compact-open topology

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### Context
#### Topology
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[[!include topology - contents]]
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#### Mapping space
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[[!include mapping space - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A natural [[topology]] on [[mapping spaces]] of [[continuous functions]], important because of its role in exhibiting [[locally compact topological spaces]] to be [[exponential object|exponentiable]], as demonstrated below, culminating in Corollary \ref{GivesExponentialObject}.

The compact-open topology on the set of [[continuous functions]] $X \to Y$ is generated by the [[base of a topology|subbasis]] of subsets $U^K \subset C(X,Y)$ that map a given [[compact subspace]] $K \subset X$ to a given [[open subset]] $U \subset Y$, whence the name.

When restricting to continuous functions between [[compactly generated topological spaces]] one usually modifies this definition to a subbase of open subsets $U^{\phi(K)}$, where now $\phi(K)$ is the [[image]] of a [[compact topological space]] under any continuous function $\phi \colon K \to X$. This definition gives a [[cartesian closed category|cartesian]] [[internal hom]] in the category of [[compactly generated topological spaces]] (see also at _[[convenient category of topological spaces]]_).

The two definitions agree when the [[domain]] $X$ is a [[compactly generated topological space|compactly generated]]  [[Hausdorff space]], but not in general. Beware that texts on compactly generated spaces nevertheless commonly say "compact-open topology" for the second definition.

## Definition

+-- {: .num_note}
###### Notation

Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[topological spaces]]. We denote by $Y^{X}$ the set of [[continuous maps]] from $(X,\mathcal{O}_{X})$ to $(Y, \mathcal{O}_{Y})$.

=--

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

Other common notations for $Y^{X}$ are $Map(X,Y)$ or $C(X,Y)$.

=--

+-- {: .num_note}
###### Notation

Let $(X, \mathcal{O}_{X})$ be a [[topological space]]. We denote by $\mathcal{O}^{c}_{X}$ the set of subsets of $X$ which are [[compact subset|compact]] with respect to $\mathcal{O}_{X}$.

=--

+-- {: .num_note}
###### Notation

Let $(X, \mathcal{O}_{X})$ be a [[topological space]]. Let $A$ be a subset of $X$. We denote by $\overline{A}$ the [[topological closure]] of $A$ in $X$ with respect to  $\mathcal{O}_{X}$.

=--

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

A topological space $(X, \mathcal{O}_{X})$ is _[[locally compact topological space|locally compact]]_ if the set of compact neighbourhoods   of points of $X$ form a [[neighbourhood basis]] of $X$. That is to say: for every $x \in X$ and every [[neighbourhood]] $N$ of $x$, there is a $V \in \mathcal{O}^{c}_{X}$ and a $U \in \mathcal{O}_{X}$ such that $x \in U \subset V \subset N$.

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

There are many variations on this definition, which can be found at _[[locally compact space]]_. These are all equivalent if $(X, \mathcal{O}_{X})$ is [[Hausdorff space|Hausdorff]]. We do _not_ however make the assumption that $(X, \mathcal{O}_{X})$ is [[Hausdorff space|Hausdorff]].

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+-- {: .num_note}
###### Notation

Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[topological spaces]]. Given $A \in \mathcal{O}^{c}_{X}$ and $U \in \mathcal{O}_{Y}$, we denote by $M_{A,U}$ the set of continuous maps $f : X \rightarrow Y$ such that $f(A) \subset U$.

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

Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[topological spaces]]. The _compact-open_ topology on $Y^{X}$ is that with [[base of a topological space|sub-basis]] given by the set of sets $M_{A,U}$ such that $A \in \mathcal{O}^{c}_{X}$ and $U \in \mathcal{O}_{Y}$.

=--

+-- {: .num_note #CompactOpenTopologyNotation}
###### Notation

Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[topological spaces]]. We shall denote the compact-open topology (def. \ref{CompactOpenTopology}) on $Y^{X}$ by $\mathcal{O}_{Y^{X}}$.

=--

## Exponentiability
 {#Exponentiability}

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

Let $(X, \mathcal{O}_{X})$ be a locally compact topological space, def. \ref{LocallyCompact}, and let $(Y, \mathcal{O}_{Y})$ be a [[topological space]]. The map $ev : X \times Y^{X} \rightarrow Y$ given by $(x,f) \mapsto f(x)$ is continuous, where $X \times Y^{X}$ is equipped with the [[product topology]] $\mathcal{O}_{X \times Y^{X}}$ with respect to $\mathcal{O}_{X}$ and $\mathcal{O}_{Y^{X}}$ (notation \ref{CompactOpenTopologyNotation}).

=--

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

By an elementary fact concerning continuous maps, it suffices to show that for any $(x,f) \in X \times Y^{X}$, and any $U \in \mathcal{O}_{Y}$ such that $f(x) \in U$, there is a $U' \in \mathcal{O}_{X \times Y^{X}}$ such that $(x,f) \in U'$, and such that $ev(U') \subset U$.

To demonstrate this, we make the following observations.

1) Since $f$ is continuous, we have that $f^{-1}(U) \in \mathcal{O}_{X}$.

2) Since $(X, \mathcal{O}_{X})$ is locally compact, we deduce from 1) that there is a $V \in \mathcal{O}^{c}_{X}$ and a $U'' \in \mathcal{O}_{X}$ such that $x \in U'' \subset V \subset f^{-1}(U)$.

3) By 2) and by definition of $\mathcal{O}_{X \times Y^{X}}$ and $\mathcal{O}_{Y^{X}}$, we have that $U'' \times M_{V, U} \in \mathcal{O}_{X \times Y^{X}}$, and that $(x,f) \in U'' \times M_{V, U}$.

4)  Let $(x',f') \in U'' \times M_{V, U}$, Since $x' \in U''$, and since $f'(U'') \subset f'(V) \subset U$, the latter inclusion holding by definition of $M_{V, U}$, we have that $f'(x) \in U$. We deduce that $ev(U'' \times M_{ V, U}) \subset U$.

By 3) and 4), we see that we can take $U'$ as the beginning of the proof to be $U'' \times M_{V , U}$.

=--

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

Let $(X, \mathcal{O}_{X})$, $(Y, \mathcal{O}_{Y})$, and $(Z, \mathcal{O}_{Z})$ be [[topological spaces]]. Let $f : X \times Y \rightarrow Z$ be a continuous map, where $X \times Y$ is equipped with the [[product topology]] $\mathcal{O}_{X \times Y}$ with respect to $\mathcal{O}_{X}$ and $\mathcal{O}_{Y}$. Then the map $\alpha_{f} : X \rightarrow Z^{Y}$ given by $x \mapsto f_{x}$ is continuous, where $f_{x} : Y \rightarrow Z$ is given by $y \mapsto f(x,y)$, and where $Z^{Y}$ is equipped with the compact-open topology $\mathcal{O}_{Z^{Y}}$.

=--

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

By an elementary fact concerning continuous maps, it suffices to show that for any $x \in X$, and any $M_{A,U} \in \mathcal{O}_{Z^{Y}}$ such that $f_{x} \in M_{A,U}$, there is a $U' \in \mathcal{O}_{X}$ such that $x \in U'$, and such that $\alpha_{f}( U' ) \subset M_{A,U}$.

To demonstrate this, we make the following observations.

1) Since $f_{x} \in M_{A,U}$, we have, by definition of $f_{x}$ and by definition of $M_{A,U}$, that $f(x,a) \in U$ for all $a \in A$.

2) Since $f$ is continuous, we have that $f^{-1}(U) \in \mathcal{O}_{X \times Y}$.

3) By 1), we have that $\{x\} \times A \subset f^{-1}(U)$.

4) Let $\mathcal{O}_{X \times A}$ denote the [[subspace topology]] on $X \times A$ with respect to $\mathcal{O}_{X \times Y}$. By 2), we have that $f^{-1}(U) \cap (X \times A) \in \mathcal{O}_{X \times A}$.

5) By 3), we have that $\{x \} \times A \subset f^{-1}(U) \cap (X \times A)$.

6) Since $A \in \mathcal{O}^{c}_{Y}$, it follows from 4), 5), and the [[tube lemma]] that there is a $U' \in \mathcal{O}_{X}$ such that $x \in U'$ and such that $U' \times A \subset f^{-1}(U) \cap (X \times A) \subset f^{-1}(U)$.

7)  We deduce from 6) that $f(U' \times A) \subset U$. This is the same as to say that $f_{x'}(a) \in U$ for all $x' \in U'$. Thus $f_{x'}$ belongs to $M_{A,U}$ for all $x' \in U'$, which is the same as to say that $\alpha_{f}(U') \subset M_{A,U}$.

We conclude that we can take the required $U'$ of the beginning of the proof to be the $U'$ of 6).

=--

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

Let $(X, \mathcal{O}_{X})$, $(Y, \mathcal{O}_{Y})$, and     $(Z, \mathcal{O}_{Z})$ be [[topological spaces]]. Suppose that $(Y, \mathcal{O}_{Y})$ is locally compact. Let $f : X \rightarrow Z^{Y}$ be a continuous map, where $Z^{Y}$ is equipped with the compact-open topology $\mathcal{O}_{Z^{Y}}$. Then the map $\alpha^{-1}_{f} : X \times Y \rightarrow Z$ given by $(x,y) \mapsto \big( f(x) \big)(y)$ is continuous, where $X \times Y$ is equipped with the [[product topology]] $\mathcal{O}_{X \times Y}$.

=--

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

We make the following observations.

1) We have that $\alpha^{-1}_{f} = ev \circ \tau \circ (f \times id)$, where $\tau : Z^{Y} \times Y \rightarrow Y \times Z^{Y}$ is given by $(f,y) \mapsto (y,f)$.

2) By Proposition \ref{evIsContinuous}, we have that $ev$ is continuous.

3) It is an elementary fact that $\tau$ is continuous.

4) Since $f$ is continuous, it follows by an elementary fact that $f \times id$ is continuous.

5) We deduce from 2) - 4) that $ev \circ \tau \circ (f \times id)$ is continuous. By 1), we conclude that $\alpha^{-1}_{f}$ is continuous.

=--

+-- {: .num_cor #GivesExponentialObject}
###### Corollary

Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$  be [[topological spaces]]. Suppose that $(X, \mathcal{O}_{X})$ is locally compact. Then $(Y^{X}, \mathcal{O}_{Y^{X}})$ together with the corresponding map $ev$ defines an [[exponential object]] in the [[category]] [[Top]] of all topological spaces.

=--

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

Follows immediately from Proposition \ref{alphafIsContinuous}, Proposition \ref{alphainvfIsContinuous}, and the fact that $Y^{X}$ and $ev$ are exhibited by the corresponding maps $\alpha_{-}$ and  $\alpha^{-1}_{-}$ of Proposition \ref{alphafIsContinuous} and Proposition \ref{alphainvfIsContinuous} to define an exponential object in the category $\mathsf{Set}$ of sets.

=--

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

A proof can also be found in [Aguilar-Gitler-Prieto 02, prop. 1.3.1](#AguilarGitlerPrieto02), or just about any half-decent textbook on point-set topology!

However, the result is almost universally stated with an assumption that $(X,\mathcal{O}_{X})$ is [[Hausdorff space|Hausdorff]] which, as the proof we have given illustrates, is not needed.

=--

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

We moreover have a homeomorphism $Map(X,Map(Y,Z))\cong Map(X\times Y,Z)$ if in addition $X$ is [[Hausdorff space|Hausdorff]]. See also [[convenient category of topological spaces]].

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## Examples

### Maps out of the point space

+-- {: .num_example #MappingSpaceOutOfPoint}
###### Example
**([[mapping space]] construction out of the [[point space]] is the identity)**

The [[point space]] $\ast$  is clearly a [[locally compact topological space]]. Hence for every [[topological space]] $(X,\tau)$ the mapping space $Maps(\ast, (X,\tau))$ exists. This is [[homeomorphism|homeomorphic]]  to the space $(x,\tau)$ itself:

$$
  Maps(\ast, (X,\tau))
    \simeq
  (X,\tau)
  \,.
$$

=--

### Maps into metric spaces

If $Y$ is a [[metric space]] then the compact-open topology on $Map(X,Y)$ is the _topology of uniform convergence on compact subsets_ in the sense that $f_n \to f$ in $Map(X,Y)$ with the compact-open topology iff for every compact subset $K\subset X$, $f_n \to f$ uniformly on $K$. If (in addition) the domain $X$ is compact then this is the _[[topology of uniform convergence]]_.

### Loop spaces and path spaces

+-- {: .num_example #LoopSpace}
###### Example
**([[loop space]] and [[path space]])**

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

1. The standard [[circle]] $S^1$ is a [[compact Hausdorff space]]
   [[open subspaces of compact Hausdorff spaces are locally compact|hence]] a [[locally compact topological space]]. Accordingly the [[mapping space]]

   $$
     \mathcal{L} X \coloneqq Maps( S^1, (X,\tau) )
   $$

   exists. This is called the _[[free loop space]]_ of $(X,\tau)$.

   If both $S^1$ and $X$ are equipped with a choice of point ("[[basepoint]]") $s_0 \in S^1$, $x_0 \in X$, then the [[topological subspace]]

   $$
     \Omega X \subset \mathcal{L}X
   $$

   on those functions which take the basepoint of $S^1$ to that of $X$, is called the _[[loop space]]_ of $X$,
   or sometimes _[[based loop space]]_, for emphasis.

1. Similarly the [[closed interval]] is a [[compact Hausdorff space]]
   [[open subspaces of compact Hausdorff spaces are locally compact|hence]] a [[locally compact topological space]] (def. \ref{LocallyCompactSpace}). Accordingly the [[mapping space]]

   $$
     Maps( [0,1], (X,\tau) )
   $$

   exists. Again if $X$ is equipped with a choice of basepoint $x_0 \in X$, then the [[topological subspace]] of those functions that take $0 \in [0,1]$ to that chosen basepoint is called the _[[path space]]_ of $(X\tau)$:

   $$
     P X \subset Maps( [0,1], (X,\tau) )
     \,.
   $$

Notice that we may encode these subspaces more abstractly in terms of [[universal properties]]:

The path space and the loop space are characterized, up to [[homeomorphisms]], as being the [[limit|limiting cones]]
in the following [[pullback]] diagrams of topological spaces:

1. [[loop space]]:

   $$
     \array{
       \Omega X &\longrightarrow& Maps(S^1, (X,\tau))
       \\
       \downarrow &(pb)& \downarrow^{\mathrlap{Maps(const_{s_0}, id_{(X,\tau)})}}
       \\
       \ast &\underset{const_{x_0}}{\longrightarrow}& X \simeq Maps(\ast,(X,\tau))
     }
     \,.
   $$

1. [[path space]]:

   $$
     \array{
       P X &\longrightarrow& Maps([0,1], (X,\tau))
       \\
       \downarrow &(pb)& \downarrow^{\mathrlap{Maps(const_x, id_{(X,\tau)})}}
       \\
       \ast &\underset{const_{x_0}}{\longrightarrow}& X \simeq Maps(\ast,(X,\tau))
     }
   $$

Here on the right we are using that the mapping space construction is a [[functor]] and we are using example \ref{MappingSpaceOutOfPoint} in the
identification on the bottom right mapping space out of the point space.

=--

## Related concepts

* [[topology of mapping spaces]]

* [[differential topology of mapping spaces]]

* [[C-k topology]]

* [[space of knots]]

* [[mapping complex]], [[simplicial mapping complex]]

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

## References

Textbook accounts:

* {#Bredon93} [[Glen Bredon]], Section VII.2 of: _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0),  [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

* [[Francis Borceux]], Sections 7.1 & 7.2 of: *Categories and Structures*, Vol. 2 of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865))

* {#AguilarGitlerPrieto02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, Sections 1.2, 1.3 of: _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

See also:

* Wikipedia, *[Compact-open topology](http://en.wikipedia.org/wiki/Compact-open_topology)*

* Ralph H. Fox, _On Topologies for Function Spaces_, Bull. AMS **51** (1945) pp.429-432. ([pdf](http://projecteuclid.org/download/pdf_1/euclid.bams/1183506987))

* Eva Lowen-Colebunders, Günther Richter, _An Elementary Approach to Exponential Spaces_, Applied Categorical Structures May 2001, Volume 9, Issue 3, pp 303-310 ([publisher](http://link.springer.com/article/10.1023%2FA%3A1011268007097))

[[!redirects compact open topology]]

[[!redirects mapping space]]
[[!redirects mapping spaces]]

[[!redirects space of maps]]
[[!redirects spaces of maps]]


compactification

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

## Idea

### General

Traditionally, a compactification of a [[topological space]] $X$ is a [[compact space]] $C$ together with an [[embedding]] $i \colon X \to C$ as a [[dense subspace]]. Well known is for instance the _[[one-point compactification]]_ of a [[locally compact Hausdorff space]] which for instance sends the real [[line]] to the [[circle]] by adding a point at infinity.

In some cases, the terminology 'compactification' is applied to generalizations where $i$ is not a dense embedding (does not map [[homeomorphism|homeomorphically]] to an image that is dense). For example, the one-point compactification of an already [[compact Hausdorff space]] adds an *isolated* point at infinity, and so is not a dense subspace. Or, if $X$ is not a Tychonoff space, then the universal map to its [[Stone-Cech compactification]] is not an embedding.

If the space has further geometric structure, the compactification is
usually required to have such a structure and embedding has to preserve it.
Many [[moduli space]]s in algebraic and differential geometry have their
natural compactifications. They are often useful because they carry natural integration which is useful in defining various invariants.

A useful intuition throughout is that a 'compactification' is a process of adding "ideal points at infinity" in some way to "complete" a space. (Compact [[regular space|regular]] spaces $X$ themselves being "complete" in a technical sense: there is a unique [[uniform space|uniform structure]] whose uniform topology is the topology on $X$, and $X$ is complete with respect to this uniformity.)

### Fiberwise

Often a space can be viewed as a total space of a bundle over some base.
We may want to embed the space into a bigger bundle, such that the induced
embedding of each fiber into the new fiber is a compactification.

This is roughly the case in most __compactifications in physics__. In most cases the space is equipped with a [[Riemannian metric]] and additional quantities for defining physics, like a [[Lagrangian density]], which possibly depend on the metric.

Then one requires that the compactified fiber is finite but small compared to some reference scale (or even viewed in a limit when the Riemannian volume tends to zero), see at _[[Kaluza-Klein mechanism]]_. Often one does not even consider a noncompact case to start with but by compactification in physics means only passing to the limit of small (Riemannian volume of) fibers.

## Definition

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

A _compactification_ of a [[topological space]] $X$
is a [[compact topological space|compact]] [[Hausdorff topological space]]
$Y$ equipped with an [[embedding]] $X \hookrightarrow Y$
such that the [[closure]] of $X$ in $Y$ is the compact space:
$\overline{X} = Y$.

An [[equivalence]] of two compactifications $Y_1$, $Y_2$ of $X$
is a [[homeomorphism]] $h \;\colon\; Y_1 \longrightarrow Y_2$
that preserves the inclusion of $X$.

=--

## Properties

### Uniqueness

In some sense the [[one-point compactification]] is the smallest
possible compactification, while the [[Stone-Cech compactification]]
is the largest. The following gives conditions that
all notions of compactification agree.

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

For $X$ a [[Tychonoff space]] the following are equivalent:

1. There is a unique (up to equivalence) compactification, def. \ref{Compactification}.

1. $X$ is already [[compact topological space|compact]] or its [[Stone-Cech compactification]] $\beta X$ adds a single point $\vert \beta X \backslash X\vert = 1$.

1. If two [[closed subsets]] of $X$ are completeley separated, then one of them is a [[compactum]].

=--

One place where this appears is ([Hewitt 47](#Hewitt47)).

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

The topological spaces satisfying the conditions of
prop. \ref{UniqueCompactificationOfAlmostCompactSpaces}
are also called [[almost compact topological space|almost compact topological spaces]].

=--

## Examples

### In topology

* [[One-point compactification]]

* [[Stone–Cech compactification]]

* [[end compactification|End compactification]]

* [[Bohr compactification]]

* the [[Fulton-MacPherson operad]] is a compactification of [[configuration spaces of points]]

### In geometry

* [[wonderful compactification]]

* [[Deligne-Mumford compactification]].

* [[Kaluza-Klein compactification]]

  * [[Freund-Rubin compactification]]

* [[conformal compactification]]

* [[projective compactification]]

## References

* Edwin Hewitt, _Certain generalizations of the Weierstrass approximation theorem_,  Duke Math. J. Volume 14, Number 2 (1947), 419-427. ([Euclid](http://projecteuclid.org/euclid.dmj/1077474139))
  {#Hewitt47}

[[!redirects compactification]]
[[!redirects compactifications]]


compactly generated (∞,1)-category

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### Context
#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#### Compact objects
+-- {: .hide}
[[!include compact object - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

For $\kappa$ a [[regular cardinal]],
an [[(∞,1)-category]] is called _$\kappa$-compactly generated_ if it is
$\kappa$-[[accessible (infinity,1)-category|accessible]] and [[locally presentable (∞,1)-category|locally presentable]].

**Terminology** The terms "$\kappa$-compactly generated (∞,1)-category" and "[[locally presentable (∞,1)-category|locally $\kappa$-presentable (∞,1)-category]]" have the same meaning. There are differences in usage, though.

 * If we "drop the $\kappa$", then a _locally presentable (∞,1)-category_ is a an (∞,1)-category which is locally $\kappa$-presentable for _some_ $\kappa$, but a _compactly generated (∞,1)-category_ is an (∞,1)-category which is locally _finitely_ presentable, i.e. locally $\kappa$-presentable for $\kappa = \aleph_0$.

* If we "leave the $\kappa$ in", the terms "$\kappa$-compactly generated (∞,1)-category" and "[[locally presentable (∞,1)-category|locally $\kappa$-presentable (∞,1)-category]]" have the same meaning. Some authors choose one term over the other. For example, in [[Higher Topos Theory]], "$\kappa$-compactly generated (∞,1)-category" is preferred. Albeit, Lurie uses $\mathrm{Pr}_\kappa$ to denote the (∞,1)-category of $\kappa$-compactly generated (∞,1)-categories.

([[Higher Topos Theory|HTT, def. 5.5.7.1]])

## Properties

### Recognition for stable $(\infty,1)$-categories
 {#RecognitionInStableCase}

Compact generation for [[stable (∞,1)-categories]] is detected already on their [[triangulated category|triangulated]] [[homotopy category of an (infinity,1)-category|homotopy category]] as discussed at _[[compactly generated triangulated category]]_

([[Higher Algebra|HA, remark 1.4.4.3]]))

## Related concepts

* [[ind-object in an (∞,1)-category]]

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

* [[compactly generated model category]]

* [[compactly generated triangulated category]]

## References

* [[Jacob Lurie]], section 5.5.7 of _[[Higher Topos Theory]]_

* [[Jacob Lurie]], section 1.4.4 of _[[Higher Algebra]]_

[[!redirects compactly generated (∞,1)-categories]]
[[!redirects compactly generated (infinity,1)-category]]
[[!redirects compactly generated (infinity,1)-categories]]


compactly generated model category

> under construction -- warning -- currently inconsistent

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###Context###
#### Model category theory
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#### Compact objects
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[[!include compact object - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

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

A [[cofibrantly generated model category|cofibrantly generated]] [[simplicial model category]] $C$ is **compactly generated** if

* there exists a [[small set]] $S \subset Obj(C)$ of [[object]]s

* such that

  1. each $K \in S$ is cofibrant;

  1. each $K \in S$ is a homotopy [[compact object]]: for all [[filtered colimit]] diagram $Y : D \to C$ the morphism

     $$
       \mathbb{L}\lim_{\to_i} C(K, Y_i)
       \simeq
       C(K, \mathbb{L}\lim_{\to_i} Y_i)
     $$

     (where $\mathbb{L}\lim_\to$ denotes the $Ho_C$ is the [[homotopy category]] of $C$) is a [[weak homotopy equivalence]] in [[sSet]];

  1. a morphism $X \to Y$ in $C$ is a [[weak equivalence]] precisely if for all $K \in S$ the induced morphism

     $$
       Ho_C(K, X) \to Ho_C(K,Y)
     $$

     is a bijection.

> Something needs to be added/fixed here!!

=--

See ([Jardine11, page 14](#Jardine)), ([Marty, def 1.7](#Marty)).

## Related concepts

* [[compactly generated triangulated category]]

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

## References

Page 88 (14) of

* {#Jardine} [[J. F. Jardine]], _Representability theorems for presheaves of spectra_ J. Pure Appl. Algebra
215 (2011) ([pdf](http://www.math.uwo.ca/~jardine/papers/preprints/rep-2010.pdf))

Def. 1.7 of

* Florian Marty, _Smoothness in relative geometry_ (2009) ([pdf](http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2010/2009_104.pdf))

A different meaning of "compactly generated model category" is used in Definition 5.9 of

* {#MMSS00} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], *[[Model categories of diagram spectra]]*, Proceedings of the London Mathematical Society Volume 82, Issue 2, 2000 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf), [doi:10.1112/S0024611501012692]( https://doi.org/10.1112/S0024611501012692))

[[!redirects compactly generated model categories]]


compactly generated topological space

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

# Contents
* table of contents
{: toc}

## Idea

{#IsCalled} A [[topological space]] is called _compactly generated_
-- also called a
"k-space"[^1] ([Gale 1950, 1.](#Gale50), following lectures by [[Witold Hurewicz|Hurewicz]] in 1948),
"Kelley space" ([Gabriel & Zisman 1967, III.4](#GabrielZisman67)), or
"kaonic space" ([Postnikov 1982, p. 34](#Postnikov82)) --
if its topology is detected by the [[continuous map|continuous]] [[images]] of [[compact Hausdorff spaces]] inside it.

As opposed to general topological spaces, compactly generated spaces form a [[cartesian closed category]] while still being general enough for most purposes of [[general topology]], hence form a [[convenient category of topological spaces]] ([Steenrod 1967](#Steenrod67)) and as such have come to be commonly used in the foundations of [[algebraic topology]] and [[homotopy theory]], especially in their modern guise as compactly generated *weakly Hausdorff* spaces, due to [McCord 1969, Sec. 2](#McCord69).

## Definition
 {#Definition}

\begin{definition}\label{kContinuousFunction}
**($k$-continuous functions)**
\linebreak
A [[function]] $f\colon X \to Y$ between [[underlying]] [[sets]] of [[topological spaces]] is called *$k$-continuous* if for all [[compact Hausdorff spaces]] $C$ and [[continuous functions]] $t \colon C \to X$ the [[composition|composite]] $f \circ t \colon C \to Y$ is [[continuous function|continuous]].
\end{definition}

\begin{proposition}\label{EquivalentConditionsForCompactGeneration}
**(equivalent characterizations of compact generation)**
\linebreak
The following conditions on a [[topological space]] $X$ are equivalent:

1. For all spaces $Y$ and all [[functions]] $f \colon X \to Y$ (of [[underlying]] [[sets]]), $f$ is [[continuous function|continuous]] if and only if $f$ is $k$-continuous (Def. \ref{kContinuousFunction}).

2. There is a [[set]] $S$ (instead of a [[proper class]]) of [[compact Hausdorff spaces]] such that the previous condition holds for all $C \in S$.

3. $X$ is an [[identification space]] of a [[disjoint union]] of compact Hausdorff spaces.

4. A [[subspace]] $U \subseteq X$ is [[open subspace|open]] if and only if the [[preimage]] $t^{-1}(U)$ under any [[continuous function]] $t \colon C \to X$ out of a [[compact Hausdorff space]] $C$ is open.

\end{proposition}

\begin{definition}\label{kSpace}
**(k-spaces)**
\linebreak
A [[topological space]] $X$ is a __$k$-space__ if any (hence all) of the conditions in Prop. \ref{EquivalentConditionsForCompactGeneration} hold.
\end{definition}
\begin{remark}
**(terminology)**
\linebreak
Some authors say that a $k$-space (Def. \ref{kSpace}) is __compactly generated__, while others reserve that term for a $k$-space which is also _[[weak Hausdorff space|weak Hausdorff]]_, meaning that the image of any $t\colon C\to X$ is closed (when $C$ is compact Hausdorff).  Some authors (especially the early authors on the subject) go on to require a [[Hausdorff space|Hausdorff]] space, but this seems to be unnecessary.
\end{remark}

## Examples
 {#Examples}

Examples of compactly generated spaces include

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

Every [[compact Hausdorff space]] is compactly generated.

=--

Note: it is not generally true that [[compact spaces]] are compactly generated, even if they are [[weakly Hausdorff space|weakly Hausdorff]]. An example is the square of the [[one-point compactification]] of the rationals $\mathbb{Q}$ with its standard topology. See for example this [MathStackExchange post](https://math.stackexchange.com/a/964546/43208).

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

Every [[locally compact Hausdorff space]] is compactly generated.

=--

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

Every [[topological manifold]] is compactly generated

=--

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

Every [[CW-complex]] is a compactly generated topological space.

=--

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

Since a CW-complex $X$ is a [[colimit]] in [[Top]] over attachments of standard [[n-disks]] $D^{n_i}$ (its cells), by the characterization of colimits in $Top$ ([prop.](Top#DescriptionOfLimitsAndColimitsInTop)) a subset of $X$ is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the $n$-disks are compact, this implies one direction: if a subset $A$ of $X$ intersected with all compact subsets is closed, then $A$ is closed.

For the converse direction, since [[a CW-complex is a Hausdorff space]] and since [[compact subspaces of Hausdorff spaces are closed]], the intersection of a closed subset with a compact subset is closed.

=--

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

Every [[first countable space]] is a compactly generated space.

=--

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

Since the topology is determined by convergent sequences = maps from one-point [[compactification]] $\mathbb{N} \cup \{\infty\}$); these include all [[Frechet–Uryson spaces]].

=--

## Properties

### Coreflection into topological spaces
 {#CoreflectionIntoTopologicalSpaces}

\begin{definition}\label{CategoryOfKSpaces}
**(category of k-spaces)**
\linebreak
We write
\[
  \label{kSpacesAsSubcategoryOfAllSpaces}
  k\Top
  \xhookrightarrow{\;\;\;\;}
  Top
\]
for the category of $k$-spaces (Def. \ref{kSpace}) with [[continuous functions]] between them, hence for the [[full subcategory]] of [[Top]] on the k-spaces.
\end{definition}

\begin{prop}\label{TheCoreflection}
The inclusion (eq:kSpacesAsSubcategoryOfAllSpaces)
is that of a  [[coreflective subcategory]]
\[
  \label{kIficationReflection}
  k Top
  \underoverset
    {\underset{k}{\longleftarrow}}
    {\overset{}{\hookrightarrow}}
    {\;\;\;\; \bot \;\;\;\;}
  Top
\]
\end{prop}
The [[coreflection]]
$k$ is sometimes called *$k$-ification* ([May 1999, p. 49](#May99)).[^3]
\begin{proof}
The reflection functor $k$ is constructed as follows:

We take $k(X) \coloneqq X$ on [[underlying]] [[sets]], and equip this with the [[topological space|topology]] whose [[closed sets]] are those whose [[intersection]] with [[compact Hausdorff space|compact Hausdorff]] subsets of (the original topology on) $X$ is [[closed subset|closed]] (in the original topology on $X$). Then $k(X)$ has all the same closed sets and possibly more, hence all the same open sets and possibly more.

In particular, the identity map $id \colon k(X)\to X$ is continuous, and forms the [[counit of an adjunction|counit]] of the [[coreflective subcategory|coreflection]].  Thus this coreflection has a counit which is both a [[monomorphisms]] as well as an [[epimorphisms]], i.e. a "[[bimorphism]]"---such a coreflection is sometimes called a "bicoreflection."
\end{proof}

\begin{remark}
There is also the category $\Top_k$ of all topological spaces and $k$-continuous maps (Def. \ref{kContinuousFunction}).  But the [[composition|composite]] sequence of inclusions
$$
  k\Top \xhookrightarrow{\;} \Top \xrightarrow{\;} \Top_k
  \,,
$$
of which the first is the  [[full subcategory|full]] inclusion (eq:kSpacesAsSubcategoryOfAllSpaces) and the second is [[bijective on objects functor|bijective on objects]] $k\Top \to Top_k$, is an [[equivalence of categories]].

Namely, the [[identity morphism]] $id \colon X \to k(X)$ is $k$-continuous, so that the [[adjunction counit]] from Prop. \ref{TheCoreflection} becomes an [[isomorphism]] in $\Top_k$.  This shows that $k\Top \to \Top_k$ is [[essentially surjective functor|essentially surjective]], and it is [[fully faithful functor|fully faithful]] since any $k$-continuous function between $k$-spaces is $k$-continuous; hence it is an equivalence.
\end{remark}

\begin{remark}
Since $k\Top \hookrightarrow \Top$ is coreflective, it follows that $k\Top$ is [[complete category|complete]] and [[cocomplete category|cocomplete]].  Its [[colimits]] are constructed as in [[Top]], but its [[limits]] are the $k$-ification (eq:kIficationReflection) of [limits in Top](Top#UniversalConstructions).

Notice that this is nontrivial already for [[products]]: the $k$-space product $X \times Y$ is the $k$-ification of the usual [[product topology]].  The $k$-space product is better behaved in many ways; e.g. it enables [[geometric realization]] to preserve products (and all [[finite limits]]), and the product of two [[CW complexes]] to be another CW complex.
\end{remark}

### Reflection into weak Hausdorff spaces
 {#ReflectionIntoWeakHausdorffSpaces}

\begin{definition}
\label{FullSubcategoryOfCGWHSpacesInsideCGSpaces}
Write
$$
  h k Top
  \xhookrightarrow{\;\;\;}
  k Top
$$
for the further [[full subcategory]] inside that of k-spaces (Def. \ref{CategoryOfKSpaces}) on those which in addition are [[weak Hausdorff spaces]].
\end{definition}

\begin{prop}\label{ReflectionOntoCGWHSpaces}
**(cgwh spaces reflective in cg spaces)**
\linebreak
The [[full subcategory]]-inclusion of [[weakly Hausdorff space|weak Hausdorff spaces]] in k-spaces (Def. \ref{FullSubcategoryOfCGWHSpacesInsideCGSpaces}) is a [[reflective subcategory]] inclusion:
$$
  h k Top
  \underoverset
    {\underset{}{\hookrightarrow}}
    {\overset{ h }{\longleftarrow}}
    {\;\;\;\; \bot \;\;\;\; }
  k Top
$$
\end{prop}
(e.g. [Strickland 2009, Prop. 2.22](#Strickland09))

\begin{remark}\label{SequencesOfCoReflections}
**(sequence of [[coreflective subcategory|(co-)]][[reflective subcategory|reflections]])**
\linebreak
In summary, Prop. \ref{TheCoreflection} and Prop. \ref{ReflectionOntoCGWHSpaces} yield a sequence of [[adjoint functors]] of this form:

$$
  h k Top
  \underoverset
    {\underset{}{\hookrightarrow}}
    {\overset{ h }{\longleftarrow}}
    {\;\;\;\; \bot \;\;\;\; }
  k Top
  \underoverset
    {\underset{k}{\longleftarrow}}
    {\overset{}{\hookrightarrow}}
    {\;\;\;\; \bot \;\;\;\;}
  Top
$$

The [[classical model structure on topological spaces]] restricts along these (co-)reflective embeddings to a [[Quillen equivalence|Quillen equivalent]] [[model structure on compactly generated topological spaces]]. See there for more.
\end{remark}

### Cartesian closure

The categories $k\Top\simeq \Top_k$ are [[cartesian closed category|cartesian closed]]. (While in [[Top]] only some objects are exponentiable, see [[exponential law for spaces]].)  For arbitrary spaces $X$ and $Y$, define the *test-open* or *compact-open topology* on $\Top_k(X,Y)$ to have the [[subbase]] of sets $M(t,U)$, for a given compact Hausdorff space $C$, a map $t\colon C \to X$, and an open set $U$ in $Y$, where $M(t,U)$ consists of all $k$-continuous functions $f\colon X \to Y$ such that $f(t(C))\subseteq U$.

(This is slightly different from the usual [[compact-open topology]] if $X$ happens to have non-Hausdorff compact subspaces; in that case the usual definition includes such subspaces as tests, while the above definition excludes them.  Of course, if $X$ itself is Hausdorff, then the two become identical.)

With this topology, $\Top_k(X,Y)$ becomes an [[exponential object]] in $Top_k$.  It follows, by [[Yoneda lemma]] arguments ([prop.](closed+monoidal+category#TensorHomIsoInternalizes)), that the bijection

$$
  k\Top(X \times Y, Z)
  \longrightarrow
  k Top\big(X,k\Top(Y,Z)\big)
$$

is actually an [[isomorphism]] in $\Top_k$, which we may call a *$k$-homeomorphism* (e.g. [Strickland 09, prop. 2.12](#Strickland09)).  In fact, it is actually a homeomorphism, i.e. an isomorphism already in $Top$.

It follows that the category $k\Top$ of $k$-spaces and continuous maps is also cartesian closed, since it is equivalent to $\Top_k$.  Its exponential object is the $k$-ification of the one constructed above for $\Top_k$.  Since for $k$-spaces, $k$-continuous implies continuous, the underlying set of this exponential space $k\Top(X,Y)$ is the set of all continuous maps from $X$ to $Y$.  Thus, when $X$ is Hausdorff, we can identify this space with the $k$-ification of the usual [[compact-open topology]] on $Top(X,Y)$.

Finally, this all remains true if we also impose the weak Hausdorff, or Hausdorff, conditions.

\begin{remark}{#LocalCartesianClosure}
**(failure of local cartesian closure)**
\linebreak
Unfortunately neither of the above categories is [[locally
cartesian closed category|locally
cartesian closed]] ([Cagliari-Matovani-Vitale 1995, p. 4](#CagliariMatovaniVitale95))

{#PartialLocalCartesianClosure} However, if $K$ is the category of not-necessarily-weak-Hausdorff k-spaces, and $A$ and $B$ are k-spaces that are weak Hausdorff, then the [[pullback]] ([[base change]]) [[functor]] $K_{/B} \to K_{/A}$ has a [[right adjoint]]. (see [Booth & Brown 1978a, Thm. 3.5 & 7.3](#BoothBrown78a); [May & Sigurdsson 2006, §1.3.7-§1.3.9](#MaySigurdsson06)).

There is still a lot of work on fibred exponential laws and their applications. One reason for the success and difficulties is that it is easy to give a topology on the space of closed subsets of a space $X$ by regarding this as the space of maps to the [[Sierpinski space]] (the set $\{0,1\}$ of [[truth value]]s in which $\{1\}$ is closed but not open). From this one can get an [[exponential law for spaces]] over $B$ if $B$ is [[To|$T_0$]], so that all fibres of spaces over $B$ are closed in their total space.  Note that weak Hausdorff implies [[T0|$T_0$]].
\end{remark}

\linebreak

### Relation to locally compact Hausdorff spaces
 {#RelationToLocallyCompactHausdorffSpaces}

\begin{prop}\label{LocallyCompactHausdorffSpacesAreCompactlyGeneratedWeaklyHausdorff}
  Every [[locally compact Hausdorff space]] is a k-space and is [[weakly Hausdorff topological space|weakly Hausdorff]]:

  $$
    LCHausSp
    \xhookrightarrow{\;}
    h k TopSp
  $$
\end{prop}
([Dugundji 1966, XI Thm. 9.3](#Dugundji66); [Strickland 2009, Prop. 1.7](compactly+generated+topological+space#Strickland09))

\begin{prop}
  The [[product topological space]] of a [[locally compact Hausdorff space]] with a k-space is already a k-space (i.e. without need of k-ification).
\end{prop}

(e.g. [Lewis 1978, Lem. 2.4](#Lewis78); [Piccinini 1992, Thm. B.6](#Piccinini92), [Strickland 2009, Prop. 2.6](compactly+generated+topological+space#Strickland09))

\begin{proposition}\label{kSpacesAreTheQuotientSpacesOfLocallyCompactHausdorffSpaces}
**([[k-spaces]] are the [[quotient topological space|quotient spaces]] of [[locally compact Hausdorff spaces]])**
\linebreak
  A [[topological space]] is a [[k-space]] (Def. \ref{kSpace})
iff it is a [[quotient topological space]] of a [[locally compact Hausdorff space]].
\end{proposition}
This is proven in [Dugundji 1966, XI Thm. 9.4](#Dugundji66) (also [Piccinini 92, Thm. B.4](#Piccinini92)) assuming Hausdorffness, and without that assumption in [Escardo, Lawson & Simpson 2004, Cor. 3.4 (iii)](#EscardoLawsonSimpson04). Moreover:
\begin{proposition}\label{kSpacesAreTheColimitsInTopOfCompactHausdorffSpaces}
**([[k-spaces]] are the [[Top#UniversalConstructions|colimits in Top]] of [[compact Hausdorff spaces]])**
\linebreak
  A [[topological space]] is a [[k-space]] (Def. \ref{kSpace})
iff it is a [[colimit]] as formed in [[Top]] (according to [this Prop.](Top#DescriptionOfLimitsAndColimitsInTop)) of a [[diagram]] of [[compact Hausdorff spaces]].
\end{proposition}
([Escardo, Lawson & Simpson 2004, Lem. 3.2 (v)](#EscardoLawsonSimpson04))

### Relation to compactly generated topological space

Insice $k$-spaces there is the further [[coreflective subcategory]] $D Top$ of [[Delta-generated topological spaces]]:

$$
  Top
  \underoverset
    { \underset{ k }{\longrightarrow} }
    { {\hookleftarrow} }
    { \;\;\;\;\;\;\bot\;\;\;\;\;\; }
  k Top_{Qu}
  \underoverset
    { \underset{ D }{\longrightarrow} }
    { {\hookleftarrow} }
    { \;\;\;\;\;\;\bot\;\;\;\;\;\; }
  D Top_{Qu}
  \,.
$$

Both of these [[adjoint functor|coreflections]] are [[Quillen equivalences]] with respect to the [[classical model structure on topological spaces]] and the induced [[model structure on compactly generated topological spaces]] and the [[model structure on Delta-generated topological spaces]] ([Gaucher 2009](#Gaucher07), [Haraguchi 2013](Delta-generated+topological+space#Haraguchi13)).

### Regularity
 {#Regularity}

\begin{proposition}
\label{kHausIsRegular}
The categories of

* [[compactly generated  Hausdorff spaces]]

* [[compactly generated weakly Hausdorff spaces]]

are both [[regular category|regular]].
\end{proposition}
([Cagliari-Matovani-Vitale 95, p. 3](#CagliariMatovaniVitale95)).

In particular this implies that in these categories [[pullback]] preserves [[effective epimorphisms]] (see [there](regular+epimorphism#InRegularCategoryRegularEpisAreStableUnderPullback)).

### Homotopy
 {#Homotopy}

\begin{proposition}\label{kIficationIsWeakHomotopyEquivalence}
  For every [[topological space]] $X$, the canonical [[continuous function]] from the $k$-ification (the [[adjunction counit]]) is a [[weak homotopy equivalence]], hence induces an [[isomorphism]] on all [[homotopy groups]]:

$$
  k(X)
    \underoverset
      {whe}
      {\;\varepsilon^k_X\;}
      {\longrightarrow}
  X
  \,,
  \;\;\;
  \text{i.e.}
  \;\;\;
  \pi_0\big(k(X)\big)
    \xrightarrow{\;\sim\;}
  \pi_0(X)
  \,,
  \;\;
  \text{and}
  \;\;
  \underset{
    {x \in X}
    \atop
    {n \in \mathbb{N}_+}
  }{\forall}
  \Big(
    \pi_n\big( k(X),\,x \big)
      \xrightarrow{\sim}
    \pi_n(X,\, x)
  \Big)
  \,.
$$
\end{proposition}
E.g. [Vogt 1971, Prop. 1.2 (h)](#Vogt71).
The proof is spelled out [here](Introduction+to+Homotopy+Theory#kificationComparisonIsWeakHomotopyEquivalence) at *[[Introduction to Homotopy Theory]]*.

## Related concepts

* [[model structure on compactly generated topological spaces]]

* [[Euclidean-generated topological spaces]] ([[Delta-generated topological spaces]])

## References
 {#References}

### k-Spaces and CG Hausdorff spaces
 {#ReferencesCGHausdorffSpaces}

The idea of compactly generated Hausdorff spaces first appears in print in:

* {#Gale50} [[David Gale]], Section 1 of: _Compact Sets of Functions and Function Rings_, Proc. AMS **1** (1950) pp.303-308. ([pdf](http://www.ams.org/journals/proc/1950-001-03/S0002-9939-1950-0036503-X/S0002-9939-1950-0036503-X.pdf), [doi:10.2307/2032373](https://doi.org/10.2307/2032373), [jstor:2032373](https://www.jstor.org/stable/2032373))

where it is attributed to [[Witold Hurewicz]], who introduced the concept in a lecture series given in Princeton, 1948-49, which Gale attended.[^2]

Early textbook accounts, assuming the Hausdorff condition:

* {#Kelley55} [[John Kelley]], p. 230 in: _General topology_, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [[ISBN:978-0-387-90125-1](https://www.springer.com/gp/book/9780387901251)]

* {#Dugundji66} [[James Dugundji]], Section XI.9 of: *Topology*, Allyn and Bacon 1966 ([pdf](https://www.southalabama.edu/mathstat/personal_pages/carter/Dugundji.pdf))

* {#GabrielZisman67} [[Pierre Gabriel]], [[Michel Zisman]], sections I.1.5.3 and III.2 of _[[Calculus of fractions and homotopy theory]]_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967)  ([pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/GZ.pdf))

also:

* [[Susan Niefield]], Section 9 of: *Cartesianness*, PhD thesis, Rutgers 1978 ([proquest:302920643](https://www.proquest.com/docview/302920643))

* [[Francis Borceux]], Section 7.2 of: *Categories and Structures*, Vol. 2 of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865))

Influential emphasis of the usefulness of the notion as providing a [[convenient category of topological spaces]]:

* {#Steenrod67} [[Norman Steenrod]], _A convenient category of topological spaces_, Michigan Math. J. 14 (1967) 133--152 ([euclid:mmj/1028999711](http://projecteuclid.org/euclid.mmj/1028999711))

Early discussion in the context of [[geometric realization of simplicial topological spaces]]:

* {#MacLane70} [[Saunders MacLane]], Section 4 of: _The Milgram bar construction as a tensor product of functors_,  In: F.P. Peterson  (eds.) *The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod's Sixtieth Birthday*, Lecture Notes in Mathematics *168*,  Springer 1970 ([doi:10.1007/BFb0058523](https://doi.org/10.1007/BFb0058523), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0058523.pdf))

and briefly in:

* {#May72} [[Peter May]], Section 1 of: *The geometry of iterated loop spaces*, Springer 1972 ([pdf](https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf), [doi:10.1007/BFb0067491](https://link.springer.com/book/10.1007/BFb0067491))

More history and early references, with emphasis on [[category theory|category-theoretic]] aspects:

* [[Horst Herrlich]], [[George Strecker]], Section 3.4 of: *Categorical topology -- Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971* ([pdf](https://link.springer.com/content/pdf/10.1007%2F978-94-017-0468-7_15.pdf)), pages 255-341 in: C. E. Aull, R Lowen (eds.), *Handbook of the History  of General Topology. Vol. 1* , Kluwer 1997 ([doi:10.1007/978-94-017-0468-7](https://link.springer.com/book/10.1007/978-94-017-0468-7))

The terminology "kaonic spaces", or rather the Russian version "каонные пространства" is used in

* {#Postnikov71} [[M M Postnikov]], *Введение в теорию Морса*, Наука 1971 ([web](http://libgen.is/book/index.php?md5=4BF450585846A0531FF485E34D062C0A))

* {#Postnikov82} [[M M Postnikov]], p. 34 of: *Лекции по алгебраической топологии. Основы теории гомотопий*, Наука 1982 ([web](http://libgen.is/book/index.php?md5=34A8C3C956EB80877F4E3CF5A297F514))

Discussion of k-spaces in the generality of subcategory-generated spaces, including [[Delta-generated topological spaces]]:

* {#Vogt71} [[Rainer M. Vogt]], *Convenient categories of topological spaces for homotopy theory*,  Arch. Math 22, 545–555 (1971) ([doi:10.1007/BF01222616](https://doi.org/10.1007/BF01222616))

* {#EscardoLawsonSimpson04} [[Martín Escardó]], [[Jimmie Lawson]], [[Alex Simpson]], Section 3 of: *Comparing Cartesian closed categories of (core) compactly generated spaces*, Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145 ([doi:10.1016/j.topol.2004.02.011](https://doi.org/10.1016/j.topol.2004.02.011))

* {#Gaucher07} [[Philippe Gaucher]], Section 2 of: *Homotopical interpretation of globular complex by multipointed d-space*, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 ([arXiv:0710.3553](https://arxiv.org/abs/0710.3553))

Proof that k-spaces form a [[regular category]]:

* {#CagliariMatovaniVitale95} F. Cagliari, S. Mantovani, [[Enrico Vitale]], *Regularity of the category of Kelley spaces*, Applied Categorical Structures volume 3, pages 357–361 (1995) ([doi:10.1007/BF00872904](https://link.springer.com/article/10.1007/BF00872904), [pdf](http://www.dm.unibo.it/~cagliari/articoli/Regularkelley.pdf))

Further accounts:

* [[George W. Whitehead]], Section I.4 of: *Elements of Homotopy Theory*, Springer 1978 ([doi:10.1007/978-1-4612-6318-0](https://link.springer.com/book/10.1007/978-1-4612-6318-0))

* [[Brian Day|Brian J. Day]], _Relationship of Spanier's Quasi-topological Spaces to k-Spaces_  , M. Sc. thesis University of Sydney 1968. ([pdf](http://www.math.mq.edu.au/~street/DayMasters.pdf))

* Peter Booth, Philip R. Heath, [[Renzo A. Piccinini]], *Fibre preserving maps and functional spaces*, Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977), pp. 158--167, Lecture Notes in Math., 673, Springer, Berlin, 1978.

* {#Piccinini92} [[Renzo A. Piccinini]], Appendix B in: *Lectures on Homotopy Theory*, Mathematics Studies **171**, North Holland 1992 ([ISBN:978-0-444-89238-6](https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/171/suppl/C))

* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], Section 0 of: _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))

  > (in a context of [[rational homotopy theory]])

* {#AGP02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, around note 4.3.22 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

* Samuel Smith, _The homotopy theory of function spaces: a survey_ ([arXiv:1009.0804](http://arxiv.org/abs/1009.0804))

### CG weak Hausdorff spaces
 {#ReferencesCGWeakHausdorffSpaces}

The idea of generalizing compact generation to weakly Hausdorff spaces appears in:

* {#McCord69} [[Michael C. McCord]], Section 2 of: *Classifying Spaces and Infinite Symmetric Products*, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298  ([jstor:1995173](https://www.jstor.org/stable/1995173), [pdf](https://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0251719-4/S0002-9947-1969-0251719-4.pdf))

where it is attributed to [[John C. Moore]].

Review in this generality of CG weakly Hausdorff spaces:

* {#Lewis78} [[Gaunce Lewis]], _Compactly generated spaces_ ([pdf](http://www.math.uchicago.edu/~may/MISC/GaunceApp.pdf)), appendix A of _The Stable Category and Generalized Thom Spectra_, PhD thesis Chicago, 1978

* {#FritschPiccinini90} [[Rudolf Fritsch]], [[Renzo Piccinini]], Appendix A.1 of: _Cellular structures in topology_, Cambridge University Press (1990) ([doi:10.1017/CBO9780511983948](https://doi.org/10.1017/CBO9780511983948), [pdf](https://epub.ub.uni-muenchen.de/4493/1/4493.pdf))

* {#May99} [[Peter May]], Chapter 5 of: _[[A concise course in algebraic topology]]_, University of Chicago Press 1999 ([ISBN: 9780226511832](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html), [pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

* {#Strickland09} [[Neil Strickland]], _The category of CGWH spaces_, 2009 ([pdf](http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf), [[StricklandCGHWSpaces.pdf:file]])

* {#Schwede12} [[Stefan Schwede]], Section A.2 of: _[[Symmetric spectra]]_ (2012)

* [[Charles Rezk]], *Compactly Generated Spaces*, 2018 ([pdf](https://faculty.math.illinois.edu/~rezk/cg-spaces-better.pdf), [[Rezk_CompactlyGeneratedSpaces.pdf:file]])

Brief review in preparation of the [[model structure on compactly generated topological spaces]]:

* {#Hovey99} [[Mark Hovey]], Def. 2.4.21 in: _[[Model Categories]]_, Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))

Review  with focus on compactly generated [[topological G-spaces]] in [[equivariant homotopy theory]] and specifically [[equivariant bundle]]-theory:

* [[Bernardo Uribe]], [[Wolfgang Lück]], Section 16 of: _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))

[[!include exponential law for parameterized topological spaces -- references]]

[^1]: The reason for  choosing the term "k-space" in [Gale 1950](#Gale50) seems to be lost in history. The "k" is not for "Kelley", as [Kelley 1955](#Kelley55) came later. It might have been an allusion to the German word *kompakt*.

[^2]: This is according to personal communication by [[David Gale]] to [[William Lawvere]] in 2003, forwarded by Lawvere to [[Martin Escardo]] at that time, and then kindly forwarded by Escardo to the nForum in 2021; see [there](https://nforum.ncatlab.org/discussion/8638/compactly-generated-topological-space/?Focus=94755#Comment_94755).

[[!redirects compactly generated topological space]]
[[!redirects compactly generated topological spaces]]
[[!redirects compactly generated space]]
[[!redirects compactly generated spaces]]

[[!redirects compactly generated Hausdorff space]]
[[!redirects compactly generated Hausdorff spaces]]

[[!redirects compactly generated weakly Hausdorff topological space]]
[[!redirects compactly generated weakly Hausdorff topological spaces]]
[[!redirects compactly generated weakly Hausdorff space]]
[[!redirects compactly generated weakly Hausdorff spaces]]

[[!redirects compactly generated weak Hausdorff topological space]]
[[!redirects compactly generated weak Hausdorff topological spaces]]
[[!redirects compactly generated weak Hausdorff space]]
[[!redirects compactly generated weak Hausdorff spaces]]

[[!redirects weakly Hausdorff compactly generated topological space]]
[[!redirects weakly Hausdorff compactly generated topological spaces]]

[[!redirects k-topological space]]
[[!redirects k-topological spaces]]

[[!redirects k-space]]
[[!redirects k-spaces]]

[[!redirects Kelley space]]
[[!redirects Kelley spaces]]

[[!redirects k-ification]]
[[!redirects k-ifications]]

[[!redirects kaonization]]
[[!redirects kaonizations]]
[[!redirects kaonisation]]
[[!redirects kaonisations]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Compact objects
+--{: .hide}
[[!include compact object - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

Let $\mathcal{T}$ be a [[triangulated category]] with [[coproducts]]. Then $\mathcal{T}$ is **compactly generated** if there is a [[set]] $\mathcal{G}$ of [[objects]] of $\mathcal{T}$ such that

1. Whenever $X$ is an object such that $\mathcal{T}(\Sigma^m G, X) = 0$ for all $G \in \mathcal{G}$ and $m \in \mathbb{Z}$, then $X = 0$.

1. All objects in $\mathcal{G}$ are [[compact object|compact]] i.e. for all $G \in \mathcal{G}$ and for every family $\{ X_i \mid i \in I\}$ of objects of $\mathcal{T}$
$$
\coprod_{i \in I} \mathcal{T}(G, X_i) \to \mathcal{T}\Big(G,\coprod_{i \in I} X_i\Big)
$$
is an [[isomorphism]].

=--

## Properties

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

If $\mathcal{T}$ is a triangulated category with coproducts, then a set of objects $\mathcal{G}$ satisfies the two conditions of def. \ref{MainDefinition} if and only if the smallest [[localizing subcategory]] of $\mathcal{T}$ that contains $\mathcal{G}$ is $\mathcal{T}$ itself.

=--

This is Lemma 2.2.1. of ([Schwede-Shipley](#SchwedeShipley)).

[[Brown representability theorem]] holds in compactly generated triangulated categories.

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

If $\mathcal{T}$ is a compactly generated triangulated category and $H : \mathcal{T}^\mathrm{op} \to Ab$ is a [[cohomological functor]], then $H$ is [[representable functor|representable]].

=--

## Examples

* The [[sphere spectrum]] is a compact generator for the [[stable homotopy category]].

* More generally, if $R$ is a [[ring spectrum]], then the homotopy category of [[module spectrum|module spectra]] over $R$ is compactly generated by $R$.

## Related concepts

* [[well-generated triangulated category]]

* [[compactly generated model category]]

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

## References

* {#SchwedeShipley} [[Stefan Schwede]], [[Brooke Shipley]],  _Stable model categories are categories of modules._ Topology 42 (2003), no. 1, 103–153.

* [[Amnon Neeman]],  _Triangulated categories_, Annals of Mathematics Studies, 148. Princeton University Press, 2001.

[[!redirects compactly generated triangulated categories]]


compactly supported cohomology

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

#Contents#
* table of contents
{:toc}

## Idea

In _compactly supported cohomology_ [[cocycles]] and [[coboundaries]] on some [[space]] are required to have [[compact support]]: to be non-trivial only over a [[compact topological space|compact subspace]]/[[compact object|compact subobject]] of the base.

## References

### General

* [[James Milne]], section 18 of _[[Lectures on Étale Cohomology]]_

### Compactly supported de Rham cohomology

Discussion of compactly supported [[de Rham cohomology]] includes for instance the following.

A standard textbook account is in

* [[Raoul Bott]], [[Loring Tu]], _Differential forms in algebraic topology_

A brief account is also in [section 11](#www.few.vu.nl/~vdvorst/DeRham.pdf#page=48) of

* Patrick Hafkenscheid, _De Rham Cohomology of smooth manifolds_ ([pdf](http://www.few.vu.nl/~vdvorst/DeRham.pdf))

[[!redirects cohomology with compact support]]


compactly supported distribution

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

#Contents#
* table of contents
{:toc}

## Idea

A _compactly supported distribution_ is a [[distribution]] whose [[support of a distribution]] is a [[compact topological space|compact]] [[subset]].

This implies that compactly supported distributions may be evaluated not just on [[bump functions]], but in fact on the larger space of all [[smooth functions]]. Indeed it turns out that the compactly supported distributions exhaust the [[continuous linear functionals]] on the space of smooth functions ([Hörmander 90, theorem 2.3.1](#Hoermander90)).

Therefore and since Schwartz's notation for the space $C^\infty(X)$ of all smooth functions on the given [[smooth manifold]] is $\mathcal{E}(X)$, the space of compactly supported distributions is often denoted by $\mathcal{E}'(X)$ ([Hörmander 90, p. 45](#Hoermander90)).

## Definition

### As continuous linear duals to smooth functions

+-- {: .num_defn #TVSOfCompactlySupportedFunctions}
###### Definition
**([[topological vector space]] of [[smooth functions]] on a [[Cartesian space]])**

For $n \in \mathbb{R}^n$,  write $C^\infty(\mathbb{R}^n)$ for the $\mathbb{R}$-[[vector space]] of [[smooth functions]] $\mathbb{R}^n \to \mathbb{R}$.

Then the vector space $C^\infty(\mathbb{R}^n)$ becomes a [[Fréchet vector space]] induced by the family of [[seminorms]]

$$
  \array{
    C^\infty(\mathbb{R}^n)
     &\overset{p_{K}^\alpha}{\longrightarrow}&
    [0,\infty)
    \\
    \Phi &\mapsto& \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x) \vert}
  }
  \,,
$$

indexed by $K \subset \mathbb{R}^n$ a [[compact subset]] and $\alpha \in \mathbb{N}^n$ a [[tuple]] encoding the degrees of the [[partial derivative]] $\partial^\alpha$.

Hence the seminorm $p_K^\alpha$ sends a [[smooth function]] $\Phi$ to the [[supremum]] over the points $x \in K$ of the [[absolute value]] of the [[partial derivative]] $\partial^\alpha \Phi$; and the [[open subsets]] defined thereby are the [[unions]] of translations of the [[neighbourhood base|base]] [[open balls]]

$$

  B^\circ_{K,\alpha,\epsilon}(0)
  =
  \left\{
    v \in C^\infty\mathbb{R}^n
    \;\vert\;
    p_K^\alpha(v)
    \lt
    \epsilon
  \right\}
$$

for $\epsilon \in (0,\infty)$.

We write

$$
  \mathcal{E}(\mathbb{R}^n) \in TopVect_{\mathbb{R}}
$$

for the resulting [[Fréchet space|Fréchet]] [[topological vector space]].

=--

+-- {: .num_defn #CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions}
###### Definition
**(compactly supported distibutions as continuous linear duals to bump functions)**

The space $\mathcal{E}'(\mathbb{R}^n)$ of _compactly supported distributions_ on $\mathbb{R}^n$ is the [[dual space]]

$$
  \mathcal{E}'(\mathbb{R}^n) \;\coloneqq\; \left(\mathcal{E}(\mathbb{R}^n)\right)^\ast
$$

to the [[topological vector space]] of [[smooth functions]] from def. \ref{TVSOfCompactlySupportedFunctions}.

=--

e.g. ([Klainerman 08, p. 9](#Klainerman08))

This means the following

+-- {: .num_prop #CharacterizationOfContinuityBySeminormBounds}
###### Proposition
**(characterization of continuity for compactly supported distributions)**

A [[linear function]]

$$
  u \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}
$$

is [[continuous function|continuous]] with respect to the [[topological space|topology]] of def. \ref{TVSOfCompactlySupportedFunctions}, hence is a compactly supported distribution (def. \ref{CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions})

$$
  \mathcal{E}(\mathbb{R}^n) \longrightarrow \mathbb{R}
$$

precisely if the following condition holds

* there exists a compact subset $K \subset \mathbb{R}^n$ and $k \in \mathbb{N}$  and $C \in (0,\infty)$ such that

  $$
    \label{BoundBySumOfNormOfDerivatives}
    \underset{\Phi \in C^\infty(\mathbb{R}^n)}{\forall}
    \left(
      \vert u(\Phi) \vert
      \;\leq\;
      C
      \underset{ {\vert \alpha \vert \leq k} }{\sum}
      \underset{x \in K}{sup} \vert \partial^\alpha \Phi \vert
    \right)
  $$

=--

(see also [Hörmander 90, (2.3.2) and theorem 2.3.1](#Hoermander90))

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

By [this prop.](locally+convex+topological+vector+space#AlternativeCharacterizationOfContinuityForLinearFunctionals) the continuity of $L$ is equivalent to there being an [[inhabited set|inhabited]] [[finite set]] $\{ (K_1, \alpha_1), \cdots, (K_r, \alpha_r) \}$ and $C \in (0,\infty)$ such that

$$
  \label{BoundByMaximumOfSeminormsOverFiniteSet}
  {\vert u(\Phi)\vert}
  \;\leq\;
  C
  \underset{i = 1, \cdots, n}{max}
  \left(
    \underset{x \in K_i}{sup} {\vert \partial^{\alpha_i} \Phi \vert}
  \right)
$$

We need to see that this is equivalent to a bound of the form (eq:BoundBySumOfNormOfDerivatives).

In one direction, assume that (eq:BoundByMaximumOfSeminormsOverFiniteSet) holds.

The union $K \coloneqq \underset{i = 1, \cdots, n}{\cup} K_i$ is still a compact
subset ([this prop.](compact+space#UnionsAndIntersectionOfCompactSubspaces)). Hence (eq:BoundByMaximumOfSeminormsOverFiniteSet) implies that

$$
  \begin{aligned}
    {\vert u(\Phi)\vert}
    & \leq
    C
    \underset{i = 1, \cdots, n}{max}
    \underset{x \in K}{sup}
    {\vert \partial^{\alpha_i} \Phi \vert}
  \end{aligned}
$$

and hence with $k \coloneqq \underset{i = 1, \cdots, n}{max} {\vert \alpha_i\vert}$ that

$$
  {\vert u(\Phi)\vert}
  \;\leq\;
  C
    \underset{{\vert \alpha\vert}  \leq k}{\sum}
    \underset{x \in K}{sup}
    {\vert \partial^{\alpha} \Phi \vert}
  \,,
$$

which is of form (eq:BoundBySumOfNormOfDerivatives).

Conversely, assume that a bound of the form (eq:BoundBySumOfNormOfDerivatives) holds. Then take the finite set of pairs $(K_i, \alpha_i)$ where all $K_i \coloneqq K$ and with all ${\vert\alpha_i\vert} \leq k$. With $N$ denoting the number of $n$-tuples $\alpha$ with ${\vert \alpha \vert} \leq k$ we then get a bound as in (eq:BoundByMaximumOfSeminormsOverFiniteSet) with coefficient $N C$.

=--

### As smooth linear duals to smooth functions
 {#AsSmoothLinearDuals}

Given that [[distributions]] are concerned with _[[smooth functions]]_ it is sometimes more natural to regard them not as [[continuous linear functionals]] as in def. \ref{CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions}, but as _smooth linear functionals_. Indeed this turns out to be equivalent (prop. \ref{CompactlySupportedDistributionsAreTheSmoothLinearFunctionals} below), if one considers an ambient context of suitably [[generalized smooth spaces]], namely [[diffeological spaces]] or more generally [[smooth sets]] or [[formal smooth sets]]. We will write $\mathbf{H}$ for any of these [[categories]] of [[generalized smooth spaces]].

We may canonially regard any [[smooth manifold]] such as the [[Cartesian space]] $\mathbb{R}^n$ as an object of $\mathbf{H}$. For $X \in \mathbf{H}$ any object, we write $[X,\mathbb{R}]$ for the [[mapping space]] (the [[internal hom]]). The underlying set is $C^\infty(X)$. If $X$ itself has $\mathbb{R}$-linear structure, we write

$$
  [X,\mathbb{R}]_{\mathbb{R}}
  \hookrightarrow
  [[X,\mathbb{R}], \mathbb{R}]
$$

for the [[subobject]] of $\mathbb{R}$-linear maps.

Concretely, for $U$ a [[smooth manifold]] (or just a [[Cartesian space]]), then the [[sheaf]] $[X,\mathbb{R}]$ assigns (see at _[[closed monoidal structure on presheaves]]_ for details)

$$
  [X,\mathbb{R}](U) = C^\infty(U \times X)
$$

and $[X,\mathbb{R}](U) \subset C^\infty(U \times X)$ is the subset of those functions $\Phi_{(-)}(-)$ such that for all $u \in U$ the function $\Phi_u \colon X \to \mathbb{R}$ is $\mathbb{R}$-linear. The [[global elements]] $\Gamma(-)$ of the [[mapping space]] constitute the ordinary [[hom set]]

$$
  \Gamma [X,\mathbb{R}] \simeq \mathbf{H}(X,\mathbb{R})
  \,.
$$

+-- {: .num_prop #CompactlySupportedDistributionsAreTheSmoothLinearFunctionals}
###### Proposition
**([[compactly supported distributions are the smooth linear functionals]])**

For $n \in \mathbb{N}$, there is a [[natural bijection]] between the underlying sets of compactly supported distributions on $\mathbb{R}^n$ (def. \ref{CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions}) and the $\mathbb{R}$-linear [[mapping space]] formed in the [[category]] $\mathbf{H}$ of either [[diffeological space]] or [[smooth sets]] or [[formal smooth sets]]:

$$
  \widetilde{(-)}
  \;\colon\;
  \mathcal{E}'(\mathbb{R}^n)
    \overset{\simeq}{\longrightarrow}
  \mathbf{H}([\mathbb{R}^n, \mathbb{R}], \mathbb{R})_{\mathbb{R}}
$$

given by sending $\mu \in \mathcal{E}'(\mathbb{R}^n)$ to the [[natural transformation]] which on a test space $U \in $ [[CartSp]] takes a smoothly $U$-parameterized function $\Phi_{(-)}(-) \colon U \times \mathbb{R}^n \to \mathbb{R}$ to its evaluation in $\mu$ pointwise in $U$:

$$
  \tilde \mu(\Phi_{(-)})(u)
  \;\coloneqq\;
  \langle \mu, \Phi_{u}\rangle
  \,.
$$

=--

([Moerdijk-Reyes 91, chapter II, prop. 3.6](#MoerdijkReyes91))

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

First consider this for the case that $\mathbf{H} = $ [[SmoothSet]] (which immediately subsumes the case that $\mathbf{H} =$ [[diffeological space|DiffelogicalSpace]]).

To see that $\widetilde{(-)}$ is well defined, we need to check that the function

$$
  \array{
     U
     &\overset{
     \tilde \mu(\Phi_{(-)})}{\longrightarrow}&
     \mathbb{R}
     \\
     u &\mapsto& \langle \mu, \Phi_u\rangle
  }
$$

is [[smooth function|smooth]]. But this follows immediately since $\langle \mu,-\rangle$ by definition is [[linear function|linear]] and [[continuous function|continuous]] ([Hörmander 90, theorem 2.1.3](#Hoermander90)).

To see that $\widetilde{(-)}$ is indeed a [[bijection]] for each $U$ it remains that every $\mathbb{R}$-linear smooth functional (morphisms of [[smooth sets]]) of the form

$$
  A \;\colon\; [\mathbb{R}^n,\mathbb{R}] \longrightarrow \mathbb{R}
$$

when restricted on [[global elements]] to a [[function]] of sets

$$
  A(\ast) \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}
$$

is [[continuous function|continuous]] with respect to the [[topological vector space]] structure from def. \ref{TVSOfCompactlySupportedFunctions} on the left.

Now by definition of the [[internal hom]] $A$ is actually "path-smooth" ([this def.](Frechet+space#PathSmoothFunction)) and hence the statement is implied by [this prop](Frechet+space#PathSmoothLinearFunctionsOnFrechetSpaceAreContinuous).

Finally to see that this argument generalizes to $\mathbf{H} = $ [[formal smooth set|FormalSmoothSet]] observe that the [[Weil algebra]] of every [[infinitesimally thickened point]] is a [[quotient ring]] of an algebra of smooth functions on some [[Cartesian space]] (by the [[Hadamard lemma]]). The previous argument now applies to representatives under this quotient coprojection and one checks that it is independent of the representative chosen.

=--

## Properties

### Fourier-Laplace transform

+-- {: .num_defn #FourierTransformOfCompactlySupportedDistribution}
###### Proposition/Definition

For $n \in \mathbb{N}$, let $u \in \mathcal{E}'(\mathbb{R}^n)$ be a compactly supported distribution on [[Cartesian space]] $\mathbb{R}^n$. Then its _[[Fourier transform of distributions]]_ is the function

$$
  \array{
    \mathbb{R}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R}
    \\
    \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right)
  }
$$

where on the right we have the application of $u$, regarded as a [[linear function]] $u \colon C^\infty(\mathbb{R}^n) \to \mathbb{R}$, to the [[exponential function]] applied to the canonical [[inner product]] $\langle -,-\rangle$ on $\mathbb{R}^n$.

This same formula makes sense more generally for [[complex numbers]] $\zeta \in \mathbb{C}^n$. This is then called the _[[Fourier-Laplace transform]]_ of $u$, still denoted by the same symbol:

$$
  \array{
    \mathbb{C}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R}
    \\
    \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right)
  }
$$

This is an [[entire analytic function]] on $\mathbb{C}^n$.

=--

([Hörmander 90, theorem 7.1.14](#Hoermander90))

+-- {: .num_theorem}
###### Theorem
**([[Paley-Wiener-Schwartz theorem]])**

For $n \in \mathbb{N}$ the vector space $C^\infty_c(\mathbb{R}^n)$ of [[compact support|compactly supported]] [[smooth functions]] ([[bump functions]]) on [[Euclidean space]] $\mathbb{R}^n$ is (algebraically and topologically) [[isomorphism|isomorphic]], via the [[Fourier-Laplace transform]] (prop. \ref{FourierTransformOfCompactlySupportedDistribution}), to the space of [[entire functions]] $F$ on $\mathbb{C}^n$ which satisfy the following estimate: there is a [[positive number|positive]] [[real number]] $B$ such that for every [[integer]] $N \gt 0$ there is a [[real number]] $C_N$ such that:

$$
  \underset{\xi \in \mathbb{C}^n}{\forall}
  \left(
    {\Vert  F(\xi) \Vert}
    \le C_N
    \left(
       1 + {\vert \xi\vert }
    \right)^{-N} e^{B \, |Im(\xi)|}
  \right)
  \,.
$$

More generally, the space of [[compactly supported distributions]] on $\mathbb{R}^n$ of [[order of a distribution|order]] $N$ is isomorphic via [[Fourier transform of distributions]] to those [[entire functions]] on $\mathbb{C}^n$ for which there exists positive [[real numbers]] $C$ and $B$ such that

$$
  \underset{\xi \in \mathbb{C}^n}{\forall}
  \left(
    {\Vert F(\xi)  \Vert}
      \le
    C_N (1 + {\vert \xi\vert })^{N} e^{ B \; |Im(\xi)|}
  \right)
  \,.
$$

(Notice that the [[Fourier transform of distributions|Fourier transform]] of a [[compactly supported distribution]] is guaranteed to be a [[smooth function]], by [this prop.](Fourier+transform+of+distributions#FourierTransformOfCompactlySupportedDistributions).)

=--

(e.g. [Hoermander 90, theorem 7.3.1](#Hoermander90))

+-- {: .num_prop #DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}
###### Proposition
**(decay of [[Fourier transform of distributions|Fourier transform]] of [[compactly supported functions]])**

A [[compactly supported distribution]] $u \in \mathcal{E}'(\mathbb{R}^n)$ is a [[non-singular distribution]] (def. \ref{NonSingularCompactlySupportedDistributions}), hence given by a [[compactly supported function]] $b \in C^\infty_{cp}(\mathbb{R}^n)$ via $u(f) = \int b(x) f(x) dvol(x)$, precisely if its [[Fourier transform of a distribution|Fourier transform]] $\hat u$ ([this def.](compactly+supported+distribution#FourierTransformOfCompactlySupportedDistribution)) satisfies the following decay property:

For all $N \in \mathbb{N}$ there exists $C_N \in \mathbb{R}_+$ such that for all $\xi \in \mathbb{R}^n$ we have that the [[absolute value]] ${\vert \hat v(\xi)\vert}$ of the Fourier transform at that point is bounded by

$$
  \label{DecayEstimateForFourierTransformOfNonSingularDistribution}
  {\vert \hat v(\xi)\vert}
  \;\leq\;
  C_N \left( 1 + {\vert \xi\vert} \right)^{-N}
  \,.
$$

=--

(e.g. [Hoermander 90, around (8.1.1)](#Hoermander90))

### Singular support and Wave front set
 {#SingularSupportAndWaveFrontSet}

+-- {: .num_defn #SingularSupportOfCompactlySupportedDistribution}
###### Definition
**([[singular support]] of a [[compactly supported distribution]])**

For $n \in \mathbb{N}$ and $u \in \mathcal{E}'(\mathbb{R}^n)$ a [[compactly supported distribution]], its _[[singular support]]_ is the [[subset]] of the [[Cartesian space]] $\mathbb{R}^n$ of those points which have no [[neighbourhood]] on which $u$ [[restriction of distributions|restricts]] to a [[non-singular distribution]]:

$$
  supp_{sing}(u)
  \;\coloneqq\;
  \left\{
    u \in \mathbb{R}^n
    \,\vert\,
    \not \left(
      \underset{U \underset{\text{nbhd}}{\supset} \{x\}}{\exists}
      \left(
        u\vert_U \in C^\infty_{cp}(U)
      \right)
     \right)
  \right\}
  \,.
$$

=--

By prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions} the [[singular support of a distribution]] (def. \ref{SingularSupportOfCompactlySupportedDistribution}) consists of those points around which the [[Fourier transform of distributions|Fourier transform of the distribution]] receives large high-[[frequency]] ("UV") contributions. But in fact prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions} allows to say more precisely _which_ high frequency Fourier modes make the distribution singular at a given point. These are said to be part of the _wave front_ of the distribution, and the collection of all of them is called the _[[wave front set]]_ of the distribution:

+-- {: .num_defn #WaveFrontSet}
###### Definition
**([[wavefront set]])**

For $n \in \mathbb{N}$ let $u \in \mathcal{E}'(\mathbb{R}^n)$ be a [[compactly supported distribution]].

For $b \in C^\infty_{cp}(\mathbb{R}^n)$ a [[compactly supported function|compactly supported]] [[smooth function]], write $b u \in \mathcal{E}'(\mathbb{R}^n)$ for the corresponding product ([this example](product+of+distributions#ProductOfADistributionWithANonSingularDistribution)).

For $x\in supp(b) \subset \mathbb{R}^n$, we say that a unit [[covector]] $\xi \in S((\mathbb{R}^n)^\ast)$ is _regular_ if there exists a [[neighbourhood]] $U \subset S((\mathbb{R}^n)^\ast)$ of $\xi$ in the [[unit sphere]] such that for all $c \xi' \in (\mathbb{R}^n)^\ast$ with $c \in \mathbb{R}_+$ and  $\xi' \in U \subset S((\mathbb{R}^n)^\ast)$ the decay estimate (eq:DecayEstimateForFourierTransformOfNonSingularDistribution) is valid for the [[Fourier transform of distributions|Fourier transform]] $\widehat{b u}$ of $b u$; at $c \xi'$. Otherwise $\xi$ is _non-regular_. Write

$$
  \Sigma(b u)
    \;\coloneqq\;
  \left\{
    \xi \in S((\mathbb{R}^n)^\ast)
    \;\vert\;
    \xi \, \text{non-regular}
  \right\}
$$

for the set of non-regular covectors of $b u$.

The _wave front set at $x$_ is the [[intersection]] of these sets as $b$ ranges over [[bump functions]] whose [[support]] includes $x$:

$$
  \Sigma_x(u)
  \;\coloneqq\;
  \underset{
    { b \in C^\infty_{cp}(\mathbb{R}^n) }
      \atop
    { x \in supp(b) }
  }{\cap}
  \Sigma(b u)
  \,.
$$

Finally the _[[wave front set]]_ of $u$ is the subset of the [[sphere bundle]] $S(T^\ast \mathbb{R}^n)$ which over $x \in \mathbb{R}^n$ consists of $\Sigma_x(U) \subset T^\ast_x \mathbb{R}^n$:

$$
  WF(u)
    \;\coloneqq\;
  \underset{x \in \mathbb{R}^n}{\cup}
  \Sigma_x(u)
  \;\subset\;
  S(T^\ast \mathbb{R}^n)
$$

Usually this is considered as the full [[conical set]] inside the [[cotangent bundle]] generated by the unit covectors under multiplication with [[positive number|positive]] [[real numbers]].

=--

([Hörmander 90, def. 8.1.2](#Hoermander90))

Hence the [[wave front set]] of a compactly supported distribution consists of all those [[direction of a vector|directions]] of [[wave vectors]] along which the [[Fourier transform of distributions|Fourier transform of the distribution]] is not a [[rapidly decreasing function]].

+-- {: .num_prop #EmptyWaveFrontSetCorrespondsToOrdinaryFunction}
###### Proposition
**([[empty set|empty]] [[wave front set]] corresponds to ordinary [[smooth functions]])**

The [[wave front set]] (def. \ref{WaveFrontSet}) of a [[compactly supported distribution]] is [[empty set|empty]] precisely if the distribution is [[non-singular distribution|non-singular]] (example \ref{NonSingularCompactlySupportedDistributions}).

=--

This is effectively the [[Paley-Wiener-Schwartz theorem]] ([Hörmander 90, below (8.1.1)](#Hoermander90)).

## Examples

+-- {: .num_defn #NonSingularCompactlySupportedDistributions}
###### Definition
**(non-singular compactly supported distributions)

For $n \in \mathbb{N}$, a [[compactly supported function|compactly supported]] [[smooth function]] $b \in C^\infty_{cp}(\mathbb{R}^n)$ (a [[bump function]]) induces a [[compactly supported distribution]]

$$
  \int_{\mathbb{R}^n} (-) b dvol_{\mathbb{R}^n}
  \;\colon\;
  C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}
$$

by [[integration]] of smooth functions against $b dvol$.

This construction defines a linear inclusion

$$
  C^\infty_{cp}(\mathbb{R}^n) \hookrightarrow \mathcal{E}'(\mathbb{R}^n)
  \,.
$$

The compactly supported distributions arising this way are called the _[[non-singular distributions]]_.

=--

* [[microcausal functional]]

## Related concepts

* [[compactly supported continuous function]]

* [[bump function]]

## References

### Traditional

Textbook accounts include

* {#Hoermander90} [[Lars Hörmander]], section 2.3 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

Lecture notes include

* {#Klainerman08} [[Sergiu Klainerman]], _Analysis_ 2008 ([pdf](https://web.math.princeton.edu/~seri/homepage/courses/Analysis2008.pdf))

### In terms of smooth toposes

Discussion of compactly supported distributions in terms morphisms out of [[internal homs]] in a [[smooth topos]] is in

* {#MoerdijkReyes91} [[Ieke Moerdijk]], [[Gonzalo Reyes]], around prop. 3.6 in chapter II of _[[Models for Smooth Infinitesimal Analysis]]_, Springer 1991

and specifically for the [[Cahiers topos]] of [[formal smooth sets]] in

* {#KockReyes03} [[Anders Kock]], [[Gonzalo Reyes]], _Some calculus with extensive quantities: wave equation_, Theory and Applications of Categories , Vol. 11, 2003, No. 14, pp 321-336 ([TAC](http://www.tac.mta.ca/tac/volumes/11/14/11-14abs.html))

* {#Kock11} [[Anders Kock]], _Commutative monads as a theory of distributions_ ([arxiv:1108.5952](http://arxiv.org/abs/1108.5952))

using results of

* {#FroelicherKriegl88} [[Alfred Frölicher]], [[Andreas Kriegl]], section 5 of  _Linear spaces and differentiation theory_, Wiley 1988 ([pdf](http://www.fuw.edu.pl/~kostecki/scans/froelicherkriegl1988.pdf))

and following the general conception of "[[intensive and extensive]]" in

* {#Lawvere86} [[William Lawvere]], Introduction to _[[Categories in Continuum Physics]], Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986

Generalization to non-compactly supported distributions is in

* {#KockReyes04} [[Anders Kock]], [[Gonzalo Reyes]], _Categorical distribution theory; heat equation_ ([arXiv:math/0407242](https://arxiv.org/abs/math/0407242))

and [[sheaf theory|sheaf theoretic]] discussion of distributions as morphisms of [[smooth spaces]] is in

* {#Paugam12} [[Frédéric Paugam]], section 3.2 of _Towards the mathematics of quantum field theory_, 2012 ([pdf](https://webusers.imj-prg.fr/~frederic.paugam/documents/enseignement/master-mathematical-physics.pdf))

[[!redirects compactly supported distributions]]


compactness and stable closure

# Compactness and stable closure
* table of contents
{: toc}

## Introduction

This page is dedicated to proving the equivalence of two notions of [[compact space|compactness]], one being the classical open-cover formulation, and the other a "stable closure" condition (the property that "$X \to 1$ is a [[closed map]]" is stable under [[base change|pullback]]).

## Compactness implies stable closure

One direction is a very classical and straightforward fact, proved in every textbook on general topology.

+-- {: .un_prop}
###### Proposition
If $X$ is compact, then for any space $Y$ the projection $\pi: X \times Y \to Y$ is a closed map.
=--

+-- {: .proof}
###### Proof
Let $C \subseteq X \times Y$ be a closed subset, and suppose that $y$ does not belong to $\pi(C)$. We want to find an open neighborhood of $y$ that does not intersect $\pi(C)$, or so that $X \times V$ does not intersect $C$. Consider the collection $\mathcal{C}$ of all open $U \subseteq X$ for which there exists an open $V \subseteq Y$ containing $y$, such that $U \times V$ does not intersect $C$. Since $y \notin \pi(C)$, for any $x \in X$ we have $(x,y) \notin C$, and since $C$ is closed in the product topology, there exist $V$ containing $y$ and $U$ containing $x$ such that $U \times V$ does not intersect $C$. Therefore, $\mathcal{C}$ covers $X$, so it has a finite subcover $U_i$. For each of the finitely many $i$ there is a corresponding $V_i$ such that $U_i \times V_i$ does not intersect $C$, and the intersection of the $V_i$ is a neighborhood of $y$ which does not intersect $\pi(C)$.
=--

## Stable closure implies compactness

The converse statement requires more ingenuity to prove. A preliminary observation is that the proof above was a bit convoluted because it was phrased throughout in terms of complements of closed sets; this suggests it would be convenient to reformulate the condition of being a closed map directly in terms of _open_ sets:

+-- {: .un_prop}
###### Proposition
A map $f: X \to Y$ is closed iff $\forall_f U$ is open in $Y$ for every open $U$ in $X$. Here $\forall_f U$ is defined by the adjunction condition
$$f^{-1}(B) \subseteq U \qquad iff \qquad B \subseteq \forall_f U$$
for every $B \subseteq Y$.
=--

+-- {: .proof}
###### Proof
The traditional formulation is that $\exists_f C$ is closed in $Y$ whenever $C$ is closed in $X$, which is the same as that $\exists_f \neg U = \neg \forall_f U$ is closed in $Y$ whenever $U$ is open in $X$, i.e., $\forall_f U$ is open in $Y$ whenever $U$ is open in $X$.
=--

In the case of a projection map $f = \pi: X \times Y \to Y$, this says

$$\{y \in Y: X \times \{y\} \subseteq U\}$$

is open in $Y$ whenever $U$ is open in $X \times Y$.

Next, a slight reformulation of the concept of compactness. Recall that a collection of subsets of $X$ is _directed_ if every finite subcollection has an upper bound. Then, a space $X$ is compact if every directed open cover $\Sigma$ of $X$ contains $X$.

+-- {: .un_theorem}
###### Theorem
If $\pi: X \times Y \to Y$ is a closed map for every space $Y$, then $X$ is compact.
=--

+-- {: .proof}
###### Proof
Let $\Sigma$ be a directed open cover of $X$. Define a space $Y$ as follows: the points of $Y$ are open sets of $X$ (so the underlying set of $Y$ is the topology $\mathcal{O}(X)$), and the open sets of $Y$ are upward-closed subsets $W$ of $\mathcal{O}(X)$ such that $\Sigma \cap W$ is nonempty whenever $W$ is nonempty.

+-- {: .un_lemma}
###### Claim
This is a topology.
=--

+-- {: .proof}
###### Proof of the claim
Clearly such $W$ are closed under arbitrary unions. If $W$ and $W'$ are open and $U \in \Sigma \cap W$ and $U' \in \Sigma \cap W'$, then any upper bound of $U$ and $U'$ in $\Sigma$ belongs to both $W$ and $W'$ since these are upward-closed.

Moreover, whenever $U$ belongs to $\Sigma$, the principal up-set $prin(U) = \{V \in \mathcal{O}(X): U \subseteq V$ is open in $Y$.
=--

Now consider the set $E = \{(x, U) \in X \times Y: x \in U\}$. Claim: this is open in $X \times Y$. Proof: for every $(x, U) \in E$, there exists $U' \in \Sigma$ such that $x \in U'$ (because $\Sigma$ is a cover), and then for $U'' = U \cap U'$, the set $U'' \times prin(U'')$ is an open set which contains $(x, U)$, and $U \times prin(U) \subseteq E$ because for every $(y, V) \in U'' \times prin(U'')$, we have $y \in V$.

By the open-set reformulation of the closed map condition, the set
$$\{V \in Y: X \times \{V\} \subseteq E\}$$
is open in $Y$, so this set is upward-closed and intersects $\Sigma$, so that $X \times \{V\} \subseteq E$ for some $V \in \Sigma$. But then $V$ is all of $X$! So $X \in \Sigma$ for any directed open cover $\Sigma$; therefore $X$ is compact.
=--

## Reference

The proof of the theorem above was extracted from

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

See also [this exchange](http://mathoverflow.net/questions/42186/does-compact-iff-projections-are-closed-require-some-form-of-choice) at Math Overflow, where the question was raised as to whether the [[axiom of choice]] (or possibly a weaker choice principle like the [[ultrafilter theorem]]) is required to prove the equivalence of these two notions of compactness (examination of the proofs above show it is not).

[[!redirects compactness and stable closure]]


compactness theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The **compactness theorem** is a fundamental result of _[[model theory]]_ on the existence of [[models]] of [[first-order theories]].

Historically it goes back to work of [[Gödel]] and Mal'cev in the 1930s.

[[Lindström's theorem]] uses the compactness and the [[Löwenheim-Skolem theorem]] to characterize classical first-order logic. The validity or non-validity of the compactness theorem in logical systems other than first-order logic plays under the name of _compactness property_ an important role in [[abstract model theory]].

## Theorem

Briefly, the compactness theorem states that a set of first-order formulas $\Phi$ has a model precisely iff every finite subset of $\Phi$ has a model.

### Remark

There are different proofs for the compactness theorem e.g. an entirely 'semantic' one using [[ultraproduct|ultraproducts]]. A particularly transparent one relies on the completeness theorem for first-order logic and the essential finitude of the notion of formal proof[^end]: Suppose $\Phi$ has no model, due to completeness this implies that there is a contradiction $\varphi$ with $\Phi\vdash\varphi$ but such a deduction of $\varphi$ can involve only a finite subset $\Phi_0\subset \Phi$ hence $\Phi_0\vdash\varphi$ and $\Phi_0$ would have no model as well.

Despite the intuitive appeal, this approach obliges one to establish the completeness theorem, which depends on fiddly details of formal proofs and deductive systems to make rigorous.

### Compactness for propositional logic

[^end]: In the German literature the compactness theorem is therefore also called 'Endlichkeitssatz'.

The motivation for the terminology stems from the observation that the compactness theorem literally expresses the compactness of a suitable topology on first-order structures.

## Illustrations

As the compactness theorem is arguably the most fundamental result of model theory, there are too many examples of its use for us to do it much justice here. But perhaps a few examples here will help illustrate some typical uses.

### Total orders {#totalorder}

+-- {: .num_prop #total}
###### Proposition
Every set $X$ can be totally ordered.
=--

Of course this trivially follows from the stronger result that every set can be [[well-ordered set|well-ordered]], if we permit the [[axiom of choice]]. But part of our point here is that we don't need the full strength of the axiom of choice: the result can be derived from the weaker "choice principle" called the [[ultrafilter theorem]], on which the compactness theorem depends.

+-- {: .proof}
###### Proof
Introduce a [[language]] (i.e., [[signature (in logic)|signature]]) with a constant symbol $c_p$ for each $p \in X$ and a binary relation symbol $L$, and for a [[theory]] take axioms $\neg (c_p = c_q)$ whenever $p \neq q$ in $X$ and the usual axioms to make $L$ a [[total order]] (reflexivity, transitivity, antisymmetry, and connectedness). This theory is finitely satisfiable: any finite subset $\Sigma$ of the axioms has a model, by interpreting each $c_x$ occurring in an inequality axiom belonging to $\Sigma$ simply as $x$, and totally ordering those finitely many $x$ in some way to interpret $L$; the remaining $c_x$ can be interpreted as the bottom element of that total order without any harm. By the compactness theorem, this theory has a model $Y$, and the restriction of the total order on $Y$ to the set of all constants $c_x$ in $Y$ is used to totally order all the $x \in X$.
=--

With slightly more effort, a similar argument can be given to show that every [[partial order]] can be extended to a total order. We just proved the case where the partial order is discrete.

#### Choice-like consequences

Again, we assume the ultrafilter principle but do not assume the [[axiom of choice]].

+-- {: .num_cor #surj}
###### Corollary
If $p: F \to X$ is [[surjection|surjective]] and every [[fiber]] is finite, then $p$ has a section $s$.
=--

+-- {: .proof}
###### Proof
Use Proposition \ref{total} to totally order $F$. Each fiber inherits a total order; being finite, it is [[well-ordered set|well-order]]. Then choose $s(x)$ for $x \in X$ to be the least element in the fiber $F_x$ (which is nonempty because $p$ is surjective).
=--

+-- {: .num_prop #countable}
###### Proposition
A countable disjoint union of finite sets is again countable.
=--

+--{: .proof}
###### Proof
Let $F_n$ be the countable family and let $F$ be the union; again we have a fibering $F \to \mathbb{N}$ with fiber $F_n$ over $n$. Using Proposition \ref{total}, totally order $F$; each $F_n$ inherits an order from $F$, and since total orders on finite sets are well-orders, we have chosen a countable family of well-orders. Now put the [[lexicographic order]] on the set of pairs $F' = \{(n, x): n \in \mathbb{N}, x \in F_n\}$. This is a well-ordering, and clearly the projection $F' \to F$ is a bijection, so we obtain a well-ordering of $F$ by transport of structure (differing, probably, from the original total ordering of $F$ which has done its job but is of no further use). Then we can define a partial map $\phi: \mathbb{N} \to F$ by induction: $\phi(0)$ is the bottom of $F$, and $\phi(n+1)$ is (if it exists) the bottom of the complement of $\phi([0, n])$. This defines a bijection from an initial segment of $\mathbb{N}$ onto $F$, which is what we wanted.
=--

+-- {: .num_cor #wkl}
###### Corollary
**(Weak König's lemma)**
Every infinite [[tree]] for which there is finite branching at each node has an infinite branch.
=--

+-- {: .proof}
###### Proof
A tree is by definition the [[category of elements]] $e = (n, x \in F(n))$ of a [[presheaf]] $F: \mathbb{N}^{op} \to Set$ where $\mathbb{N}$ has its usual order and $F(0)$ is terminal. A *descendant* of an element $e = (n, x \in F(n))$ is an element that maps to $e$, i.e., an element $(m, y \in F(m))$ with $m \geq n$ and $F(m \geq n)(y) = x$. A *child* of $e$ is such a descendant with $m = n+1$.

By the branching hypothesis, the set $E_n$ of elements $e = (n, x)$ is finite.
By Proposition \ref{countable}, the set of nodes or elements $E = \sum_n E_n$ is countable, i.e., is enumerated by some bijection $\mathbb{N} \to E$. Then an infinite branch (a [[global section]] $s$ of $F$) is given by recursion as follows: $s(0)$ is the root (the unique element of $E_0$). The root has infinitely many descendants since the tree is infinite. Given that $s(n) \in F(n)$ has infinitely many descendants, let $s(n+1)$ be the first of its children in the enumeration that also has infinitely many descendants (such children exist since $s(n)$ has only finitely many children by the branching hypothesis).
=--

Weak König's lemma lends its name to one the so-called Big Five systems of [[reverse mathematics]], namely $WKL_0$, and (over a certain weak fragment of second-order arithmetic) is equivalent to many results in classical elementary analysis.

+-- {: .num_remark #weak}
###### Remark
Actually, the WKL as stated above is stronger than another result that goes by that name, namely that an infinite subtree of the infinite binary tree $2^{\lt \omega}$ (consisting of finite 0-1 sequences or words, where the root is the empty word and the parent of a nonempty word is obtained by leaving off the last binary digit) has an infinite branch. This is provable in ZF without choice or other choice-free forms of set theory; the key point is that $2^{\lt \omega}$, and subtrees thereof, already have enumerations ready to hand; we don't need an argument like Proposition \ref{countable} to construct one. From there, the recursive construction of a branch in the proof of Corollary \ref{wkl} can be performed without making choices.
=--

## Related concepts

* [[Löwenheim-Skolem theorem]]

* [[Skolem's paradox]]

* [[Lindström's theorem]]

* [[Deligne completeness theorem]]

* [[Stone duality]]

## References

* Wikipedia, _[compactness theorem](https://en.wikipedia.org/wiki/Compactness_theorem)_

[[!redirects compactness theorems]]