenko08))

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

The [[set]] of $H$-[[cosets]] is a [[cover]] of $G$ by [[disjoint subsets|disjoint]] [[open subsets]]. One of these cosets is $H$ itself and hence it is the [[complement]] of the [[union]] of the other cosets, hence the complement of an open subspace, hence closed.

=--

## Examples

### Stabilizer subgroup

\begin{prop}\label{StabilizerSubgroupsOfContinuousActionsOnT1SpacesAreClosed}
**([[stabilizer subgroups]] of [[topological G-space|continuous actions]] on [[T1-spaces]] are closed)**
  Let $G$ be a [[topological group]] and let $X \in G Actions(TopologicalSpaces)$ a [[topological G-space]] whose underlying [[topological space]] is a [[T1-space]], e.g. a [[Hausdorff space]].

Then for all [[points]] $x \in X$  their [[isotropy groups]], hence their [[stabilizer subgroup]]

  $$
    G_x
    \;\coloneqq\;
    Stab_G(x)
    \;\subset\;
    G
    \,,
  $$

  is a closed subgroup (Def. \ref{ClosedSubgroup}).

\end{prop}
\begin{proof}
  We may understand $G_x$ as the [[pre-image]]

$$
  G_x
  \;\simeq\;
  \big(
    \rho(-)(x)
  \big)^{-1}
  \big(
    \{x\}
  \big)
$$

of the [[singleton set|singleton]] [[subset]] $\{x\} \subset X$ under the [[continuous function]] which sends $x$ to its [[image]] under the given $G$-[[action]]:

$$
  \array{
    G
    &\overset{\rho(-)(x)}{\longrightarrow}&
    X
    \\
    g &\mapsto& g \cdot x
    \,.
  }
$$

Since this is a [[continuous function]], and since $x \in X$ is a [[closed point]] by assumption that $X$ is [[T1 topological space|T1]], hence since $\{x\} \subset X$ is a [[closed subset]], it follows that $G_x \subset G$ is a closed subset, since [continuous preimages of closed subsets are closed](continuous+map#PropertiesPreservedByAContinuousFunction).
\end{proof}

### Localic subgroups

Any localic subgroup of a [[localic group]] is closed (see [this Theorem](https://ncatlab.org/nlab/show/localic+group#LocalicSubgroupsAreClosed)).

## Properties

### For Lie groups
 {#ForLieGroups}

\begin{prop}\label{CartanClosedSubgroupTheorem}
**([[Cartan's closed subgroup theorem]])**

If $H \subset G$ is a [[closed subgroup]] of a ([[finite number|finite]] [[dimension of a manifold|dimensional]]) [[Lie group]], then $H$ is a sub-Lie group, hence a [[smooth manifold|smooth]] [[submanifold]] such that its group operations are [[smooth functions]] with respect to the the [[submanifold]] [[smooth structure]].
\end{prop}

### Local sections over coset space

\begin{prop}\label{CosetSpaceCoprojectionsWithLocalSections}
**([[coset space coprojections with local sections]])**\linebreak
Let $G$ be a [[topological group]] and $H \subset G$ a [[subgroup]].

Sufficient conditions for the [[coset space]] [[coprojection]] $G \overset{q}{\to} G/H$ to admit [[local sections]], in that there is an [[open cover]] $\underset{i \in I}{\sqcup}U_i \to G/H$ and a [[continuous section]] $\sigma_{\mathcal{U}}$ of the [[pullback]] of $q$ to the cover,

$$
  \array{
    &&
    G_{\vert \mathcal{U}}
    &\longrightarrow&
    G
    \\
    &
    {}^{\mathllap{
      \exists \sigma
    }}
    \nearrow
    &
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    \mathllap{ \exists \; }
    \underset{i \in I}{\sqcup} U_i
    &=&
    \underset{i \in I}{\sqcup} U_i
    &\longrightarrow&
    G/H
    \mathrlap{\,,}
  }
$$

include the following:

* $G$ is any [[topological group]]

  and $H$ is a [[compact Lie group]]

  (in particular for $H$ a [[closed subgroup]] if $G$ itself is a [[compact Lie group]], since [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]).

  ([Gleason 50, Thm. 4.1](#Gleason50))

or:

* The underlying [[topological space]] of $G$ is

  1. [[locally compact topological space|locally compact]];

  1. [[separable metric space]];

  1. of [[finite number|finite]] [[dimension of a separable metric space]]

  (e.g. if $G$ is a [[Lie group]])

  and $H \subset G$ is a [[closed subgroup]].

  ([Mostert 53, Theorem 3](#Mostert53))

\end{prop}

## Related concepts

* [[maximal compact subgroup]]

* [[locally compact topological group]]

* [[compact Lie group]]

## References

* {#Gleason50} [[Andrew Gleason]], _Spaces With a Compact Lie Group of Transformations_, Proceedings of the American Mathematical Society Vol. 1, No. 1 (Feb., 1950), pp. 35-43 ([jstor:2032430](https://www.jstor.org/stable/2032430), [doi:10.2307/2032430](https://doi.org/10.2307/2032430))

* {#Mostert53} [[Paul Mostert]], _Local Cross Sections in Locally Compact Groups_, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 ([jstor:2032540](https://www.jstor.org/stable/2032540), [doi:10.2307/2032540](https://doi.org/10.2307/2032540))

* {#ArhangelskiiTkachenko08} Alexander Arhangel'skii, Mikhail Tkachenko, _Topological Groups and Related Structures_, Atlantis Press 2008 ([doi:10.2991/978-94-91216-35-0](https://doi.org/10.2991/978-94-91216-35-0))

[[!redirects closed subgroups]]


closed subscheme

A morphism $i = (i,i^\sharp):Y\hookrightarrow X$ of algebraic [[schemes]] is a __[[closed immersion of schemes|closed immersion]]__ if, as a topological space, $i(Y)$ is a closed subspace of $X$, $i$ is a homeomorphism on the image, and the [[comorphism]] $i^\sharp:\mathcal{O}_X\to i_*\mathcal{O}_Y$ is a surjective map of rings.

A closed subscheme of $X$ is an [[equivalence class]] of closed immersions into $X$ (morphisms of schemes $i:Y\to X$ and $i':Y'\to X$ are equivalent if there is an isomorphism $eq:Y\cong Y'$ of schemes such that $i'\circ eq = i$)

In an equivalent description of closed subschemes, there is a sheaf of ideals $\mathcal{I}\subset\mathcal{O}_X$ such that the quotient sheaf $\mathcal{O}_X/\mathcal{I}$ is *supported* on the image set $i(Y)\cong Y$. One says that $\mathcal{I}$ is the [[defining sheaf of ideals]] (or in more French style, *the ideal of definition*) of the subscheme $Y$. The structure sheaf $\mathcal{O}_Y$ of the subscheme $Y$ is the quotient sheaf $\mathcal{O}_X/\mathcal{I}$ restricted to $Y$. The [[conormal sheaf]] of the closed immersion $Y\hookrightarrow X$ is the quasicoherent sheaf of modules defined by formula $\mathcal{I}/\mathcal{I}^2$.

In the affine case, one can take $X=Spec\,R$ and $Y=Spec\,R/I$ where $I=\Gamma_X(\mathcal{I})$ is the __defining ideal__ of the closed affine subscheme $Y$. The underlying set of $Y$ is precisely the set $V(I)$ of all prime ideals in $R$ which contain $I$.

See also [[open subscheme]].

category: algebraic geometry
[[!redirects closed immersion of schemes]]
[[!redirects closed subschemes]]


closed subspace

> This entry is about [[closed subsets]] of a [[topological space]]. For other notions of "closed space" see for instance _[[closed manifold]]_.

***

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

# Contents
* table of contents
{: toc}

## Idea

A [[subset]] $C$ of a [[topological space]] (or more generally a [[convergence space]]) $X$ is _closed_ if its [[complement]] is an [[open subset]], or equivalently if it contains all its [[limit points]]. When equipped with the [[subspace topology]], we may call $C$ (or its inclusion $C \hookrightarrow X$) a *closed subspace*. More abstractly, a [[subspace]] $A$ of a [[space]] $X$ is __closed__ if the [[inclusion function|inclusion map]] $A \hookrightarrow X$ is a [[closed map]].

The collection of closed subsets of a space $X$ is closed under arbitrary [[intersections]]. If $A \subseteq X$, then the intersection of all closed subsets containing $A$ is the smallest closed subset that contains $A$, called the _[[topological closure]]_ of $A$, and variously denoted $Cl(A)$, $Cl_X(A)$, $\bar{A}$, $\overline{A}$, etc. It follows that $A \subseteq B$ implies $Cl(A) \subseteq Cl(B)$ and $Cl(Cl(A)) = Cl(A)$, so that $A \mapsto Cl(A)$ forms a [[Moore closure]] operator on the power set $P(X)$.

Since closed subsets are closed with respect to finite unions, we have $Cl(A \cup B) = Cl(A) \cup Cl(B)$.

A _topological closure operator_ is a Moore closure operator $Cl: P(X) \to P(X)$ that preserves finite unions ($Cl(0) = 0$ and $Cl(A \cup B) = Cl(A) \cup Cl(B)$). It is easy to see that all such closure operators come from a topology whose closed sets are the fixed points of $Cl$.

(There is a lot more to say, about [[convergence spaces]], [[smooth spaces]], [[schemes]], etc.)

## Definition

+-- {: .num_defn #ClosedSubset}
###### Definition
**([[closed subsets]])**

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

1. A [[subset]] $S \subset X$ is called a _closed subset_ if its [[complement]] $X \setminus S$ is an  _[[open subset]]_:

   $$
     \left(
       S \subset X\,\, \text{is closed}
     \right)
     \phantom{AA}
       \Leftrightarrow
     \phantom{AA}
     \left(
       X\setminus S \, \subset X \,\, \text{is open}
     \right)
     \,.
   $$

   > graphics grabbed from [Vickers 89](#Vickers89)

1. If a [[singleton]] subset $\{x\} \subset X$ is closed, one says that $x$ is a _[[closed point]]_ of $X$.

1. Given any subset $S \subset X$, then its _topological closure_ $Cl(S)$ is the smallest closed subset containing $S$:

   $$
     Cl(S)
       \;\coloneqq\;
     \underset{ {C \subset X\, \text{closed}  }   \atop {S \subset C }  }{\cap} \left( C \right)
     \,.
   $$

1. A subset $S \subset X$ such that $Cl(S) = X$ is called a _[[dense subset]]_ of $(X,\tau)$.

=--

## Properties

### Basic properties

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

[[subsets are closed in a closed subspace precisely if they are closed in the ambient space]]

=--

+-- {: .num_lemma #UnionOfOpensGivesClosure}
###### Lemma
**(alternative characterization of topological closures)**

Let $(X,\tau)$ be a [[topological space]] and let $S \subset X$ be a [[subset]] of its underlying
set. Then a point $x \in X$ is contained in the topological closure $Cl(S)$ (def. \ref{ClosedSubset})
precisely if every [[open neighbourhood]] $U_x \subset X$ of $x$ [[intersection|intersects]] $S$:

$$
  \left(
     x \in Cl(S)
  \right)
   \phantom{AA}
     \Leftrightarrow
   \phantom{AA}
   \not\left(
     \underset{ {U \subset X \setminus S}  \atop  { U \subset X \, \text{open}  } }{\exists}
      \left(
        x \in U
      \right)
   \right)
   \,.
$$

=--

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

Due to [[de Morgan duality]] we may rephrase the definition of the [[topological closure]]
as follows:

$$
  \begin{aligned}
    Cl(S)
    & \coloneqq
    \underset{ {S \subset C }  \atop { C \subset X\,\text{closed} } }{\cap}
    \left(C \right)
    \\
    & =
    \underset{ { U \subset X \setminus S } \atop {U \subset X\, \text{open}}  }{\cap} \left( X \setminus U \right)
    \\
    & = X \setminus
    \left(
       \underset{ {U \subset X \setminus S} \atop { U \subset X\, \text{open} }}{\cup} U
    \right)
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop #ClosureOfAFiniteUnionIsUnionOfTheClosures}
###### Proposition
**(closure of a finite union is the union of the closures)**

For $I$ a [[finite set]] and $\{U_i \subset X\}_{i \in I}$ a finite set of subsets of a topological space, we have

$$
  Cl(\underset{i \in I}{\cup}U_i)
  =
  \underset{i \in I}{\cup} Cl(U_i)
  \,.
$$

=--

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

By lemma \ref{UnionOfOpensGivesClosure} we use that a point is in the closure of a set precisely if every open neighbourhood of the point intersects the set.

Hence in one direction

$$
  \underset{i \in I}{\cup} Cl(U_i)
  \subset
  Cl(\underset{i \in I}{\cup}U_i)
$$

because if every neighbourhood of a point intersects some $U_i$, then every neighbourhood intersects their union.

The other direction

$$
  Cl(\underset{i \in I}{\cup}U_i)
  \subset
  \underset{i \in I}{\cup} Cl(U_i)
$$

is equivalent by [[de Morgan duality]] to

$$
  X \setminus
  \underset{i \in I}{\cup} Cl(U_i)
  \subset
  X \setminus
  Cl(\underset{i \in I}{\cup}U_i)
$$

On left now we have the point for which there exists for each $i \in I$ a neighbourhood $U_{x,i}$ which does not intersect $U_i$. Since $I$ is finite, the intersection $\underset{i \in I}{\cap} U_{x,i}$ is still an open neighbourhood of $x$, and such that it intersects none of the $U_i$, hence such that it does not intersect their union. This implis that the given point is contained in the set on the right.

=--

### In metric spaces
 {#InMetricSpaces}

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

Using [[classical logic]] then:

Let $(X,d)$ be a [[metric space]], regarded as a [[topological space]] via its [[metric topology]],
and let $V \subset X$ be a [[subset]]. Then the following are equivalent:

1. $V \subset X$ is a [[closed subspace]].

1. For every [[sequence]] $x_i \in V \subset X$ with elements in $V$, which [[convergence|converges]] as a sequence in $X$ it also converges in $V$.

=--

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

First assume that $V \subset X$ is closed and that $x_i \overset{i \to \infty}{\longrightarrow} x_{\infty}$ for some
$x_\infty \in X$. We need to show that then $x_\infty \in V$. Suppose it were not, then
$x_\infty \in X\backslash V$. Since by definition this [[complement]] $X \backslash V$ is an [[open subset]],
it follows that there exists a [[real number]] $\epsilon \gt 0$ such that the [[open ball]]
around $x$ of radius $\epsilon$ is still contained in the complement: $B^\circ_x(\epsilon) \subset X \backslash V$.
But since the sequence is assumed to converge in $X$, this means that there exists $N_\epsilon$ such that all
$x_{i \gt N_{\epsilon}}$ are in $B^\circ_x(\epsilon)$, hence in $X\backslash V$. This contradicts the assumption that
all $x_i$ are in $V$, and hence we have [[proof by contradiction|proved by contradiction]] that $x_\infty \in V$.

Conversely, assume that for all sequences in $V$ that converge to some $x_\infty \in X$ then $x_\infty \in V \subset W$.
We need to show that then $V$ is closed, hence that $X \backslash V \subset X$ is an open subset, hence that for every
$x \in X \backslash V$ we may find a real number $\epsilon \gt 0$ such that the [[open ball]] $B^\circ_x(\epsilon)$ around $x$
of radius $\epsilon$ is still contained in
$X \backslash V$. Suppose on the contrary that such $\epsilon$ did not exist. This would mean that for each $k \in \mathbb{N}$
with $k \geq 1$ then the [[intersection]] $B^\circ_x(1/k) \cap V$ is [[inhabited|non-empty]]. Hence then we could choose
points $x_k \in B^\circ_x(1/k) \cap V$ in these intersections. These would form a sequence which clearly converges to
the original $x$, and so by assumption we would conclude that $x \in V$, which violates the assumption that $x \in X \backslash V$.
Hence we [[proof by contradiction|proved by contradiction]] $X \backslash V$ is in fact open.

=--

### Relation to interior subspaces
 {#RelationToInteriorSubspaces}

+-- {: .num_lemma #RelationClAndInt}
###### Lemma

Let $(X,\tau)$ be a [[topological space]] and let $S \subset X$ be a [[subset]].
Then the [[topological interior]] $Int(S)$ of $S$  equals the [[complement]] of the
[[topological closure]] $Cl(X\backslash S)$ of the complement of $S$:

$$
  Int(S)
    =
  X \backslash Cl\left(
    X \backslash S
  \right)
  \,.
$$

=--

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

By taking [[complements]] once more, the statement is equivalent to

$$
  X \backslash Int(S)
  =
  Cl( X \backslash S )
  \,.
$$

Now we compute:

$$
  \begin{aligned}
    X \backslash Int(S)
    & =
    X \backslash \left(  \underset{{U \, open} \atop {U \subset S}}{\cup}U \right)
    \\
    & = \underset{U \subset S}{\cap} X \backslash U
    \\
    & = \underset{{C\, closed} \atop {C \supset X \backslash S}}{\cap} C
    \\
    & = Cl(X \backslash S)
  \end{aligned}
$$

=--

### Relation to compact subspaces

The relation of closed subspaces to [[compact subspaces]] is expressed by the following statements

* [[Heine-Borel theorem]]

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

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

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

### Kuratowski's closure-complement problem

This mildly amusing curiosity asks how many set-theoretic operations on a topological space $X$ are derivable from closure $C$ and [[complementation]] $\neg$ and applying finite composition. The answer is that at most 14 operations are so derivable (and there are examples showing this number is achievable). As the proofs below indicate, this bare fact has little to do with topology; it has more to do with general Moore closures and how they interact with complements (using [[classical logic]]).

Let $P(X)$ denote the [[power set]] (ordered by [[subset|inclusion]]) and $M$ the [[monoid]] of [[endofunctions]] $P(X) \to P(X)$ with order defined pointwise. Then $C^2 = C$ and $\neg^2 = 1$ in $M$, with $C$ order-preserving and $\neg$ order-reversing. Also

* $I \coloneqq \neg C \neg$ is the [[interior]] operation, with $I \leq Id$.

+-- {: .num_prop}
###### Proposition
$C \neg C \neg$ is [[idempotent element|idempotent]].
=--

+-- {: .proof}
###### Proof
$I (C \neg C) \leq (C \neg C)$, i.e., $\neg C \neg C \neg C \leq C \neg C$. Applying the order-preserving operation $C$ to both sides together with the fact that $C^2 = C$, this gives

$$C\neg C \neg C \neg C \leq C C \neg C = C \neg C.$$

Since $I C \leq C$, we have also $\neg C \neg C \leq C$. Applying the order-reversing operation $C \neg C$ to both sides, we obtain

$$C \neg C = C \neg C C \leq C \neg C (\neg C \neg C).$$

Combining the two displayed inequalities gives $C \neg C = C \neg C \neg C \neg C$, and then multiplying this on the right by $\neg$, the proposition follows.
=--

+-- {: .num_prop}
###### Proposition
Let $K$ be the monoid presented by two generators $C, \neg$ and subject to the relations $C^2 = C$, $\neg^2 = 1$, and $C \neg C \neg C \neg C = C \neg C$. Then $K$, called the _Kuratowski monoid_, has at most 14 elements.
=--

+-- {: .proof}
###### Proof
We may apply an obvious reduction algorithm on the set of words in two letters $C, \neg$, in which a word is reduced by replacing any substring $C C$ by $C$ and any substring $\neg\neg$ by an empty substring, so that any word which cannot be further reduced must be alternating in $C, \neg$. This leads to a list of 14 words

$$1, \qquad \neg, \qquad C, \qquad \neg C, \qquad C \neg, \qquad \neg C \neg, \qquad C \neg C, $$

$$\,$$

$$\neg C \neg C, \qquad C \neg C \neg, \qquad \neg C \neg C \neg, \qquad C \neg C \neg C, \qquad \neg C \neg C \neg C, \qquad C \neg C \neg C \neg, \qquad \neg C \neg C \neg C \neg $$

with any further alternating words reducible by replacing a substring $C \neg C \neg C \neg C$ by $C \neg C$. Thus each element in the monoid $K$ is represented by one of these 14 words.
=--

These 14 words actually name distinct set-theoretic operations $P(X) \to P(X)$ for a judicious choice of space $X$; as a corollary, the Kuratowski monoid $K$ has exactly 14 elements. For instance (courtesy of Wikipedia), taking $X = \mathbb{R}$ with its standard topology, the orbit of the element $(0, 1) \cup (1, 2) \cup \{3\} \cup ([4, 5] \cap \mathbb{Q})$ under the monoid action consists of 14 distinct elements.

+-- {: .num_remark}
###### Remark
At most 7 operations are possible with interior and closure, corresponding to the covariant Kuratowski operations. Thus there is a 7-element submonoid $K_{cov} \hookrightarrow K$. Spaces $X$ for which the topological action $K_{cov} \to Set(P(X), P(X))$ is *not* injective are of some structural interest; for instance, the spaces for which $C I C = I C$ are the [[extremally disconnected spaces]], whereas spaces for which $I C = C$ are those where the open sets are equivalence classes for some equivalence relation (partition spaces). Those for which $I = C$ are [[discrete spaces]].
=--

+-- {: .num_remark}
###### Remark
A more manifestly topological consideration is what happens when we throw [[joins]] (or [[meets]]) into the $I, C$ mix. Briefly, at most 13 subsets can be obtained by starting with a subset $A \in P(X)$ and generating new subsets by taking closures, interiors, and unions; the order structure of these 13 subsets coincides with the free cocompletion of the finite ordered monoid $K_{cov}$ with respect to nonempty joins. Here we must use distributivity of $C$ over joins.
=--

For some details on these remarks (and quite a bit more), see [Gardner and Jackson, 2008](http://nzjm.math.auckland.ac.nz/images/6/63/The_Kuratowski_Closure-Complement_Theorem.pdf) and [Sherman 2004](http://people.virginia.edu/~des5e/papers/14-sets.pdf). An example of a non-topological Moore closure where the 14 operations are all distinct is given [here](http://math.stackexchange.com/a/1912101/43208).

## Generalizations

### Locales

In [[locale]] theory, every open $U$ in a locale $X$ defines a closed [[sublocale]] $\mathsf{C} U$ which is given by the __closed [[nucleus]]__
$$ j_{\mathsf{C} U}\colon V \mapsto U \cup V .$$
The idea is that $\mathsf{C}U$ is the part of $X$ which does not involve $U$ (hence the notation $\mathsf{C}U$, or any other notation for a [[complement]]), and we may identify $V$ with $U \cup V$ when we are looking only away from $U$.

The sublocale $\mathsf{C}U$ is literally a complement of $U$ in the lattice of sublocales of $X$, i.e. $U\cap \mathsf{C}U = \emptyset$ and $U\cup \mathsf{C}U = X$ as sublocales.  Moreover, if $X$ is a ([[sober space|sober]]) topological space regarded as a locale, then the locale $U$ is also spatial, and so is $\mathsf{C}U$, corresponding exactly to the topological closed set $X\setminus U$.  (The fixed points of $j_{\mathsf{C}U}$ can be identified with the open sets containing $U$, which are bijectively related to the open subsets of $X\setminus U$.)  Thus there is really only one notion of "closed subspace" whether we regard $X$ as a space or as a locale (at least as long as $X$ is sober).

### Constructive mathematics

In [[constructive mathematics]], however, there are many possible *inequivalent* definitions of a closed subspace, including:

1. A subspace $C\subset X$ is closed if it is the complement of an open subspace, i.e. if $C = X\setminus U$ for some open subspace $U$;
1. A subspace $C\subset X$ is closed if its complement $X\setminus C$ is open;
1. A subspace $C\subset X$ is closed if it contains all its [[limit points]], i.e. if for any $x\in X$ such that $U\cap C$ is [[inhabited set|inhabited]] for all [[neighborhoods]] $U$ of $x$, we have $x\in C$.

Definition (1) coincides with definition (2) or (3) only if excluded middle holds, since under (2) or (3) every subspace of a [[discrete space]] is closed, while under (1) the only closed subspaces are those that are complements, and if every proposition is a negation then the [[law of double negation]] follows.  On the other hand, Definition (2) is clearly too strong, because even [[closed intervals]] $[a,b]$ in the [[real numbers]] can't be proved constructively to satisfy it (though they do satisfy definitions (1) and (3)).  Note also that Definition (1) always implies Definition (3), since if $C = X\setminus U$ and every open neighborhood of $x$ intersects $C$, then $x\notin U$ and thus $x\in C$.  Thus it is not unreasonable (see also below) to define:

* $C\subset X$ is **strongly closed** if it is the complement of an open subspace.
* $C\subset X$ is **weakly closed** if it contains all its [[limit points]].

This constructive variety of notions of closed subspace gives rise to a corresponding variety of notions of [[Hausdorff space]] when applied to the diagonal subspace.

Note also that neither of the sensible constructive definitions behaves quite like closed subspaces do classically.  In particular, neither of them is apparently stable under finite unions (though the too-strong Definition (2) is).  The situation is better for locales; see below.

### Locales in constructive mathematics

Of the "topological" definitions of closed subspace above, it is "strongly closed" (and the too-strong Definition (2)) that seem closest to the localic one.  However, in the topological definitions we may not have $X = U \cup (X\setminus U)$ or $X = (X\setminus C) \cup C$ even as sets, whereas it remains true constructively that $X = U \cup \mathsf{C}U$ in the lattice of sublocales.  In fact, we have the following:

+-- {: .un_theorem}
###### Theorem
The following are equivalent:

1. The law of [[excluded middle]].
1. Every closed sublocale of a spatial locale is spatial.
1. Every closed sublocale of a [[discrete locale]] is spatial.
=--
+-- {: .proof}
###### Proof
We remarked above that (1)$\Rightarrow$(2), and of course (2)$\Rightarrow$(3).  So assume (3).  Every spatial sublocale is a union of its points, and in a discrete space points are open; thus if closed sublocales are spatial, they are also open.  Since $X = U \cup \mathsf{C}U$ is constructively true, it follows that every open set is complemented in the open-set lattice of any discrete locale, which is to say that all powersets are [[Boolean algebras]], i.e. excluded middle holds.
=--

On the other hand, there is a localic notion of *weakly closed sublocale* that is closely related to topologically weakly closed subspaces (so that the above notion of "closed sublocale" --- the formal complement of an open sublocale --- could also be called *strongly closed*).  It is the specialization of the notion of [[fiberwise closed sublocale]] to locale maps $X\to 1$ into the terminal object.

+--{: .un_defn}
###### Definition
A sublocale $C\subseteq X$ is **weakly closed** if it is not [[strongly dense sublocale|strongly dense]] in any larger sublocale of $X$.  That is, if whenever $D$ is a sublocale of $X$ such that $C\subseteq D$ by a strongly dense inclusion, then $C=D$.
=--

Since strong and weak denseness coincide classically, so do strong and weak closedness.  And as we expect, strong closedness implies weak closedness, since strong density implies weak density.  Moreover, both of them are better-behaved than the corresponding topological notions.  For instance, strongly and weakly closed sublocales are both stable under finite unions (in the lattice of sublocales), even constructively.

Both strongly and weakly closed sublocales are "correct" notions; they are simply different.  Some classical theorems about closed sublocales are constructively about strongly closed ones, while others (such as the theorem that any [[subgroup]] of a [[localic group]] is closed) are about weakly closed ones.

However, as we saw above, not every strongly closed sublocale (hence not every weakly closed sublocale either) can be spatial.  But it is shown at [[strongly dense sublocale]] that a subspace of a space is localically strongly dense if and only if it is topologically strongly dense.  This leads us to guess:

+--{: .un_theorem}
###### Conjecture
A subspace $C\subseteq X$ of a (sober) topological space $X$ is topologically weakly closed if and only if it is the spatial coreflection of a weakly closed sublocale.
=--

In one direction this is easy: suppose $C$ is topologically weakly closed, and let $D$ be its localic weak closure.  This is, by definition, the largest sublocale of $X$ in which $C$ is localically strongly dense.  Now let $E$ be the spatial coreflection of $D$; since $C$ is spatial we have $C\subseteq E$, and since $C$ is localically strongly dense in $D$, it is also so in $E$, and hence topologically strongly dense.  But $C$ is also topologically weakly closed, hence $C=E$ and is the spatial coreflection of the weakly closed sublocale $D$.

The other direction is harder.

## Related concepts

* [[closed point]]

* [[irreducible closed subspace]]

* [[open subset]]

* [[clopen subset]]

* [[locally closed set]]

* [[dense subspace]]

* [[limit point]]

* [[closed cover]]

* [[Moore closure]]

* [[closed subtopos]]

* [[co-Heyting boundary]]

* [[closed morphism]]

* [[proper morphism]]

## References

* [[Kazimierz Kuratowski|C. Kuratowski]], _Sur l'opération_ $\bar{A}$ _de l'analysis situs_ , Fund. Math. **III** (1922) pp.192-195. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3121.pdf))

Weakly closed sublocales are discussed in

* [[Sketches of an Elephant]], C1.1 and C1.2

* [[Peter Johnstone]], *A constructive 'closed subgroup theorem' for localic groups and groupoids*, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1989),  Volume: 30, Issue: 1, page 3-23 [link](https://eudml.org/doc/91430)

* M. Jibladze and [[Peter Johnstone]], *The frame of fibrewise closed nuclei*, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1991), Volume: 32, Issue: 2, page 99-112, [link](https://eudml.org/doc/91478)

* [[Peter Johnstone]], *Fiberwise separation axioms for locales*

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

[[!redirects weakly closed subspace]]
[[!redirects weakly closed subspaces]]
[[!redirects weakly closed subset]]
[[!redirects weakly closed subsets]]
[[!redirects weakly closed set]]
[[!redirects weakly closed sets]]

[[!redirects strongly closed subspace]]
[[!redirects strongly closed subspaces]]
[[!redirects strongly closed subset]]
[[!redirects strongly closed subsets]]
[[!redirects strongly closed set]]
[[!redirects strongly closed sets]]

[[!redirects topological closure]]
[[!redirects topological closures]]

[[!redirects closed sublocale]]
[[!redirects closed sublocales]]
[[!redirects closed nucleus]]
[[!redirects closed nuclei]]

[[!redirects strongly closed sublocale]]
[[!redirects strongly closed sublocales]]
[[!redirects strongly closed nucleus]]
[[!redirects strongly closed nuclei]]

[[!redirects weakly closed sublocale]]
[[!redirects weakly closed sublocales]]
[[!redirects weakly closed nucleus]]
[[!redirects weakly closed nuclei]]

[[!redirects weakly closed]]

[[!redirects closed space]]
[[!redirects closed spaces]]
[[!redirects closed topological space]]
[[!redirects closed topological spaces]]


closed subspaces of compact Hausdorff spaces are equivalently compact subspaces

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

#Contents#
* table of contents
{:toc}

## Statement

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

Let $(X,\tau)$ be a [[compact topological space|compact]] [[Hausdorff topological space]] and let $Y \subset X$
be a [[topological subspace]]. Then the following are equivalent:

1. $Y \subset X$ is a [[closed subspace]];

1. $Y$ is a [[compact topological space]].

=--

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

The two directions to be proven are

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

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

See the proofs there.

=--

## Related entries

* [[subsets are closed in a closed subspace precisely if they are closed in the ambient space]]

* [[Heine-Borel theorem]]

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

* [[Hausdorff implies sober]]

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

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


closed subspaces of compact spaces are compact

[[!redirects closed subsets of compact spaces are compact]]

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

# Contents
* table of contents
{: toc}

## Statement

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

Let $(X,\tau)$ be a [[compact topological space]], and let $Y \subset X$ be a  [[closed subspace|closed]] [[topological subspace]]. Then also $Y$ is [[compact topological space|compact]].

=--

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

Let $\{V_i \subset Y\}_{i \in I}$ be an [[open cover]] of $Y$. We need to show that this has a finite sub-cover.

By definition of the [[topological subspace|subspace topology]], there exist open subsets $U_i$ of $X$ with

$$
  V_i = U_i \cap Y
  \,.
$$

By the assumption that $Y$ is closed, the [[complement]] $X \setminus Y$ is an open subset of $X$, and therefore

$$
  \{ X \setminus Y \subset X\} \cup \{ U_i \subset X \}_{i \in I}
$$

is an [[open cover]] of $X$. Now by the assumption that $X$ is compact, this latter cover has a finite subcover,
hence there exists a [[finite set|finite]] [[subset]] $J \subset I$ such that

$$
  \{ X \setminus Y \subset X\} \cup \{ U_i \subset X \}_{i \in J \subset I}
$$

is still an open cover of $X$, hence in particular intersects to a finite open cover of $Y$.
But since $Y \cap ( X \setminus Y ) = \empty$, it follows that indeed

$$
  \{V_i \subset Y\}_{i \in J \subset I}
$$

is a cover of $Y$, and is indeed a finite subcover of the original one.

=--

## Related statements

* [[compact subspaces in Hausdorff spaces are separated by neighbourhoods from points]]

* [[open subspaces of compact Hausdorff spaces are locally compact]]

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

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

* [[compact Hausdorff spaces are normal]]

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

* [[Hausdorff implies sober]]

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

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


closed subtopos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

The concept of a **closed subtopos** generalizes the concept of a [[closed subspace]] from topology to toposes.

## Definition

Let $U$ be a [[subterminal object]] of a [[topos]] $\mathcal{E}$.  Then $c_U(V) \coloneqq U\cup V$ defines a [[Lawvere-Tierney topology]] on $\mathcal{E}$, whose corresponding [[subtopos]] is called the _closed subtopos_ associated to $U$.

The reflector into the sheaves can be constructed explicitly as the [[join]] with $U$, i.e. $C_U(X)$ is the [[pushout]] of $X$ and $U$ under $X\times U$.

A _topology_ $j$ that is of this form for some [[subterminal object]] $U$ is called _closed_.

## Example

In case $\mathcal{E}=Sh(X)$ is the topos of sheaves on a topological space $X$, a subterminal object is just an open subset $U$ of $X$ and the closed subtopos corresponding to it is equivalent to $Sh(X\setminus U)$.

## Remark

The [[subterminal object]] $U$ in $\mathcal{E}$ is associated with an [[open subtopos]] $Sh_{o(U)}(\mathcal{E})$ as well e.g. in the case of $\mathcal{E}=Sh(X)$ on a space $X$ this yields $Sh(U)$.

Moreover, given a [[Lawvere-Tierney topology]] $j$ on a topos $\mathcal{E}$ with corresponding subtopos $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$, we a get a canonical [[subterminal object]] $ext(j)$  associated to $j$ by taking the $j$-closure of $O\rightarrowtail 1$. The corresponding closed and open subtoposes associated to $ext(j)$ provide a '_closure_' $Sh_j(\mathcal{E})\hookrightarrow Sh_{c(ext(j))}(\mathcal{E})$, called _l'adhérence_ in ([[SGA4]], p.461), respectively, an '_exterior_' $Sh_{o(ext(j))}(\mathcal{E})$ for $Sh_j(\mathcal{E})$.

## Properties

+-- {: .num_prop}
###### Proposition
Let $U$ a [[subterminal object]] and $Sh_{c(U)}(\mathcal{E})$ and $Sh_{o(U)}(\mathcal{E})$ the corresponding closed, resp. open subtoposes. Then $Sh_{c(U)}(\mathcal{E})$ and $Sh_{o(U)}(\mathcal{E})$ are [[complement|complements]] for each other in the [[lattice of subtoposes]].
=--

For the proof see ([Johnstone (2002)](#Johnstone), pp.212,215).

Closed subtoposes that are [[Boolean topos|Boolean]] are parametrized by automorphisms of the [[subobject classifier]] as the next proposition documents:

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

Let $\mathcal{E}$ be a topos. Then automorphisms of $\Omega$ correspond bijectively to closed Boolean subtoposes. The group operation on $Aut(\Omega)$ corresponds to symmetric difference of subtoposes.

=--

This result appears in [Johnstone (1979)](#Johnstone79).

The following result is a part of the so called [[(dense,closed)-factorization]].

+-- {: .num_prop #dense-closed}
###### Proposition
Let $i:Sh_{c(U)}(\mathcal{E})\hookrightarrow\mathcal{E}$ be dominant and a closed inclusion at the same time. Then $i$ is an isomorphism.
=--

**Proof**: Recall that $X\in\mathcal{E}$ are in the closed subtopos precisely when they satisfy $X\times U\cong U$ with $U$ the [[subterminal object]] associated to $i$. But $i$ is dominant, or what comes to the same for inclusions: dense, hence $\emptyset$ is in $Sh_{c(U)}(\mathcal{E})$ and therefore $\emptyset\times U\cong U$ . From this follows $U\cong\emptyset$, which in turn implies that all $X\in\mathcal{E}$ are in $Sh_{c(U)}(\mathcal{E})$ . $\qed$

This says that the only closed subtopos of a topos $\mathcal{E}$ that is dense, is $\mathcal{E}$ itself.

It follows e.g. that [[Boolean toposes|Boolean toposes]] have no non-trivial dense subtoposes since all their subtoposes are closed (and open, as well). Of course, this follows also from the fact that a Boolean topos coincides with its [[double negation|double negation subtopos]] and the latter is the _smallest dense_ subtopos!

## Related pages

* [[(dense,closed)-factorization]]
* [[open subtopos]]
* [[locally closed subtopos]]
* [[dense subtopos]]
* [[Artin gluing]]
* [[closed subspace]]

## References

* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (Exposé IV 9.3.4-9.4., pp.456ff)

* [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint 2014, pp.93-95)

* {#Johnstone79}[[Peter Johnstone]], _Automorphisms of $\Omega$_ , Algebra Universalis **9** (1979) pp.1-7.

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_, Oxford UP 2002. (A4.5., pp.204-220)

[[!redirects closed subtoposes]]
[[!redirects closed topology]]


closed time-like curve

<div class="rightHandSide toc">
[[!include physicscontents]]
</div>

# Clossed time-like curves
* automatic table of contents goes here
{:toc}

## Idea

Physical [[observer]]s in [[spacetime]] seem to travel only slower than the [[speed of light]].  Mathematically, when we represent [[spacetime]] by a [[Lorentzian manifold]], the paths of physical observers must be _timelike_.  Our intuition is that travel along a timelike curve will always move either forwards or backwards in time (and it is the former that matches our subjective experience).  Naïvely, we may think that travelling along timelike curves prevents observers from engaging in time travel.

However, in principle it is quite possible for a time-like curve to be closed!  Locally, we are always travelling into the future when following such a curve (in the future-oriented direction), but after some time we find ourselves back where we started, in the past.  This is not possible in [[Minkowski space]], or indeed in any spacetime with a global time coordinate, but it is allowed under certain circumstances in [[general relativity]].

In science-fiction descriptions of time travel through wormholes, the time travelling follow a portion of a closed time-like curves (CTC) through the wormhole.  (However, this really only works for the you-can't-change-the-past sort of time travel, and science fiction rarely limits itself to that.)

See [[Lorentzian manifold]] for a precise mathematical definition.

## Physicality

As with any time travel, there are numerous paradoxes that arise when considering such curves and many physicists doubt their existence.

Here are some references on the Deutsch--Bacon consistency condition for CTCs:

* Dave Bacon, _Quantum Computational Complexity in the Presence of Closed Timelike Curves_ ([pdf](http://arxiv.org/pdf/quant-ph/0309189v3))

* Charles H. Bennett, Debbie Leung, Graeme Smith, John A. Smolin, _Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems?_ ([pdf](http://arxiv.org/pdf/0908.3023))

* Ian T. Durham, _Quantum communication on closed time-like curves_ ([pdf](http://arxiv.org/pdf/0803.3287v3))

* T.C. Ralph, _Unitary Solution to a Quantum Gravity Information Paradox_ ([pdf](http://arxiv.org/pdf/0708.0449v1))

[[!redirects closed time-like curve]]
[[!redirects closed time-like curves]]
[[!redirects closed timelike curve]]
[[!redirects closed time like curve]]
[[!redirects closed timelike curves]]
[[!redirects closed time like curves]]


closed-projection characterization of compactness

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

# Contents
* table of contents
{:toc}

## Statement and proof

There are a number of [[category theory|categorically-flavored]] ways of characterizing [[compact spaces]]; for example, as [[relational beta-modules]] $X$ where the beta-module structure $\xi: \beta(X) \nrightarrow X$ is an [[entire relation]] (satisfies $1_{\beta(X)} \leq \xi^\circ \xi$). One of the more useful characterizations, as emphasized by [[Walter Tholen]], is given by the following elementary statement (also one of the definitions listed in [[compact space]]):

+-- {: .num_theorem #char}
###### Theorem
A [[topological space]] $X$ is [[compact topological space|compact]] if and only if for every space $Y$, the [[projection map]] $\pi \colon Y \times X \to Y$ out of the [[product topological space]] is a [[closed map]].
=--

In this article we will collect a number of proofs and points of view on this result, and show how it makes possible a very direct hands-on proof of the [[Tychonoff theorem]], without the need for infrastructure such as [[ultrafilters]] or [[nets]]. This type of proof may be adapted for use in undergraduate classrooms.

The "only if" half is covered in essentially every textbook and should be routine for any working mathematician, although it seems proofs often use needlessly many indices and look as though they tacitly invoke the [[axiom of choice]]. Our proof should help circumvent that impression.

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

If $X, Y$ are [[topological spaces]] with $X$ [[compact topological space|compact]], then the [[projection map]] $\pi \colon Y \times X \to Y$ out of the [[product topological space]] is a [[closed map]].

=--

+-- {: .proof}
###### Proof
Let $C \subseteq Y \times X$ be a [[closed subset]]; we must show the [[direct image]] $\pi(C)$ is closed in $Y$. Suppose $y \notin \pi(C)$; equivalently, suppose that the [[pre-image]] $\pi^{-1}(y) = \{y\} \times X$ does not intersect $C$. By the nature of closed subsets ([this lemma](Introduction+to+Topology+--+1#UnionOfOpensGivesClosure)) it is sufficient that we find an [[open neighborhood]] $V$ of $y$ that does not [[intersection|intersect]] $\pi(C)$, or equivalently such that $(V \times X) \cap C = \emptyset$. Consider the collection of all open $U \subseteq X$ such that $(W \times U) \cap C = \emptyset$ for some [[open neighborhood]] $W$ of $y$. This collection [[open cover|covers]] $X$, since $(y, x) \notin C$ for every $x \in X$ and by closure of $C$ we can therefore find a neighborhood $W \times U$ of $(y, x)$ that doesn't intersect $C$. By compactness of $X$, this [[open cover]] has a [[finite cover|finite]] subcover, say $U_1, \ldots, U_n$. For each of these $U_i$ there is a corresponding $W_i$ such that $(W_i \times U_i) \cap C = \emptyset$ (in fact there is a unique largest such $W_i$: just take the union of all such). Now put $V \coloneqq \bigcap_{i=1}^n W_i$. We have for each $i$ between $1$ and $n$ that $V \times U_i \subseteq W_i \times U_i \subseteq \neg C$ (the [[complement]] of $C$ in $X \times Y$), hence

$$V \times X \;=\; V \times \bigcup_{i=1}^n U_i \;=\; \bigcup_{i=1}^n V \times U_i \subseteq \neg C$$

so that $(V \times X) \cap C = \emptyset$.
=--

The "if" half is not quite so routine. We will first give a very direct proof that doesn't involve any fancy machinery (such as [[ultrafilters]]), although the fancy machinery does help give a better conceptual understanding of what is really going on, as we explain below.

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

If $X$ is a [[topological space]] with the property that the [[projection]] $\pi \colon Y \times X \to Y$ out of the [[product topological space]] is a [[closed map]] for every space $Y$, then $X$ is [[compact topological space|compact]].

=--

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

Given a space $X$ satisfying this property, let $(C_i)_{i \in I}$ be any collection of [[closed subsets]] of $X$ having the [[finite intersection property]], that any intersection of finitely many $C_i$ is [[inhabited set|inhabited]]. Compactness of $X$ will follow if we can show $\bigcap_{i \in I} C_i$ is inhabited (by [this prop.](compact+space#fip)).

Given that collection, construct a space $Y$ whose underlying set ${|Y|}$ is formed by adjoining a point $\infty$ to the underlying set of $X$ (so ${|Y|} = {|X|} \sqcup \{\infty\}$), and topologized by taking [[sub-base of a topology|subbasis]] elements to consist of _any_ subset of $X$ (not necessarily open) and any subset of the form $C_i \cup \{\infty\}$. (Thus we are embedding the "[[discrete topological space|discretification]]" of $X$ in a larger space $Y$.) Let $K \subseteq Y \times X$ be the [[topological closure]] of the set $\Delta = \{(x, x): x \in {|X|}\}$. By hypothesis, $\pi(K)$ is closed in $Y$, and contains $X$. But then it must also contain $\infty$ because the [[finite intersection property]] of the collection $(C_i)$ guarantees that $\infty$ is not an [[open point]] of $Y$. It follows that $K$ contains a point of the form $(\infty, x)$.

This means that every [[neighborhood]] $(C_i \cup \{\infty\}) \times U$ of $(\infty, x)$ intersects the [[diagonal]] $\Delta$ in an inhabited set. In other words, $C_i \cap U$ is inhabited. But this means that $x \in C_i$ for every $i$, for if $x \notin C_i$ for some $i$, then by closure of $C_i$ we could find an open neighborhood $U$ of $x$ such that $C_i \cap U = \emptyset$. Hence $x \in \bigcap_{i \in I} C_i$, making this intersection inhabited, as required.
=--

+-- {: .num_remark #Acknowledgement}
###### Remark
I ([[Todd Trimble]]) learned of this particular arrangement of proof during an email exchange with [[Tom Leinster]], who told me he learned it from [[Ioan James]]. I don't know who first invented it though. Pedagogically it is attractive because it avoids any need to discuss nets or ultrafilters or filters; it's just straight-up topology of the kind taught at typical American undergraduate institutions.
=--

### Variant proofs

There are actually quite a few proofs of Proposition \ref{if} that appear in the literature. The proof given [above](#ifProof) might have been the result of a gradual whittling down from more abstract, high-powered-looking proofs.

1. One "canonical" proof that a seasoned mathematician might dream up goes as follows. Consider $Y = \beta({|X|})$, the space of [[ultrafilters]] on the underlying set ${|X|}$ of $X$, or the [[Stone-Cech compactification]] of $X$ (retopologized as a discrete space). Let $K \subseteq Y \times X$ be the convergence relation (i.e., $(U, x) \in K$ iff the ultrafilter $U$ on the set $X$ converges to the point $x$ in the space $X$). One readily checks that $K$ is closed. Thus $\pi(K) \subseteq Y$ is closed by hypothesis. Also observe that the "diagonal" $\Delta = \{(prin(x), x): x \in X\}$, where $prin(x)$ is the principal ultrafilter generated by $x$, sits inside $K$ since $prin(x)$ converges to $x$. So $\pi(K)$ also contains all principal ultrafilters which form a dense subset of $Y$. Being closed and dense, $\pi(K)$ is all of $Y$. But this just means that for every ultrafilter $U$ there is some $(U, x) \in K$, i.e., every ultrafilter converges to some point. This classically implies compactness of $X$.

1. On reflection, we didn't really have to consider the entire space $\beta({|X|})$; we can just work with one ultrafilter at a time. So if we want to prove that every ultrafilter $U$ converges, we can just look at the subset ${|X|} \cup \{U\} \hookrightarrow \beta({|X|})$ with the [[subspace topology]] coming from $\beta({|X|})$ and run through the same diagonal proof. This gives us the proof found in Bourbaki's General Topology: given an ultrafilter $U$ on the underlying set of $X$, form a new space $Y$ whose underlying set is ${|X|} \sqcup \{\infty\}$ and topologized by saying all subsets of $X$ are open, and $A \sqcup \{\infty\}$ is an open neighborhood of $\infty$ if $A \in U$. This is just a direct description of the subspace topology coming from $\beta({|X|})$, and the rest of the proof runs parallel to the proof given in 1.

1. On further reflection, the ultrafilter convergence characterization of compact spaces does rely on the [[ultrafilter theorem]] (that every proper filter can be extended to an ultrafilter), which involves the [[axiom of choice]] or some choice-y principle outside ZF. If one doesn't like this, one can always use just [[filters]] instead, and use the characterization of compact spaces that says that every proper filter on ${|X|}$ admits a [[cluster point]]. This characterization doesn't require any form of Choice. So, one would start with $Y = {|X|} \sqcup \{\infty\}$ but use a given proper filter $F$ to describe the neighborhoods of $\infty$; otherwise the description of $Y$ looks just the same as before.

1. On still further reflection, one doesn't have to mention filters or cluster points at all. Instead one can work directly with a collection of sets $C_i$ satisfying the finite intersection property, and secretly think of that as generating a filter (a proper one by the FIP) and then follow 3. to define the topology of the space $Y$. As it happens, a cluster point of that filter is just a point lying in the intersection of all the $C_i$. This brings us to the proof we gave of Proposition \ref{if}.

1. {#StillOtherCleverConstructions} Still other clever constructions of $Y$ are available. One has been described by [[Martín Escardó]] ([Escardó 09](Escardo09)), who credits [[Leopoldo Nachbin]]. One nice feature is that Escardó favors a point of view where instead of defining a continuous map $f: X \to Z$ to be closed if the direct image mapping or [[existential quantification]] $\exists_f: P(X) \to P(Z)$ preserves closed subsets, he uses instead the [[universal quantification]] $\forall_f: P(X) \to P(Z)$ and asks that this preserve *open sets*. This strategy circumvents the apparent use of [[excluded middle]] in proving Proposition \ref{onlyif}: he proves it directly and constructively. In any case, he considers a [[directed set|directed]] [[open cover]] $\Sigma$ of the given space $X$, then takes $Y = \mathcal{O}(X)$, the set of open sets of $X$, and topologizes $Y$ by taking its inhabited open sets to be upward-closed subsets $W \subseteq \mathcal{O}(X)$ such that $W \cap \Sigma$ is inhabited. Using the open-set formulation of closed maps, he then argues that $X$ belongs to $\Sigma$, which proves compactness. (This argument was also reprised [here](https://ncatlab.org/toddtrimble/published/Characterizations+of+compactness#escardo).)

## Consequences

### Tychonoff theorem
 {#TychonoffTheorem}

It has been observed by [Clementino and Tholen](#CT1996) that the closed-projection formulation of compactness (theorem \ref{char}) allows a fairly efficient proof of the [[Tychonoff theorem]], saying that the [[product topological space]] of any [[set]] of [[compact topological space]] is itself compact. First a very routine lemma.

+-- {: .num_lemma #LemmaTychonoff}
###### Lemma
Let $(X_i)_{i: I}$ be a family of [[topological spaces]]. Then for a point $x$ and subset $A$ of the [[product topological space]] $\prod_{i: I} X_i$, we have $x \in Cl(A)$ (the [[topological closure]] of $x$) if, for every [[finite set|finite]] $F \subseteq I$, we have $\pi_F(x) \in Cl(\pi_F(A))$ under the [[projection]] map $\pi_F \colon \prod_{i: I} X_i \to \prod_{i: F} X_i$.
=--

+-- {: .proof #LemmaTychonoffProof}
###### Proof
Suppose $x \notin Cl(A)$, so that there is an [[open neighborhood]] $U$ of $x$ such that $U \cap A = \emptyset$ (by [this lemma](Introduction+to+Topology+--+1#UnionOfOpensGivesClosure)). Sets of the [[pre-image]] of the form $\pi_i^{-1}(U_i)$ for open $U_i \subseteq X_i$ form a [[subbase|subbasis]] of the product topology, so there is a basis element $\pi_F^{-1}(V)$ (for some $V = U_{i_1} \times \ldots \times U_{i_n}$ and $F = \{i_1, \ldots, i_n\}$) that is a neighborhood of $x$ and contained in $U$, so we have $A \cap \pi_F^{-1}(V) = \emptyset$ or $A \subseteq \neg \pi_F^{-1}(V) = \pi_F^{-1}(\neg V)$ ([[pre-image]] of the [[complement]]) or what is the same, $\pi_F(A) \subseteq \neg V$ or $\pi_F(A) \cap V = \emptyset$. Since $V$ is a neighborhood of $\pi_F(x)$ not intersecting $\pi_F(A)$, we conclude $\pi_F(x) \notin Cl(\pi_F(A))$. In summary we have shown that $(x \notin Cl(A)) \Rightarrow \underset{ {F \subset  I} \atop { \text{finite} }}{\exists} \left( \pi_F(x) \notin Cl(\pi_F(A))\right)$. By [[contraposition]], this proves the claim.
=--

+-- {: .num_theorem}
###### Theorem
**([[Tychonoff theorem]])**

The [[cartesian product]] of a [[small set|small]] family of [[compact topological spaces]] with the [[product topology]] is itself compact.

=--

+-- {: .proof}
###### Proof
Let $(X_\alpha)_{\alpha \lt \kappa}$ be a family of compact spaces indexed by an [[ordinal]] $\kappa$. According to Theorem \ref{char}, it is enough to show that the [[projection]]

$$Y \times \prod_{\alpha \lt \kappa} X_\alpha \to Y$$

is a [[closed map]] for any space $Y$. We do this by [[induction]] on $\kappa$. The case $\kappa = 0$ is trivial.

It will be convenient to introduce some notation. For $\gamma \leq \kappa$, let $X^\gamma$ denote the product $Y \times \prod_{\alpha \lt \gamma} X_\alpha$ (so $X^0 = Y$ in this notation), and for $\beta \leq \gamma$ let $\pi_\beta^\gamma \;\colon\; X^\gamma \to X^\beta$ be the obvious projection map. Let $K \subseteq X^\kappa$ be closed, and put $K_\beta \coloneqq Cl(\pi_{\beta}^\kappa(K))$. In particular $K_\kappa = K$ since $K$ is closed, and we are done if we show $\pi_0^\kappa(K) = K_0$.

Assume as inductive hypothesis that starting with any $x_0 \in K_0$ there is $x_\beta \in K_\beta$ for all $\beta \lt \kappa$ such that whenever $\beta \lt \gamma \lt \kappa$, the compatibility condition $\pi_\beta^\gamma(x_\gamma) = x_\beta$ holds. In particular, $\pi_0^\beta(x_\beta) = x_0$ for all $\beta \lt \kappa$, and we are now trying to extend this up to $\kappa$.

If $\kappa = \beta + 1$ is a [[successor ordinal]], then the projection

$$\pi_\beta^\kappa \;\colon\; X^\beta \times X_\beta \to X^\beta$$

is a closed map since $X_\beta$ is compact. Thus $\pi_\beta^\kappa(K) = Cl(\pi_\beta^\kappa(K)) = K_\beta$ since $K$ is closed, so there exists $x_\kappa \in K$ with $\pi_\beta^\kappa(x_\kappa) = x_\beta$, and then

$$\pi_0^\kappa(x_\kappa) = \pi_0^\beta \pi_\beta^\kappa (x_\kappa) = \pi_0^\beta(x_\beta) = x_0$$

as desired.

If $\kappa$ is a [[limit ordinal]], then we may regard $X^\kappa$ as the [[inverse limit]] of spaces $(X^\beta)_{\beta \lt \kappa}$ with the obvious transition maps $\pi_\beta^\gamma$ between them. Hence the tuple $(x_\beta)_{\beta \lt \kappa}$ defines an element $x_\kappa$ of $X^\kappa$, and all that remains is to check that $x_\kappa \in K$. But since $K$ is closed, lemma \ref{LemmaTychonoff} says that it is sufficient to check that for every finite set $F$ of ordinals below $\kappa$, that $\pi_F(x_\kappa) \in Cl(\pi_F(K))$ (as a subspace of $\prod_{\alpha \in F} X_\alpha$). But for every such $F$ there is some $\beta \lt \kappa$ that dominates all the elements of $F$. One then checks

$$\pi_F(x_\kappa) = \pi_F^\beta \pi_\beta^\kappa(x_\kappa) = \pi_F^\beta(x_\beta) \in \pi_F^\beta(K_\beta) = \pi_F^\beta(Cl(\pi_\beta^\kappa(K))) \subseteq Cl(\pi_F^\beta \pi_\beta^\kappa(K)) = Cl(\pi_F(K))$$

where the inclusion indicated as $\subseteq$ just results from continuity of $\pi_F^\beta$. This completes the proof.
=--
### Tube lemma
 {#TubeLemma}

The following useful lemma was effectively proved already within the proof of prop. \ref{onlyif}, but it is worth making it explicit:

+-- {: .num_lemma #tube}
###### Lemma
**([[tube lemma]])**

Let

1. $X$ be a [[topological space]],

1. $Y$ a [[compact topological space]],

1. $x \in X$ a point,

1. $W \underset{\text{open}}{\subset} X \times Y$ an open subset in the [[product topology]],

such that the $Y$-[[fiber]] over $x$ is contained in $W$:

$$
  \{x\} \times Y \;\subseteq\; W
  \,.
$$

Then there exists an [[open neighborhood]] $U_x$ of $x$ such that the "tube" $U_x \times Y$ around the fiber $\{x \} \times Y$ is still contained:

$$
  U_x \times Y \subseteq W
  \,.
$$

=--

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

Let

$$
  C \coloneqq (X \times Y) \backslash W
$$

be the [[complement]] of $W$. Since this is closed, by prop. \ref{onlyif} also its projection $p_X(C) \subset X$ is closed.

Now

$$
  \begin{aligned}
    \{x\} \times Y \subset W
      & \;\Leftrightarrow\;
    \{x\} \times Y \; \cap \; C = \emptyset
    \\
    & \;\Rightarrow\;
    \{x\} \cap  p_X(C) = \emptyset
  \end{aligned}
$$

and hence by the closure of $p_X(C)$ there is (by [this lemma](Introduction+to+Topology+--+1#UnionOfOpensGivesClosure)) an open neighbourhood $U_x \supset \{x\}$ with

$$
  U_x \cap p_X(C) = \emptyset
  \,.
$$

This means equivalently that $U_x \times Y \cap C = \emptyset$, hence that $U_x \times Y \subset W$.

=--

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

More [[category theory|category theoretically]], the tube lemma \ref{tube} becomes the statement that for $X$ compact, the map $\forall_\pi: P(Y \times X) \to P(Y)$ takes opens to opens, as mentioned [above](#StillOtherCleverConstructions).

=--

## References

* {#CT1996} [[Maria Manuel Clementino]] and [[Walter Tholen]], *Tychonoff's theorem in a category*, Proc. Amer. Math. Soc., Vol. 124 No. 11 (Nov. 1996), 3311-3314. ([pdf](http://www.ams.org/journals/proc/1996-124-11/S0002-9939-96-03435-1/S0002-9939-96-03435-1.pdf))

* {#Escardo09} [[Martín Escardó]], *Intersections of compactly many open sets are open*, preprint dated May 27, 2009. ([pdf](http://www.cs.bham.ac.uk/~mhe/papers/compactness-submitted.pdf))

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

Much of the material of the present article was earlier collected in

* [[Todd Trimble]], *Characterizations of compactness*, personal nLab webpage. ([link](https://ncatlab.org/toddtrimble/published/Characterizations+of+compactness))


closure algebra

# Closure algebras
* table of contents
{: toc}

## Idea

Closure algebras are type one [[modal algebras]], in which  the single operator behaves like a [[topological closure|closure]] operation in a [[topological space]].

## Definitions

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

A **closure algebra** is a [[Boolean algebra]] with operator, $(\mathbb{B}, m)$, which satisfies:
for all $x$, $x + m m x \leq m x$.
=--

In general, if $(\mathbb{B}, m)$ is a closure algebra and $x \in B$, we say that $x$ is _closed_ if $m x = x$ and _open_ if $l x = x$, where $l$ is the [[de Morgan duality|dual]] operator of $m$.

A closure algebra is sometimes written in terms of $l$ instead of $m$ and is then called an **interior algebra**.

## Examples

Let $X$ be a [[topological space]] and $\mathbb{P}(X)$ the [[powerset]] Boolean algebra of the underlying set of $X$.  Set $m T$ to be the [[topological closure]] of the set $T \subseteq X$ in the topology of $X$, then $(\mathbb{P}(X), m)$ is a closure algebra.

## Properties

*  If $\mathfrak{B} = (\mathbb{B}, m)$ is a closure algebra, let $Open(\mathfrak{B})$ be the set of open elements in $\mathfrak{B}$, then $Open(\mathfrak{B})$ has the natural structure of a [[Heyting algebra]].  Moreover any Heyting algebra can be represented as the algebra of open elements of a closure algebra.

*  Closure algebras underly the algebraic semantic models of the [[epistemic logic]] $S4$.

*  The algebraic semantics of $S4(n)$ uses [[polyclosure algebra]]s. Here there are many different closure operators on the Boolean algebra.

[[!redirects closure algebra]]
[[!redirects closure algebras]]
[[!redirects interior algebra]]
[[!redirects interior algebras]]