continuous functor

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## Definition

In ordinary ([[Set]]-based) [[category theory]], a [[functor]] $F \colon  C \to D$ is **continuous** if it [[preserved limit|preserves]] all [[small diagram|small]] [[limits]] that exist in its [[domain]] $C$, i.e. given any [[small category]] [[diagram]] $A \colon E \to C$ in $C$ for which $\lim A$ exists, with universal [[cone]] $\lim A \to A$, application of $F$ to that cone results in a universal cone $F(\lim A) \to F \circ A$, thus making $F(\lim A)$ the limit of $F \circ A$.

Put slightly differently, the limit $\lim (F \circ A)$ exists and the natural [[map]] induced by the [[universal property]] of the limit

$$
  F(\lim A) \longrightarrow \lim (F\circ A)
$$

is an [[isomorphism]]. In many and perhaps most cases, one refers to continuous functors $F \colon C \to D$ in cases where one already knows $C$ and $D$ are [[complete categories]], i.e., where $\lim A$ and $\lim F \circ A$ exist as a matter of course.

In such cases, all limits can be obtained from (small) [[products]] and binary [[equalizers]] (see [here](limit#ConstructionFromProductsAndEqualizers)), and so it follows that a functor from a [[complete category]] is continuous if and only if it [[preserved limoit|preserves]] all products and all binary equalizers.

## Examples

\begin{example}
The archetypical example is the [[Set]]-valued [[hom-functor]]: its continuity in both arguments (cf. *[[hom-functors preserve limits]]*) is indeed equivalent to the very definition of [[limit]]: for
$F \colon D^{op} \to C$ a [[diagram]] and $c \in C$, the [[covariant functor|covariant]] [[hom-functor]] $Hom_C(c,-) : C \to Set$ satisfies by definition of [[limit]]
$$
  Hom_C(c, \lim F)
  \simeq
  \lim Hom(c,F(-))
  \,.
$$
Likewise for [[contravariant functor|contravariant]] representable functors on $C$, i.e., functors of the form $C(-, c): C^{op} \to Set$, which take limits in $C^{op}$ to limits in $Set$, or [[colimits]] in $C$ to limits in $Set$.
\end{example}

\begin{example}
Every [[right adjoint functor]] is continuous, since [[right adjoints preserve limits]].
\end{example}

## In enriched category theory

One can also formulate limit preservation in terms of [[representable functors]]. This is especially appropriate in [[enriched category]] theory, where ordinary limits in terms of cones may no longer suffice and one should use [[weighted limits]] instead. Thus, working over a [[base of enrichment]] $V$, assumed to be a [[Bénabou cosmos]] for convenience, suppose given a ([[enriched functor|$V$-enriched]]) functor $F \colon C \to D$, a small weight $W\colon J \to V$, and a diagram $A \colon J \to C$ over that weight. Assume the [[weighted limit]] $\{W, A\}$ exists in $C$, i.e., suppose we have a representing object for the $V$-functor $c \mapsto V^{J}(W, C(c, A(-)))$, giving an isomorphism

$$C(c, \{W, A\}) \cong V^{J}(W, C(c, A(-)))$$

natural in $c$. In this situation, playing the role analogous to a universal cone is a universal $V$-[[natural transformation]]

$$W \to C(\{W, A\}, A(-))$$

and we say $F \colon C \to D$ *preserves* the weighted limit if the composite

$$W \to C(\{W, A\}, A(-)) \to D(F\{W, A\}, (F \circ A)(-))$$

induces, à la [[Yoneda lemma|Yoneda]], a $V$-natural isomorphism

$$D(d, F\{W, A\}) \cong V^J(W, D(d, (F \circ A)(-))).$$

Then we say that $F \colon C \to D$ is *$V$-continuous* if it preserves all weighted limits that exist in $C$.

## Relation to other concepts

* The [[adjoint functor theorem]] states that any continuous functor between [[complete category|complete]] categories has a [[adjoint functor|left adjoint]] if it satisfies a certain 'small solution set' criterion.

* If $C$ has [[finite  limits]], then functors commuting
with these _finite_ limits are precisely what are
called left [[exact functors]], or "finitely continuous" functors.

## Warnings

1. It is not enough to demand that there exists an abstract isomorphism $F(\lim A) \cong \lim (F\circ A)$.

2. Topologists sometimes use "continuous functor" to mean a [[Top]]-[[enriched functor]], since a functor between topologically [[enriched categories]] is enriched iff its actions on hom-spaces are [[continuous functions]].

3. Sheaf-theorists sometimes say "continuous functor" for a [[cover-preserving functor]] between [[sites]], with the intuition being that it generalizes the [[inverse image]] induced by a [[continuous function]] of topological spaces.

4. H. Bass in his treatment of [[K-theory]] uses the older term 'right continuous functor' for the dual notion of [[cocontinuous functor]] in a version which is [[additive functor|additive]]. If the domain of an additive functor which commutes with direct sums is a [[cocomplete category]], then the functor automatically has [[adjoint functor|right adjoint]]. Following this fact, some people in ring theory and noncommutative geometry use the simple term 'continuous functor' for a functor with a right adjoint (even if the domain [[abelian category]] is not cocomplete). In general, of course, this is just a bit more than *co*continuous in the standard sense.

## Related concepts

* [[cocontinuous functor]]

* [[complete category]]

* [[adjoint functor theorem]]

[[!include properties of functors -- contents]]

## References

* [[Peter Freyd]], [[Max Kelly]], *Categories of continuous functors*, J. Pure. Appl. Algebra **2** (1972) 169-191 [doi:10.1016/0022-4049(72)90001-1]

* [[Max Kelly]], _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

[[!redirects continuous functors]]


continuous homomorphisms of Lie groups are smooth

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## Statement

\begin{proposition}
Let $G, H \,\in\, Grp(SmthMfd) \xrightarrow{\;Grp(undrl)\;} Grp(TopSp)$ be [[Lie groups]] ([[finite number|finite]] [[dimension of a manifold|dimensional]]) with [[underlying]] [[topological groups]] denoted by the same symbol.

Then every [[continuous function|continuous]] [[group homomorphism]] $G \xrightarrow{\;} H$ is [[smooth function|smooth]].

In other words, the [[function]] of [[hom-sets]]

$$
  Grp(SmthMfd)
  (G,\,H)
  \xrightarrow{undrl_1}
  Grp(TopSp)
  (G,\,H)
$$

is a [[bijection]].
\end{proposition}

This follows by applying [[Cartan's closed subgroup theorem]] to the [[graph of a function|graph]] of the homomorphism.

(e.g. [Wang 2013, Cor. 1.7](#Wang13), [Hall 2015, Cor. 3.50](#(Hall15))

## Related concepts

* [[Cartan's closed subgroup theorem]]

* [[smooth Oka principle]]

## References

Textbook accounts:

* {#Hall15} [[Brian C. Hall]], Cor. 3.50 in: *Lie Groups, Lie Algebras, and Representations*, Springer 2015 ([doi:10.1007/978-3-319-13467-3](https://doi.org/10.1007/978-3-319-13467-3))

Lecture notes:

* {#Wang13} [[Zuoqin Wang]], Cor. 1.7 in: *Cartan's closed subgroup theorem*, Lecture 11 in: [李群引论](http://staff.ustc.edu.cn/~wangzuoq/Courses/13F-Lie/Lie.html), 2013 ([pdf](http://staff.ustc.edu.cn/~wangzuoq/Courses/13F-Lie/Notes/Lec%2011.pdf), [[Wang_CartanClosedSubgroupTheorem.pdf:file]])


continuous images of compact spaces are compact

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## Idea

The [[image]] under a [[continuous function]] of a [[compact topological space]] is itself compact (cor. \ref{ContinuousImageOfACompactSpaceIsCompact} below.)

This is a generalization of the _[[extreme value theorem]]_ in [[analysis]].

## Statement

In fact the following more general statement holds

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

Let $f \colon (X,\tau_X) \longrightarrow (Y,\tau_Y)$ be a [[continuous function]] between [[topological spaces]] such that

1. $(X,\tau_X)$ is a [[compact topological space]];

1. $f \colon X \to Y$ is a [[surjective function]].

Then also $(Y,\tau_Y)$ is [[compact topological space|compact]].

=--

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

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

By the continuity of $f$ the [[pre-images]] $f^{-1}(U_i)$ form an [[open cover]] $\{f^{-1}(U_i) \subset X\}_{i \in I}$ of $X$. Hence by compactness of $X$, there exists a [[finite set|finite]] [[subset]] $J \subset I$ such that
$\{f^{-1}(U_i) \subset X\}_{i \in J \subset I}$ is still an open cover of $X$. Finally, by surjectivity of $f$ it follows that

$$
  \begin{aligned}
    Y
      & = f(X)
    \\
      & = f\left( \underset{i \in J}{\cup} f^{-1}(U_i) \right)
    \\
      & = \underset{i \in J}{\cup} U_i
  \end{aligned}
$$

where we used that [[images of unions are unions of images]].

This means that also $\{U_i \subset Y\}_{i \in J \subset I}$ is still an open cover of $Y$, and in particular a finite subcover of the original cover.

=--

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

If $f \colon X \longrightarrow Y$ is a [[continuous function]] out of a [[compact topological space]] $X$ which is not necessarily [[surjective function|surjective]], then we may consider its [[image]] factorization

$$
  f \;\colon\; X \longrightarrow im(f) \hookrightarrow Y
$$

with $im(f)$ regarded as a [[topological subspace]] of $Y$. Now by construction $X \to \im(f)$ is surjective, and so lemma \ref{ContinuousSurjectionOutOfCompactSpaceHasCompactCodomain} implies that $im(f)$ is compact.

=--

## Related statements

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

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

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

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

[[!redirects continuous image of a compact space is compact]]

[[!redirects continuous surjections out of compact spaces have compact codomain]]


continuous logic

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## Idea

Continuous logic is a logic whose truth values can take continuous values in $[0,1]$. The main variant used in model theory is motivated by the model theory of Banach spaces and similar structures. The language has connectives for each continuous function $c: [0,1]^n \to [0,1]$ and the quantifiers are interpreted as infimum and supremum. The models of this logic are bounded complete metric structures equipped with uniformly continuous maps and $[0,1]$-valued predicates.

As well as satisfying a form of [[completeness theorem]],

>...continuous first-order logic satisfies suitably phrased forms of the compactness theorem, the Löwenheim-Skolem  theorems, the diagram arguments, Craig's interpolation theorem, Beth's definability theorem, characterizations of quantifier elimination and model completeness, the existence of saturated and homogeneous  models results, the omitting types theorem, fundamental results of stability theory, and nearly all other results of elementary model theory. ([Yaacov & Pedersen 10](#YaacovPed10))

## In terms of enriched category theory

Building on the proposal of [[Lawvere]] to understand a form of [metric space](metric+space#LawvereMetricSpace) as a category [[enriched category theory|enriched]] in the [[monoidal category|monoidal]] [[poset]] $([0, \infty], \geq)$, there have been attempts to consider continuous logic as a similarly enriched logic. In ([Albert & Hart](#AlbertHart)), the authors develop a parallel for [[conceptual completeness]] via the notion of a *metric* [[pretopos]].

[[Simon Cho]] argues that the object of truth values of continuous logic, $[0,1]$, should be seen as a "continuous subobject classifier" in a similar sense to the [[subobject classifier]] of topos theory where subobjects are classified via [[pullbacks]] ([Cho 19](#Cho19)).

## References

Continuous logic is introduced in

* C. W. Henson, J. Iovino, _Ultraproducts in analysis_ in: Analysis and Logic, London Math. Soc. Lecture Note Series __262__, Cambridge University Press 2002.

motivated by

* Chen-Chung Chang, Jerome H. Keisler, _Continuous Model Theory_,
Annals of Mathematics Studies __58__, 1966.

A recent version is in

* [[Itaï Ben Yaacov]], _Uncountable dense categoricity in cats_, J. Symb. Logic __70__, 829--860, 2005
* [[Itaï Ben Yaacov]], _Continuous first-order logic and logical stability_, [pdf](http://math.univ-lyon1.fr/~begnac/articles/cfo.pdf)
* {#YaacovPed10} [[Itaï Ben Yaacov]], Arthur Pedersen, _A proof of completeness for continuous first order logic_, Journal of Symbolic Logic 75, 168–190, 2010, ([arXiv:0903.4051](https://arxiv.org/abs/0903.4051)).
* [[Itaï Ben Yaacov]], Alex Usvyatsov. _Logic of metric spaces and Hausdorff CATs_
* [[Itaï Ben Yaacov]], Alexander Berenstein, Ward C. Henson, Alexander Usvyatsov,
_Model Theory for metric structures_, in Model Theory with Applications to Algebra and Analysis, Volume 2, Cambridge University Press, 2008, [pdf](http://matematicas.uniandes.edu.co/~aberenst/mtfms.pdf)
* Alexander Berenstein, [[Andrés Villaveces]], _Hilbert spaces with random predicates_, [pdf](http://matematicas.uniandes.edu.co/~aberenst/amalg10.pdf)
* [[Jerome Keisler]], _Model Theory for Real-valued Structures_, ([arXiv:2005.11851](https://arxiv.org/abs/2005.11851))

A discussion of [[conceptual completeness]] in the setting of continuous logic is found in

* {#AlbertHart} Jean-Martin Albert, Bradd Hart. _Metric logical categories and conceptual completeness for first order continuous logic_, ([arXiv:1607.03068](https://arxiv.org/abs/1607.03068))

A treatment of metric space semantics for continuous logic as a variety of [[enriched categories]] is given in

* {#Cho19} [[Simon Cho]], _Categorical semantics of metric spaces and continuous logic_, ([arXiv:1901.09077](https://arxiv.org/abs/1901.09077))

See also his thesis

* [[Simon Cho]], _Continuity in enriched categories and metric model theory_, ([thesis](http://math.lsa.umich.edu/~simoncho/thesis.pdf))

For [[syntax-semantics duality]] in the case of [[infinitary logic|infinitary]] continuous logic, see

* Ruiyuan Chen, _Representing Polish groupoids via metric structures_, ([arXiv:1908.03268](https://arxiv.org/abs/1908.03268))

An older and rather _different system_ also called continuous logic of a Russian school is surveyed

* Vitaly I. Levin, _Basic concepts of continuous logic_, Studies in logic grammar and rethoric, 11 (24) 2007, [pdf](http://math.univ-lyon1.fr/~begnac/articles/cfo.pdf)

There is a version of [[abstract elementary classes]] in the setting of continuous logic, [[metric abstract elementary class]]es.

An analysis which avoids enriched logic is

* Daniel Figueroa, [[Benno van den Berg]], _A topos for continuous logic_ ([arXiv:2107.10543](https://arxiv.org/abs/2107.10543))


continuous map

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# Continuous maps
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## Idea

There are multiple different concepts of continuous maps, even in classical mathematics, depending upon which foundation one is using.

### In traditional topology

A [[function]] $f \colon X \to Y$ is called _continuous_ if its values $f(x)$ do not "jump" with variation of its argument $x$, unless $x$ itself "jumps".  Roughly speaking, if $x_1 \approx x_2$, then $f(x_1) \approx f(x_2)$.  (This can be made into a precise definition in [[nonstandard analysis]] if care is taken about the domains of these variables.)

In order to make this precise (in standard analysis) one needs some concept of [[neighbourhoods]] of elements of $X$ and $Y$.

For instance if $X$ and $Y$ carry structure of [[metric spaces]], then one may say that $f$ is continuous if for every point $x \in X$ and for every small [[open ball]] around its image $f(x)$ in $Y$, there exists a sufficiently small open ball around $x \in X$  which is still mapped by $f$ into that target open ball.  This definition turns out to have more elegant formulation that needs to mention neither the points of $x$ nor the radii of open balls around points: the [[metric]] induces a concept of [[open subsets]] and $f$ is continuous precisely if [[preimages]] under $f$ of [[open subsets]] in $Y$ are still open subsets in $X$.

This then is the general definition of continuity of a function $f$ between [[topological spaces]]:

A function between [[topological spaces]] is _continuous_ precisely if its [[preimages]] of [[open subsets]] are again open subsets.

Continuous maps are the [[homomorphisms]] between [[topological spaces]].
In other words, the collection of [[topological spaces]] forms a [[category]], often denoted _[[Top]]_, whose [[morphisms]] are the continuous functions.

Further generalization of the concept of continuity exists, for instance to
_[[locales]]_ (and then to [[toposes]]) or to _[[convergence spaces]]_.  (See also at _[[continuous space]]_.)

### In cohesive homotopy type theory

There is also a concept of continuous function in [[cohesive homotopy type theory]], a two-tiered type theory with two types (usual types) and (cohesive types), where the continuous functions are precisely the functions between cohesive types, and the not-necessarily continuous functions are the functions between usual types.

## Definitions

### For the set of real numbers

Given a bounded or unbounded [[open interval|open]] or [[closed interval]] in the [[Dedekind real numbers]] $I$ with [[injection]] $i:I \to \mathbb{R}$ A function $f:I \to \mathbb{R}$ is continuous if the [[graph of a function|graph]] of $f$ defined by the [[function]] $r_f:I \to \mathbb{R}^2$, $r_f(x) \coloneqq (i(x), f(x))$, has an [[image]] whose [[shape]] is [[contractible]]

$$\esh \mathrm{im}(r_f) \simeq \mathbb{1}$$

This definition is the formal definition that most closely adheres to the intuitive idea of a continuous function as introduced in an introductory college algebra class or textbook: as a function which could be drawn on a sheet of paper without picking the writing utensil up.

### The epsilontic definition for metric spaces
 {#EpsilonticDefinition}

We state the definition of continuity in terms of [[epsilontic analysis]], definition \ref{EpsilonDeltaDefinitionOfContinuity} below. First recall the relevant concepts:

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

A _[[metric space]]_ is

1. a [[set]] $X$ (the "underlying set");

1. a [[function]] $d \;\colon\; X \times X \to [0,\infty)$ (the "distance function") from the [[Cartesian product]] of the set with itself to the [[nonnegative number|non-negative]] [[real numbers]]

such that for all $x,y,z \in X$:

1. $d(x,y) = 0 \;\Leftrightarrow\; x = y$

1. (symmetry) $d(x,y) = d(y,x)$

1. ([[triangle inequality]]) $d(x,y)+ d(y,z) \geq d(x,z)$.

=--

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

Every [[normed vector space]] $(V, {\Vert -\Vert})$
becomes a [[metric space]] according to def. \ref{MetricSpace} by setting

$$
  d(x,y) \coloneqq {\Vert x-y\Vert}
  \,.
$$

=--

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

Let $(X,d)$, be a [[metric space]]. Then for every element $x \in X$ and every  $\epsilon \in \mathbb{R}_+$ a [[positive number|positive]] [[real number]], write

$$
  B^\circ_x(\epsilon)
    \;\coloneqq\;
  \left\{
    y \in X \;\vert\; d(x,y) \lt \epsilon
  \right\}
$$

for the [[open ball]] of [[radius]] $\epsilon$ around $x$.

=--

+-- {: .num_defn #EpsilonDeltaDefinitionOfContinuity}
###### Definition
**(epsilontic definition of continuity)**

For $(X,d_X)$ and $(Y,d_Y)$ two [[metric spaces]] (def. \ref{MetricSpace}), then a [[function]]

$$
  f \;\colon\; X \longrightarrow Y
$$

is said to be _continuous at a point $x \in X$_ if for every $\epsilon \gt 0$ there exists $\delta\gt 0$ such that

$$
  d_X(x,y) \lt \delta \;\Rightarrow\; d_Y(f(x), f(y)) \lt \epsilon
$$

or equivalently such that

$$
  f(B_x^\circ(\delta)) \subset B^\circ_{f(x)}(\epsilon)
$$

where $B^\circ$ denotes the [[open ball]] (definition \ref{OpenBalls}).

The function $f$ is called just _continuous_ if it is continuous at every point $x \in X$.

=--

This definition is equivalent to a more abstract one, which does not explicitly refer to points or radii anymore:

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

Let $(X,d)$ be a [[metric space]] (def. \ref{MetricSpace}). Say that

1. A _[[neighbourhood]]_ of a point $x \in X$ is a [[subset]] $x \in U \subset X$ which contains some [[open ball]] $B_x^\circ(\epsilon)$ around $x$ (def. \ref{OpenBalls}).

1. An _[[open subset]]_ of $X$ is a [[subset]] $U \subset X$ such that for every for $x \in U$ it also contains a [[neighbourhood]] of $x$.

=--

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

The collection of open subsets in def. \ref{OpenSubsetsOfAMetricSpace} constitutes a _[[topological space|topology]]_ on the set $X$, making it a _[[topological space]]_. This is called the _[[metric topology]]_. Stated more concisely: the [[open balls]] in a metric space constitute the [[basis of a topology]] for the [[metric topology]].

=--

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

A [[function]] $f \colon X \to Y$ between [[metric spaces]] (def. \ref{MetricSpace}) is continuous in the [[epsilontic analysis|epsilontic]] sense of def. \ref{EpsilonDeltaDefinitionOfContinuity} precisely if it has the property that its [[pre-images]] of [[open subsets]] of $Y$ (in the sense of def. \ref{OpenSubsetsOfAMetricSpace}) are open subsets of $X$.

=--

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

First assume that $f$ is continuous in the epsilontic sense. Then for $O_Y \subset Y$ any [[open subset]] and $x \in f^{-1(O_Y)}$ any point in the pre-image, we need to show that there exists a [[neighbourhood]] of $x$ in $U$. But by assumption there exists an [[open ball]] $B_x^\circ(\epsilon)$ with $f(B_X^\circ(\epsilon)) \subset O_Y$. Since this is true for all $x$, by definition this means that $f^{-1}(O_Y)$ is open in $X$.

Conversely, assume that $f^{-1}$ takes open subsets to open subsets. Then for every $x \in X$ and $B_{f(x)}^\circ(\epsilon)$ an [[open ball]] around its image, we need to produce an open ball $B_x^\circ(\delta)$ in its pre-image. But by assumption $f^{-1}(B_{f(x)}^\circ(\epsilon))$ contains a [[neighbourhood]] of $x$ which by definition means that it contains such an open ball around $x$.

=--

### For topological spaces

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

A [[function]] $f \;\colon\; X\to Y$ between [[topological spaces]] is a __continuous map__ (or is said to be *continuous*) if for every [[open subset]] $U \subset Y$, the [[preimage]] $f^{-1}(U)$ is an open subset $X$.

=--

In [[nonstandard analysis]], this is equivalent to

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

A [[function]] $f \;\colon\; X\to Y$ between [[topological spaces]] is a __continuous map__ (or is said to be *continuous*) if for every [[standard point]] $x_1$ and every [[hyperpoint]] $x_2$, if $x_1$ and $x_2$ are [[adequality|adequal]] (infinitely close, or in other words if $x_2$ is in the [[halo]] of $x_1$), then $f(x_1)$ and $\multiscripts{^*}f{}(x_2)$ are adequal (where $\multiscripts{^*}f{}$ is the [[nonstandard extension]] of $f$).  Equivalently, $f$ is continuous iff $\multiscripts{^*}f{}$ is [[microcontinuous function|microcontinuous]].

=--

### Further variants

A function $f$ between [[convergence spaces]] is __continuous__ if for any [[filter]] $F$ such that $F \to x$, it follows that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{F(A) \;|\; A \in F\}$.

A __continuous map__ between [[locales]] is simply a [[frame]] [[homomorphism]] in the opposite direction.  Equivalently (via the [[adjoint functor theorem]]), it may be defined as a homomorphism of [[inflattices]] whose [[left adjoint]] preserves finitary [[meets]] (and hence is a frame homomorphism).

## Properties

Since continuity is defined in terms of *preservation of property* (namely preserving "openness" under preimages), it is natural to ask what other properties they preserve.
Also, when a property is not always preserved it is useful to label those maps which do preserve it for closer study.

### Properties preserved

\begin{prop}\label{PropertiesPreservedByAContinuousFunction}

Under a [[continuous function]]:

1. the [[preimage]] of an [[open subset]] is open;

1. the [[preimage]] of a [[closed subset]] is closed;

1. the [[image]] of a [[connected space|connected subset]] is again connected;

1. theimage of a [[compact space|compact subset]] is again compact (see at _[[continuous images of compact spaces are compact]]_).

\end{prop}

### Special maps

1. The preimage of a compact set need not be compact; a continuous map for which this is true is known as a __[[proper map]]__.

2. The image of an open set need not be open; a continuous map for which this is true is said to be an __[[open map]]__.  (Technically, an open map is any [[function]] with just this property.)

3. The image of an closed set need not be closed; a continuous map for which this is true is said to be an __[[closed map]]__.  (Technically, a closed map is any function with just this property.)

4. A continuous map of topological spaces which is invertible as a function of sets is a __[[homeomorphism]]__ if the [[inverse function]] is a continuous map as well.

### Special cases in specific contexts

Although these don't make sense for arbitrary topological spaces (convergence spaces, locales, etc), they are special kinds of continuous maps in contexts such as [[metric spaces]]:

* [[uniformly continuous maps]],
* [[Lipschitz maps]],
* [[short maps]],
* [[differentiable maps]],
* [[smooth maps]].

## In constructive mathematics

Various notions of continuous function are used in [[constructive mathematics]].  A function $f$ (say [[real number|real]]-valued and defined on a real [[interval]]) is:

* _pointwise-continuous_ if it [[pointwise continuous function|pointwise continuous]] in the usual [[epsilon-delta]] (or equivalently [[open-subset]]) sense;
* _uniformly continuous_ if it [[uniformly continuous map|uniformly continuous]] in the usual epsilon-delta (or equivalently [[entourage]]-theoretic) sense;
* _Bishop-continuous_ or _locally uniformly continuous_ if it is pointwise continuous and furthermore, the restriction to any closed and bounded interval is uniformly continuous;
* _Bridges-continuous_ if ... (this one\'s kind of complicated).

In [[classical mathematics]], these are all equivalent when the domain is itself a closed and bounded interval, and all of them except for uniform continuity are equivalent in general.  The same equivalences hold in [[intuitionistic mathematics]], thanks to the [[fan theorem]].  But no two of these are equivalent in [[Russian constructivism]].

In fact, assuming that $\mathbb{R}$ is defined as the set of located [[Dedekind cuts]], there is the following negative result by [[Frank Waaldijk]] ([Waaldijk2003](#Waaldijk2003)): Without the [[fan theorem]], there is no notion of continuity for set-theoretic functions in [[constructive mathematics]], spelled "kontinuity" in the following, such that all of the following desiderata are met:

* A function $[0,1] \to \mathbb{R}$ is kontinuous if and only if it is [[uniformly continuous]] in the usual sense.
* The composition of kontinuous functions is kontinuous.
* The function $\mathbb{R}^+ \to \mathbb{R}, x \mapsto 1/x$ is kontinuous.

The key problem is that a uniformly continuous, [[positive number|positive]]-valued function defined on $[0,1]$ might fail to be bounded below by a positive number, since the interval $[0,1]$ might fail to be [[compact space|compact]], yet its reciprocal (if also uniformly continuous) must be bounded above.

Waaldijk's negative result can be circumvented by dropping the insistence on points and instead working with maps between [[locales]], [[toposes]], or formal spaces as studied in [[formal topology]].

## Related concepts

* [[pointwise continuous function]]

* [[degree of a continuous function]]

* [[equicontinuous family of functions]]

* [[analytic function]]

* [[differentiable function]]

* [[smooth function]]

* [[convex function]]

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

* [[bounded function]]

* [[compactly supported function]]

* [[measurable function]]

* [[rapidly decreasing function]]

* [[function with rapidly decreasing partial derivatives]]

## Literature

Discussion in [[constructive mathematics]]:

* {#Waaldijk2003} [[Frank Waaldijk]], _On the foundations of constructive mathematics – especially in relation to the theory of continuous functions_, 2003 ([pdf](http://www.fwaaldijk.nl/foundations%20of%20constructive%20mathematics.pdf))

[[!redirects continuous map]]
[[!redirects continuous maps]]
[[!redirects continuous function]]
[[!redirects continuous functions]]

[[!redirects continuity]]


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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop}
###### Proposition
**(continuous metric space valued function on compact metric space is uniformly continuous)**

Let $K$ and $Y$ be two [[metric spaces]] regarded as [[topological spaces]] via their [[metric topology]], such that $K$ [[compact topological space|compact]].

Then every [[continuous function]] of the form

$$
  K \overset{f}{\longrightarrow} Y
$$

is [[uniformly continuous function|uniformly continuous]].

In particular since [[subspaces]] of metric spaces canonically are metric spaces, it follows that if $X$ is a metric space

=--

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

Assume on the contrary that it were not. This would mean that for all $\epsilon \in (0,\infty)$ there would for all $n \in \mathbb{N}$ be points $x_n,y_n \in K$ with $d_K(x_n,y_n) \lt 1/(n+1)$ but $d_Y(f(x_n),f(y_n)) \gt \epsilon$.

Since  in compact metric spaces every [[sequence]] has a [[convergence|converging]] subsequence ([[sequentially compact metric spaces are equivalently compact metric spaces|here]]), it follows that there is a  converging subsequences $(x_{n_k})_{k \in \mathbb{N}}$. Since $d_K(x_{n_k} y_{n_k}) \lt 1/(n+1)$ it follows that also $(y_{n_k})_{k \in \mathbb{N}}$ is a converging subsequence which converges to the same point:

But since continuous functions preserve [[limits of sequences]], it would follow that

$$
  \underset{k \to \infty}{\lim} f(x_{n_k})
  \;=\;
  \underset{k \to \infty}{\lim} f(y_{n_k})
  \,.
$$

This however would contradict the assumption that $d_Y(f(x_k), f(y_k)) \gt \epsilon$. Hence we have a [[proof by contradiction]].

=--

## Related statements

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

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

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

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

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

* [[Lebesgue number lemma]]

[[!redirects continuous function on compact space is uniformly continuous]]


continuous multilinear operator

+-- {: .rightHandSide}
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###Context###
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Multilinear maps between [[locally convex topological vector spaces]] are important in many respects not least because of their use in infinite dimensional calculus as the homes of higher derivatives.  However, outside the realm of [[normed vector spaces]], there is no natural choice of topology on the spaces of multilinear maps.

## Definition

The definition of an $n$-linear map is straightforward and is an application of the definition at [[bilinear map]].

+-- {: .num_defn #mult}
###### Definition
Let $V$ and $W$ be [[vector spaces]] and $n \in \mathbb{N}$.  The space of $n$-linear mappings $V \to W$, written $L^n(V,W)$, is the space of set maps $V^n \to W$ with the property that they are linear in each variable.

For $n = 1$ we write $L(V,W)$ and we extend the notation to $n = 0$ where by $L^0(V,W)$ we mean just $W$.
=--

When $V$ and $W$ are [[LCTVS]] we can restrict to continuous maps.

+-- {: .num_defn #cmult}
###### Definition
Let $V$ and $W$ be [[locally convex topological vector spaces]] and $n \in \mathbb{N}$.  The space of continuous $n$-linear mappings $V \to W$, written $\mathcal{L}^n(V,W)$, is the space of continuous maps $V^n \to W$ with the property that they are linear in each variable.

For $n = 1$ we write $\mathcal{L}(V,W)$ and we extend the notation to $n = 0$ where by $\mathcal{L}^0(V,W)$ we mean just $W$.
=--

There are a variety of topologies that one can put on the spaces $\mathcal{L}^n(V,W)$.  The most common are the topologies of uniform convergence on some family of [[bounded sets]] of the source.

+-- {: .num_defn #cmtop}
###### Definition
Let $V$ and $W$ be [[LCTVS]] and $n \in \mathbb{N}$.  Let $\mathcal{S}$ be a collection of [[bounded subsets]] of $V$ which covers $V$.  Define $\Lambda_{\mathcal{S}}$ to be the topology on $\mathcal{L}^n(V,W)$ given by uniform convergence on $\mathcal{S}$.  That is, it is the [[LCTVS]] structure on $\mathcal{L}^n(V,W)$ defined by the family of [[semi-norms]]:
$$
\rho(u)_{\sigma,S} \coloneqq \sup \{\sigma(u(h_1,\dots,h_n)) : h_i \in S\}
$$
where $S \in \mathcal{S}$ and $\sigma \colon W \to \mathbb{R}$ is a continuous semi-norm on $W$.

We write $\mathcal{L}^n_{\mathcal{S}}(V,W)$ for the [[LCTVS]] with underlying vector space $\mathcal{L}^n(V,W)$ and topology $\Lambda_{\mathcal{S}}$.
=--

The commonest families in use are:

1. The finite subsets.  This resulting topology is known as the **simple** topology and the usual notations are $\Lambda_s$ and $\mathcal{L}_s(V,W)$.
1. The compact subsets, with notations $\Lambda_k$ and $\mathcal{L}_k(V,W)$.
1. The [[precompact subsets]], with notations $\Lambda_{p k}$ and $\mathcal{L}_{p k}(V,W)$.
1. The [[bounded subsets]], with notations $\Lambda_b$ and $\mathcal{L}_b(V,W)$.

Of all the topologies definable using Definition \ref{cmtop}, the topology $\Lambda_s$ is the coarsest and $\Lambda_b$ the finest.

The topology of simple convergence ($\Lambda_s$) also makes sense on the spaces $L^n(V,W)$ and we write $L_s^n(V,W)$ for the resulting LCTVS.  We have that $\mathcal{L}_s^n(V,W) \subseteq L^n_s(V,W) \subset W^{V^n}$ (with the product topology) are topological embeddings.

## Properties

1. For any suitable family of [[bounded sets]], $\mathcal{S}$, the natural map $\mathcal{L}_{\mathcal{S}}^{n+m}(V,W) \to \mathcal{L}_{\mathcal{S}}^n(V,\mathcal{L}_{\mathcal{S}}^m(V,W))$ is a topological embedding.

1. If $V$ is [[metrisable]], $W$ arbitrary, and $\mathcal{S}$ contains all [[compact]] sets then $\mathcal{L}_{\mathcal{S}}^{n+m}(V,W) \to \mathcal{L}_{\mathcal{S}}^n(V,\mathcal{L}_{\mathcal{S}}^m(V,W))$ is an isomorphism.

1. If $V$ is [[metrisable]] and [[barrelled]], $W$ arbitrary, and $\mathcal{S}$ arbitrary then $\mathcal{L}_{\mathcal{S}}^{n+m}(V,W) \to \mathcal{L}_{\mathcal{S}}^n(V,\mathcal{L}_{\mathcal{S}}^m(V,W))$ is an isomorphism.

## Separate Continuity

The definition of $\mathcal{L}^n(V,W)$ requires the maps $V^n \to W$ to be continuous in the product topology.  On the other hand, the linearity is required only on a per-factor basis.  It is possible to make the continuity requirement also work on a per-factor basis using the notion of [[hypocontinuity]].  For this we need to fix the type of continuity first.

+-- {: .num_defn #smult}
Let $V$ and $W$ be [[LCTVS]] and $n \in \mathbb{N}$.  Let $\mathcal{S}$ be a family of [[bounded subsets]] of $V$ which cover $V$.  We define $\mathcal{H}^n_{\mathcal{S}}(V,W)$ inductively by:

1. $\mathcal{H}^0(V,W) \coloneqq W$
2. $\mathcal{H}^n_{\mathcal{S}}(V,W) \coloneqq \mathcal{L}_{\mathcal{S}}(V, \mathcal{H}^{n-1}_{\mathcal{S}}(V,W))$.
=--

## Properties

1. $\mathcal{L}^n_{\mathcal{S}}(V,W) \to \mathcal{H}^n_{\mathcal{S}}(V,W)$ is a topological embedding.
1. If $\mathcal{S} \subseteq \mathcal{S}'$ then $\mathcal{H}^n_{\mathcal{S}'}(V,W)$ injects continuously into $\mathcal{H}^n_{\mathcal{S}}(V,W)$.
2. $\mathcal{H}^n_s(V,W)$ consists of the [[separately continuous]] multilinear maps $V^n \to W$.
3. If $V$ is [[metrisable]] and $\mathcal{S}$ contains all compact sets, or if $V$ is [[metrisable]] and [[barrelled]], then for any $W$ and $n$, $\mathcal{L}^n_{\mathcal{S}}(V,W) = \mathcal{H}^n_{\mathcal{S}}(V,W)$.

## Convergence Structures

A [[topological vector space]] defines a [[convergence vector space]] with the same underlying vector space and the same continuous mappings (that is to say, the functor from topological vector spaces to convergence vector spaces is full and faithful and respects the vector space functor).
The construction of the spaces of continuous multilinear operators makes sense for convergence vector spaces as well as for topological ones.
The resulting spaces of multilinear operators has several useful structures as a [[convergence vector space]] which are not topological.

+-- {: .num_defn #cvs}
###### Definition
Let $V$ and $W$ be [[convergence vector spaces]].  The **continuous convergence** structure on $\mathcal{L}^n(V,W)$ is the coarsest convergence structure which makes the evaluation map $\mathcal{L}^n(V,W) \times V^n \to W$ continuous.  The resulting convergence space (which is a convergence vector space) is written $\mathcal{L}^n_c(V,W)$.

The structure of **bounded convergence** (resp. **quasi-bounded convergence**) on $\mathcal{L}^n(V,W)$ is defined by the following requirement: for every filter $\mathcal{F}$ on $\mathcal{L}^n(V,W)$ and every $u \in \mathcal{L}^n(V,W)$ then $\mathcal{F} \to u$ if and only if $(\mathcal{F} - u)(\mathcal{B}^n) \to 0$ in $W$ for every bounded (resp. quasi-bounded) filter $\mathcal{B}$ on $V$.
The resulting convergence vector space is written $\mathcal{L}_b^n(V,W)$ (resp. $\mathcal{L}_{q b}^n(V,W)$.
=--

Recall that a filter is *bounded* if it contains a bounded set and is *quasi-bounded* if the filter $\mathcal{V} \cdot \mathcal{F} \to 0$ where $\mathcal{V}$ is the filter of $0$-neighbourhoods in $\mathbb{R}$.

If both $V$ and $W$ are LCTVS then the two notions of bounded convergence agree.


continuous poset

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### Context
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Continuous posets
* table of contents
{: toc}

## Definitions

Let $P$ be a [[poset]] with [[directed joins]].

### Elementary definition

+-- {: .un_defn}
###### Definition
For $a, b \in P$, we say that $a$ is **way below** $b$ and write $a \ll b$ if whenever $S \subseteq P$ is a [[directed subset]] and $b \leq \bigvee S$ (where $\bigvee S$ denotes the [[join]] of $S$), then there exists $s \in S$ with $a \leq s$.
=--

+-- {: .un_defn}
###### Definition
We say that $P$ is **continuous** if for every $a\in P$, the subset
$$ \Downarrow (a) \coloneqq \{ b \in P | b \ll a \} $$
is directed and has join $a$.
=--

If $P$ also has finite joins, hence is a [[suplattice]], then $\Downarrow(a)$ is automatically directed, so the condition reduces to $\bigvee (\Downarrow(a)) = a$; in this case $P$ is called a **continuous lattice**.

### Adjoint characterization

Let $Idl(P)$ denote the poset of [[ideals]] in $P$, i.e. subsets that are downward-closed and upwards-directed.

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

A poset $P$ has directed joins if and only if the [[principal-ideal]] map $\downarrow : P \to Idl(P)$, defined by
$$ \downarrow(a) \coloneqq \{ b \in P | b \leq a \} ,$$
has a [[left adjoint]], which must be $\bigvee$.
=--

+-- {: .un_theorem}
###### Theorem
A poset $P$ with directed joins is continuous if and only if $\bigvee: Idl(P) \to P$ has its own left adjoint, which must be $\Downarrow$.
=--

This characterization generalizes directly to the notion of [[continuous category]].

## Morphisms

Since a continuous poset must have directed joins, the obvious [[morphisms]] in a [[category]] whose [[objects]] are continuous posets would be the [[Scott-continuous functions]], that is those that preserve directed joins.  (These always preserve the order; that is, they are [[monotone functions]].)

Between continuous lattices, we may use the same morphisms; or we may more generally use those Scott-continuous functions that preserve the [[semilattice]] structure of finitary [[joins]], in other words, the [[suplattice]] morphisms that preserve all joins.  However, because a suplattice is a [[complete lattice]], another common choice is to use the Scott-continuous functions that are also [[inflattice]] morphisms, that is those that also preserve all [[meets]].  Another choice might be the complete-lattice morphisms, those that preserve all meets and all joins.

## Properties

* Continuous lattices are those [[complete lattices]] for which taking [[suprema]] of directed subsets commutes with taking [[infima]] of arbitrary subsets.

* The [[forgetful functor]] $U$ from the category of continuous lattices to the [[category of sets]] is [[monadic functor|monadic]] if we use Scott-continuous inflattice morphisms. Here the left adjoint of $U$ takes a set $X$ to the lattice of [[filters]] on $X$ (that is filters in the [[power set]] Boolean algebra $P X$). For more, see [[filter monad]].

* The category of continuous lattices is [[cartesian closed category|cartesian closed]] if we use all Scott-continuous functions.  This category was used by [[Dana Scott]] to construct models of the untyped [[lambda calculus]].

* Every continuous lattice is a [[Baire lattice]].

## Examples

* In [Freyd08](#Freyd08) in [[constructive mathematics]] an [[interval coalgebra]] is a [[continuous poset]] and the [[unit interval]] is the terminal interval coalgebra.

* A [[locale]] is called [[locally compact locale|locally compact]] just when the corresponding [[frame]] is a continuous lattice. This is equivalently to being an [[exponentiable object|exponentiable]] in the [[category of locales]]. A [[continuous map]] between such locales is [[proper map|proper]] iff its [[direct image]] function (which is always an [[inflattice]] morphism) is [[Scott topology|Scott-continuous]].

* As a consequence the [[category of open subsets|lattice of open subsets]] of a topological space is a continuous lattice if and only if the [[sobrification]] of the topological space is [[locally compact]] (i.e. the topology has a basis of compact neighborhoods).

## Related pages

* [[continuous category]]
* [[locally compact locale]]

* Continuous posets can be generalized to [[continuous algebras]] for any [[lax-idempotent 2-monad]].

## References

* Rudolf-E. Hoffmann, _Continuous posets and adjoint sequences_, Semigroup Forum **18** (1979) pp. 173-188. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN001253433))

 * Karl H. Hofmann, _A note on Baire spaces and continuous lattices_ 1980. Bulletin of the Australian Mathematical Society, 21(2), pp. 265-279.

See section 30 of

* {#Freyd08} [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

[[!redirects continuous poset]]
[[!redirects continuous posets]]
[[!redirects continuous lattice]]
[[!redirects continuous lattices]]

[[!redirects way-below relation]]
[[!redirects way below relation]]
[[!redirects way-below containment]]
[[!redirects way below containment]]
[[!redirects way-below]]
[[!redirects way below]]


continuous predicate

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Continuous_predicate).


continuous space

Analogously to a [[smooth space]], a __continuous space__ is any notion of [[space]] between which the [[morphisms]] are [[continuous maps]].  Examples include:

* [[convergence spaces]],
* [[topological spaces]],
* [[locales]].

Contrast this with, for example:

* [[uniform spaces]] (morphisms are [[uniformly continuous map]]s),
* [[smooth spaces]] (morphisms are [[smooth maps]]),
* (weak) [[homotopy types]] (objects are still (nice) topological spaces, but now morphisms are [[homotopy]] classes of continuous maps).

[[!redirects continuous space]]
[[!redirects continuous spaces]]


continuous truth

# Continuous truth
* table of contents
{: toc}

## Idea

Continuous truth provides a [[topos]] [[categorical semantics]] for Brouwer's continuity principles in [[intuitionistic mathematics]].  The topos is intuitively the [[gros topos]] of [[sheaves]] on some category of spaces.

## Related entries

* [[topological site]]

## References

* Mike Fourman, _Continuous Truth II: Reflections_ [PDF](http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2013-continuous-truth-II.pdf)

* van der Hoeven, Moerdijk: On choice sequences determined by spreads.

* Xu, Escardo: A constructive model of uniform continuity. 2013 [PDF](http://www.cs.bham.ac.uk/~mhe/papers/xu-escardo.pdf)

[[!redirects continuous truth]]
[[!redirects Continuous truth]]
[[!redirects Continuous Truth]]


continuous étale cohomology

## Related concepts

* [[étale cohomology]]

## References

* [[Uwe Jannsen]], _Continuous étale cohomology_, Mathematische Annalen
March 1988, Volume 280, Issue 2, pp 207-245

[[!redirects continuous etale cohomology]]


continuous-trace algebra

Let $X$ be a locally compact Hausdorff space. An
algebra of continuous trace over $X$
is a $C^*$-algebra $A$ with dual space $\hat{A} = X$, such
that Fell's condition holds: for each $x_0\in X$, there is $a\in A$ such that $x(a)$ is a rank-one projection for each $x$ in a neighbourhood of $x_0$. This holds if and only if the set of all $a\in A$ for which the map $\pi\mapsto
tr (\pi(a)\pi(a)^*)$ is finite and continuous on $\hat{A}$ is dense in $A$. A Fell algebra is a $C^*$-algebra $A$
which satisfies Fell's condition but $X$ is not necessarily Hausdorff.

A continuous trace $C^*$-algebra is, to some extent, an operator algebraic counterpart to the theory of [[Azumaya algebra]]s. [[Dixmier-Douady class]] has been designed originally to give invariant of such operator algebras.

* I. Raeburn, D. Williams, _Morita equivalence and continuous-trace $C^*$-algebras_, AMS Monographs __60__ (1998) xiv+327 pp.
* Jonathan Rosenberg, _Continuous-trace algebras from the bundle theoretic point of view_, J. Austral. Math. Soc. Ser. A __47__ (1989), no. 3, 368–381 [MR91d:46090](http://www.ams.org/mathscinet-getitem?mr=1018964) [doi](http://dx.doi.org/10.1017/S1446788700033097)
* chapter 9 in Joachim Cuntz, Ralph Meyer, Jonathan M. Rosenberg, _Topological and bivariant K-theory_, Oberwolfach Seminars
* Alex Kumjian, Paul Muhly, Jean Renault, Dana Williams:
_The Brauer group of a locally compact groupoid_,
Amer. J. Math. __120__ (1998) 901-954 [ps](http://wolfweb.unr.edu/homepage/alex/pub/kmrw.ps)

category: operator algebras, noncommutative geometry


continuously differentiable map

+-- {: .rightHandSide}
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### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Once beyond the realm of [[normed vector spaces]], the various ways of defining [[differentiation]] diverge.  This is particularly evident if one considers the slightly stronger notion of **continuous differentiability** wherein the assignment of the derivative must also be continuous.

One can make a reasonable start by saying that for a function $f \colon E \supseteq U \to F$ to be continuously differentiable then it must at least satisfy the notion of [[Gâteaux differentiability]], and one can throw in the requirement that the assignment of the directional derivative be continuous and linear (this is known as *Gâteaux--Lévy differentiability*).  Thus one obtains a map $D f \colon U \to \mathcal{L}(E,F)$.  However, outside the realm of normed vector spaces there is not a unique topology on $\mathcal{L}(E,F)$ and thus one can come up with a variety of meanings for the phrase "$D f$ is continuous".

## Finite Dimensions

Let us remind ourselves of the situation in finite dimensions.

+-- {: .num_defn #ctsd}
###### Definition
A function $f \colon \mathbb{R}^m \supseteq U \to \mathbb{R}^n$, where $U$ is open, is said to be **continuously differentiable**, or of class $C^1$, if there is a continuous map $D f \colon U \to \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)$ with the property that for each $x \in U$ and $h \in \mathbb{R}^m$ then

$$
\lim_{t \to 0} \frac{f(x + t h) - f(x) - t D f(x)h}{t} = 0
$$
=--

Note that for $x \in U$ and $h \in \mathbb{R}^n$ there is an open interval $(-\epsilon,\epsilon)$ with the property that for $t \in (-\epsilon,\epsilon)$ then $x + t h \in U$ and so the limit makes sense.

## Infinite Dimensions

In infinite dimensions the difficulty with extending the standard definition is that of the topology on continuous linear maps.  This becomes more evident with higher derivatives.  Thus the definition depends on such a choice.  In addition, one needn't use a topology but can make sense of the definition with a [[convergence structure]] on the space of linear maps.

+-- {: .num_defn #ctsdinf}
###### Definition
Let $E$ and $F$ be [[locally convex topological vector spaces]].  Let $U \subseteq E$ be an open set.  Let $\mathcal{L}(E,F)$ be the space of continuous linear maps from $E$ to $F$.  Let $\Lambda$ be a [[convergence structure]] on $\mathcal{L}(E,F)$.  A continuous function $f \colon U \to F$ is said to be **differentiable of class $C^1_\Lambda$** if there exists a continuous mapping $D f \colon U \to \mathcal{L}_\Lambda(E,F)$, called the **derivative** of $f$, such that for every $x,h \in U \times E$ then

$$
\lim_{t \to 0} \frac{f(x + t h) - f(x)}{t} = D f(x) h
$$

We define $C^1_\Lambda(X,F)$ to be the set of functions $f \colon X \to F$ which are of class $C^1_\Lambda$.
=--

## Convergence Structures

There are a variety of convergence structures and topologies on $\mathcal{L}(E,F)$ that can be used.  Some of them with particular properties are gathered in the list below.  For these, the notation is condensed slightly as indicated.

In the following, given [[semi-norms]] $\alpha$ on $E$ and $\beta$ on $F$ we define a semi-norm $\rho_{\beta,\alpha}$ on $\mathcal{L}(E,F)$ by

$$
\rho_{\beta,\alpha}(u) \coloneqq \sup \{\beta(u h) | \alpha(h) \le 1\}
$$

1. $C^1_\Theta$: the translation-invariant convergence structure with filters:

    $\mathcal{F} \to 0$ if there is a semi-norm $\alpha$ on $E$ such that for each semi-norm $\beta$ on $F$ and $\epsilon \gt 0$ there is some $Q \in \mathcal{F}$ with $\sup_{u \in Q} \rho_{\beta,\alpha}(u) \le \epsilon$.

2. $C^1_\Delta$: (Marinescu's convergence structure) the colimit in the category of [[convergence vector spaces]] of the following family of spaces.  Let $\Phi$ denote the family of mappings from the set of continuous semi-norms on $F$ to that on $E$.  For $\phi \in \Phi$ define:

$$
\mathcal{L}_\phi(E,F) \coloneqq \{ u \in \mathcal{L}(E,F) | \rho_{\beta,\phi(\beta)} \lt \infty\; \text{for all}\; \beta\}
$$

3. $C^1_\Pi$: the compatible convergence structure with filters:

    $\mathcal{F} \to 0$ if for each semi-norm $\beta$ on $F$ there is a semi-norm $\alpha$ on $E$ such that for each $\epsilon \gt 0$ there is some $Q \in \mathcal{F}$ with $\sup\{\rho_{\beta,\alpha}(u) | u \in Q\} \le \epsilon$.

4. $C^1_{q b}$: the *quasi-bounded* convergence structure.  This has filters:

    $\mathcal{F} \to 0$ if $\mathcal{F}(\mathcal{B}^n) \to 0$ for every quasi-bounded filter $\mathcal{B}$ on $E$.

    Recall that a filter $\mathcal{B}$ on $E$ is *quasi-bounded* if $\mathcal{V} \cdot \mathcal{B} \to 0$ where $\mathcal{V}$ is the neighbourhood filter of $0$ in $\mathbb{R}$.

5. $C^1_c$: the *continuous convergence structure*.  This is the coarsest convergence structure on $\mathcal{L}(E,F)$ which makes the evaluation map $\mathcal{L}(E,F) \times E \to F$ continuous.

6. $C^1_{\mathcal{S}}$: the topology of uniform convergence on a family $\mathcal{S}$ of [[bounded subsets]] of $E$, in particular:
   1. $C^1_b$: the topology of uniform convergence on [[bounded subsets]] of $E$,
   2. $C^1_{p k}$: the topology of uniform convergence on [[precompact subsets]] of $E$,
   3. $C^1_k$: the topology of uniform convergence on compact subsets of $E$,
   4. $C^1_s$: the topology of uniform convergence on finite subsets of $E$.

The relationships between the various definitions of $C^1_?$ are displayed in the following diagram.

+-- {: style="text-align: center"}
<svg width="286" height="425" xmlns="http://www.w3.org/2000/svg">
 <g>
  <title>Layer 1</title>
  <foreignObject height="25" width="40" font-size="16" id="svg_1" y="0" x="0">
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=--

Let us extract some particular cases:

1. $E$ normable, $F$ arbitrary:
    1. $C^1_\Theta$, $C^1_\Delta$, $C^1_\Pi$, $C^1_{q b}$, $C^1_b$ are equivalent
    2. $C^1_c$, $C^1_{p k}$, $C^1_k$ are equivalent
2. $E$ Banach space, $F$ arbitrary:
    1. $C^1_\Theta$, $C^1_\Delta$, $C^1_\Pi$, $C^1_{q b}$, $C^1_b$ are equivalent
    2. $C^1_c$, $C^1_{p k}$, $C^1_k$, $C^1_s$ are equivalent
3. $E$ Fréchet space, $F$ arbitrary:
    1. $C^1_{q b}$, $C^1_b$ are equivalent
    2. $C^1_c$, $C^1_{p k}$, $C^1_k$, $C^1_s$ are equivalent
4. $E$ Fréchet--Schwartz, $F$ arbitrary: $C^1_\Pi$, $C^1_{q b}$, $C^1_c$, $C^1_b$, $C^1_{p k}$, $C^1_k$, $C^1_s$ are equivalent
5. $E$ finite dimensional and $F$ arbitrary, or $E$ Fréchet--Schwartz and $F$ normable: All equivalent

## Chain Rule

An important question to ask of the various definitions of continuously differentiable is whether they satisfy the chain rule.  The following result provides the basis for this.

+-- {: .num_lemma #lemchain}
###### Lemma
Let $E$, $F$, $G$ be LCTVS, let $U \subseteq E$ and $V \subseteq F$ be open sets, let $\Lambda$ and $\Lambda'$ be convergence structures on $\mathcal{L}(E,F)$ and $\mathcal{L}(E,G)$ respectively.

Assume that the composition map:

$$
\mathcal{L}_k(F,G) \times \mathcal{L}_\Lambda(E,F) \to \mathcal{L}_{\Lambda'}(E,G)
$$

is continuous.

Let $f \colon U \to F$ and $g \colon V \to G$ be functions with $f(U) \subseteq V$.  Suppose that $f$ is of class $C^1_\Lambda$ and $g$ of class $C^1_k$.  Then $g \circ f$ is of class $C^1_{\Lambda'}$.
=--

+-- {: .num_cor #corchain}
###### Corollary
The chain rule holds for each of $C^1_c$, $C^1_{q b}$, $C^1_\Pi$, $C^1_\Delta$, $C^1_\Theta$.

The following partial chain rules also hold:

1. $C^1_\Pi(F,G) \times C^1_{\mathcal{S}}(E,F) \to C^1_{\mathcal{S}}(E,G)$
1. If $E$ is metrisable, $C^1_k(F,G) \times C^1_s(E,F) \to C^1_s(E,G)$
1. If $E$ is metrisable, $C^1_k(F,G) \times C^1_k(E,F) \to C^1_k(E,G)$
=--

## Higher Derivatives

A minor wriggle enters the story with higher derivatives due to the fact that the higher derivatives are multilinear maps $E^n \to F$ and so not only are there different notions of convergence to put on these spaces, there are also different possible meanings of the statement that these are continuous.  When dealing with one of the topologies (defined by some family of bounded sets), we will end up with derivatives in $\mathcal{H}^n_{\mathcal{S}}(E,F)$ rather than $\mathcal{L}^n_{\mathcal{S}}(E,F)$ in the notation of [[continuous multilinear operator]].

However, we can start with a very weak notion to get the ball rolling.  Let $L^n(E,F)$ be the space of all (not necessarily continuous) $n$-linear maps $E^n \to F$.  We equip it with the topology of simple convergence.

+--{: .num_defn #wdiff}
###### Definition
Let $E$ and $F$ be LCTVS, $U \subseteq E$ an open subset, $p \in \mathbb{N}$.  A function $f \colon U \to F$ is said to be **weakly $p$-times differentiable** if there exist functions $D^k f \colon U \to L^k(E,F)$ for $k = 0, 1, \dots, p$ such that $D^0 f = f$ and for each $x \in U$, $h \in E$, and $k = 0, 1, \dots, p-1$ then

$$
\lim_{t \to 0} \frac{D^k f(x + t h) - D^k f(x)}{t} = D^{k+1} f(x)h.
$$
=--

Note that we don't assume that the $D^k f$ are continuous.  If some are continuous then some nice properties ensue.  For example, if $D^p f$ is continuous then each $D^k f$ for $k \le p$ is totally symmetric.

From the definition of weakly $p$-times differentiable we can define the various classes of continuously $p$-times differentiable.  For the definition of $\mathcal{H}^n_{\mathcal{S}}(E,F)$ see the page on [[continuous multilinear operator]].

+-- {: .num_defn #chdiff}
###### Definition
Let $E$ and $F$ be LCTVS, $U \subseteq E$ an open subset, $p \in \mathbb{N}$.  Let $\mathcal{S}$ be a family of bounded sets in $E$ which covers $E$.  A function $f \colon U \to F$ is said to be **differentiable of class $C^p_{\mathcal{S}}$** if $f$ is weakly $p$-times differentiable and such that for $k = 0,1,\dots, p$ then:

1. $D^k f(U) \subseteq \mathcal{H}^k_{\mathcal{S}}(E,F)$, and
2. $D^k f \colon U \to \mathcal{H}^k_{\mathcal{S}}(E,F)$ is continuous.
=--

Using convergence structures, we have:

+-- {: .num_defn #cvdiff}
###### Definition
Let $E$ and $F$ be LCTVS, $U \subseteq E$ an open subset, $p \in \mathbb{N}$.  Let $\Lambda$ be one of the convergence structures $\Lambda_c$, $\Lambda_{q b}$, $\Pi$, $\Delta$, or $\Theta$ on $\mathcal{L}^p(E,F)$.  A function $f \colon U \to F$ is said to be **differentiable of class $C^p_{\mathcal{S}}$** if $f$ is weakly $p$-times differentiable and such that for $k = 0,1,\dots, p$ then:

1. $D^k f(U) \subseteq \mathcal{L}^k_{\Lambda}(E,F)$, and
2. $D^k f \colon U \to \mathcal{L}^k_{\Lambda}(E,F)$ is continuous.
=--

For fixed $p$, the relationships between the various $C^p_?$ are the same as for $p = 1$.  For varying $p$ we have:

+-- {: .num_lemma #diffp}
###### Lemma
Let $E$ and $F$ be LCTVS, $U \subseteq E$ an open set.  If $f \colon U \to E$ is of class $C^{p+1}_c$ then $f$ is of class $C^p_\Pi$.  Whence we have:

$$
C^{p+1}_c(X,F) \subseteq C^p_\Pi(X,F) \subseteq C^p_{q b}(X,F) \subseteq C^p_b(X,F)
$$

If $E$ is metrisable (resp. Fréchet) and $f$ is of class $C^{p+1}_k$ (resp. $C^{p+1}_s$) then $f$ is of class $C^p_\Pi$.
=--

## Smooth Functions

We make the obvious definition for a [[smooth function]]:

+--{: .num_defn #smooth}
###### Definition
Let $\Lambda$ be one of the convergence structures $\Lambda_s$, $\Lambda_k$, $\Lambda_{p k}$, $\Lambda_b$, $\Lambda_c$, $\Lambda_{q b}$, $\Pi$, $\Delta$, or $\Theta$.  A function $f \colon U \to F$ is said to be **differentiable of class $C^\infty_\Lambda$** if it is of class $C^p_\Lambda$ for all $p \in \mathbb{N}$.
=--

We have the following table of relationships.

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=--

Therefore, for [[Banach spaces]] all the notions of "smooth function" collapse, whilst for [[Fréchet spaces]] all of the notions coarser than $\Pi$ collapse.

## Related concepts

* [[continuous function]], [[differentiable function]], **continuously differentiable function**, [[smooth function]], [[analytic function]]

## References

* Keller, H. H. (1974). _Differential calculus in locally convex spaces_. Lecture Notes in Mathematics, Vol. 417. Berlin: Springer-Verlag. [MR0440592](http://www.ams.org/mathscinet-getitem?mr=440592)

[[!redirects continuous differentiability]]
[[!redirects continuously differentiable]]
[[!redirects continuously differentiable map]]
[[!redirects continuously differentiable maps]]
[[!redirects continuously differentiable function]]
[[!redirects continuously differentiable functions]]


continuum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _continuum_ is in general something opposite to a [[discrete object|discrete]].
There are several notions of continuum in mathematics:

* Often **the continuum** is taken to refer to the [[real line]]; consequently the *cardinality of the continuum* referse to the [[cardinality]] of the [[set]] of [[real numbers]]. There is a related **[[continuum hypothesis]]** in [[set theory]].

* A **metric continuum** is any [[compact space|compact]] [[connected space|connected]] [[metric space]]. Often in [[general topology]], a **continuum** is taken to mean a compact [[separation axiom|Hausdorff]] connected space.

In [[physics]] one may also mean by _a continuum_ a medium which spreads the physical quantities spatially with some finite density, unlike the physics of a system of [[particles]], where an infinite ([[delta-distribution]]-like) density is attached to a discrete system of points. See at _[[geometry of physics]]_ for more on the relation of the continuum in form of the real line to physics.

## In cohesive homotopy type theory
 {#InCohesiveHomotopyTypeTheory}

One can [[axiom|axiomatize]] aspects of the notion of _line continuum_ in [[cohesive homotopy type theory]]. There the idea of an [[object]] $\mathbb{A}^1$ all whose [[generalized element|points]] are, while different, connectable by _[[continuous function|continuous]]_ [[path groupoid|paths]] (and uniquely so, up to suitable [[homotopy]]) is encoded in asking that after applying the [[fundamental infinity-groupoid in a locally infinity-connected (infinity,1)-topos|fundamental ∞-groupoid functor]] $\mathbf{\Pi}$ to it, the result is something [[(-1)-connected|contractible]]

$$
  \mathbf{\Pi}(\mathbb{A}^1) \simeq *
  \,.
$$

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

In the [[model]] of [[cohesive homotopy type theory]] called [[Smooth∞Grpd]] we have a [[full and faithful functor|full and faithful]] embedding of [[smooth manifolds]]. Therefore we can embed the [[integers]] $\mathbb{Z}$, the [[rational numbers]] $\mathbb{Q}$ as well as the [[real numbers]] $\mathbb{R}$, all equipped with their canonical [[smooth manifold]] structure. This is [[discrete object|discrete]] for the first two, but not for the last one, and homotopy cohesion can detect this:

$$
  \mathbf{\Pi}(\mathbb{Z}) \simeq \mathbb{Z}
  \,;
$$

$$
  \mathbf{\Pi}(\mathbb{Q}) \simeq \mathbb{Q}
  \,;
$$

but

$$
  \mathbf{\Pi}(\mathbb{R}) \simeq *
  \,.
$$

This reflects the fact that the points of $\mathbb{R}$ form a continuum, but those of $\mathbb{Z}$ and $\mathbb{Q}$ do not.

Also the [[complex numbers]] $\mathbb{C}$ with their canonical manifold structure of course form a continuum in this sense

$$
  \mathbf{\Pi}(\mathbb{C}) \simeq *
  \,.
$$

But the [[real line]] in fact "exhibits the cohesion" (in the sense of [this definition](cohesive+%28infinity%2C1%29-topos#A1ExhibitingCohesion) by [this discussion](cohesive+%28infinity%2C1%29-topos+--+structures#RealLineIsTheContinuum)) in that the [[shape modality]] $\Pi$ is equivalent to [[localization of an (infinity,1)-category|localization]] at $(-) \times \mathbb{R}^1 \longrightarrow (-)$

$$
  \Pi \simeq L_{\mathbb{R}^1}
  \,.
$$

This is not true in smooth cohesion for $\mathbb{R}^1$ replaced by $\mathbb{C}^1$ and in this sense $\mathbb{R}^1$ is "the real continuum" here (in both sense of the word "real").

=--

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

For a [[ring object]] $\mathbb{A}^1$ to be geometrically contractible, $\Pi(\mathbb{A}^1) \simeq *$, it is sufficient that there be a map $i$ from a [[bi-pointed type]] $(left, right, I)$ to the bipointed type $(0, 1, \mathbb{A}^1)$ such that $I$ is geometrically contractible.

Hence in words: "If in the ring $\mathbb{A}^1$ the elements 0 and 1 are path-connected, then $\mathbb{A}^1$ is already contractible, hence is a line continuum."

=--

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

Under the given assumptions we obtain a [[commuting diagram]] of the form

$$
  \array{
    \mathbb{A}^1
    \\
    \downarrow & \searrow^{\mathrlap{ 0}}
    \\
    I \times \mathbb{A}^1 &\stackrel{ mult\circ (i,id)}{\to}& \mathbb{A}^1
    \\
    \uparrow & \nearrow_{\mathrlap{id}}
    \\
    \mathbb{A}^1
  }
  \,.
$$

Since $\Pi$ preserves products, it sends this to a diagram of the form

$$
  \array{
      & \nearrow \searrow^{\mathrlap{0}}
     \\
    \Pi(\mathbb{A}^1) &\Downarrow_{\simeq}& \Pi(\mathbb{A}^1)
    \\
    & \searrow \nearrow_{\mathrlap{id}}
  }
  \,,
$$

which exhibits a contracting [[homotopy]] of $\Pi(\mathbb{A}^1)$.

=--

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

The standard unit intervals $[0,1] \hookrightarrow \mathbb{R} \in $ [[TopMfd]] $\hookrightarrow$ [[ETop∞Grpd]] and $[0,1] \hookrightarrow \mathbb{R} \in $ [[SmthMfd]] $\hookrightarrow$ [[Smooth∞Grpd]]
satisfy the assumptions of prop. \ref{ContractibilityOfAFromThatOfI}.

=--

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

The $I$ of prop. \ref{ContractibilityOfAFromThatOfI} is in general not an [[interval type]], but only its image $\Pi(I)$ is. See the discussion _[Geometric spaces and their cohesive homotopy types](cohesive%20homotopy%20type%20theory#GeometricSpacesAndTheirHomotopyTypes)_ at _[[cohesive homotopy type theory]]_ for more on this.

=--

## References
 {#References}

* {#Johnstone08} [[Peter Johnstone]], _Topos-theoretic models of the continuum_ 2008 ([slides](http://www.cs.ox.ac.uk/quantum/slides/clap2-peterjohnstone.pdf), [lecture video](https://www.youtube.com/watch?feature=player_embedded&v=pKWYa9sc5UI))

[[!redirects continua]]

[[!redirects metric continuum]]
[[!redirects metric continua]]