uniform convergence space

# Uniform convergence spaces
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## Idea

Uniform convergence spaces are simultaneously a generalisation of [[uniform spaces]] and a specialisation with [[extra structure]] of [[convergence spaces]].  That is, we have [[functors]]
$$ Unif \to Unif Conv \to Conv ,$$
both [[faithful functor|faithful]] and with $Unif \to Unif Conv$ also [[full functor|full]].  (Here we take the [[morphisms]] of $Unif Conv$ and $Unif$ to be the [[uniformly continuous maps]], while $Conv$ must use the [[continuous maps]].)

## Motivation

The [[uniform property|uniform properties]] of a [[metric space]] $X$ are determined entirely by the [[equivalence relation]] between [[infinite sequences]] through which sequences $a$ and $b$ are related iff the distance between them tends to zero:
$$ \lim_{n \to \infty} {\|a_n - b_n\|} = 0 .$$

An analogous result holds for a [[uniform space]] $X$ if one generalises from sequences to [[nets]], as well as for [[metric spaces]] in [[constructive mathematics]] without [[weak countable choice]].  However, the relevant equivalence relation is not (directly) between two arbitrary nets but rather between two nets with the same [[directed set]] as [[domain]].  (This is because the limit above refers to a single index $n$; if we changed this to the limit as $m, n \to \infty$ of ${\|a_m - b_n\|}$, then we would get a [[partial equivalence relation]] that applies only to [[Cauchy sequences]], and this would recover only the [[Cauchy structure]] and not the entire [[uniform structure]].)

Rather than speak of two nets with the same domain, let us speak equivalently of a single net in the [[cartesian square]] $X \times X$.  Then for a more manifestly [[size issue|small]] definition, let us speak of a [[filter]] on $X \times X$.  So a uniform convergence structure on a [[set]] $X$ will be a collection of filters on $X \times X$.  We will require this collection to satisfy certain properties; if we imposed enough properties, we could recover a definition of uniform space (or even of [[metrizable uniform space]]), but some less stringent requirements yield the definition of uniform convergence space below.

## Definitions

Let $X$ be an [[abstract set]], and let $U$ be a collection of [[filters]] on the [[cartesian square]] $X \times X$, called the __asymptotic filters__.  We will consider various properties of such a structure below.

A collection that satisfies all of (1--6) below is a __uniform convergence structure__.  If it satisfies at least (2--6), then it is is a __[[base]]__ for a uniform convergence structure; if it satisfies at least (4--6), then it is a __[[subbase]]__ for a uniform convergence structure.  That said, we could start with a perfectly arbitrary collection and generate a subbase, then a base, then an entire structure.

If we use (6′) instead of (6), then this is a __pointwise uniform convergence structure__.  If we drop (4), then we have a __quasiuniform convergence structure__.  These also can have bases and subbases and be generated from an arbitrary collection of asymptotic filters.

A (pointwise) (quasi)-**uniform convergence space** is a set $X$ equipped with such a structure.

Here are the relevant properties:

1.  _Isotone_:  If $F \subseteq G$ (as filters on $X \times X$) and $F$ is asymptotic, then $G$ is asymptotic.

2.  _Proper_:  There is an asymptotic filter.  If (1) also holds, then we may equivalently say that the [[improper filter]] on $X \times X$ is asymptotic.  This is automatic if (6) holds but is independent of the other conditions; it *almost* follows from (6′), but only if $X$ is [[inhabited set|inhabited]].

3.  _Filtered_:  If $F$ and $G$ are asymptotic filters, then there is an asymptotic filter contained in both $F$ and $G$.  If (1) also holds, then we may equivalently say that the [[intersection]] $F \cap G$ is asymptotic.

4.  _Symmetric_:  If $F$ is an asymptotic filter, then there is an asymptotic filter $G$ with the property that, whenever $R \in G$, its [[inverse relation|inverse]] (as a [[binary relation]] on $X$) $R^{-1}$ belongs to $F$.  If (1) also holds, then we may equivalently say that the filter $\{ R^{-1} \;|\; R \in F \}$ is asymptotic.

5.  _Transitive_:  If $F$ and $G$ are asymptotic filters, then there is an asymptotic filter $H$ with the property that, whenever $R \in H$, there are a [[natural number]] $n$, an $n$-[[tuple]] $(S_1, \ldots, S_n) \in F^n$, and an $n$-tuple $(T_1, \ldots, T_n) \in G^n$ such that $R \supseteq (S_1 \circ T_1) \cap \cdots \cap (S_n \circ T_n)$ (where $\circ$ indicates [[composition]] of [[binary relations]] on $X$).  If (1) also holds, then we may equivalently say that the filter generated by $\{ S \circ T \;|\; S \in F,\; T \in G \}$ is asymptotic.

6.  _Reflexive_:  There is an asymptotic filter $F$ with the property that each $R \in F$ is [[reflexive relation|reflexive]] as a binary relation on $X$.  If (1) also holds, then we may equivalently say that the collection of all reflexive relations (which is a filter) is asymptotic.

Although (6) as given here is in the original definition and is the proper nullary analogue of the binary (5), a weaker version is now more popular:

*  (6′).  _Pointwise reflexive_:  For every point $a$ of $X$, there is an asymptotic filter $F$ with the property that each $R \in F$ satisfies $(a,a) \in R$ (so that $R$ is reflexive at $a$).  If (1) also holds, then we may equivalently say that the collection of all relations reflexive at $a$ (which is a filter, indeed the [[principal ultrafilter]] at $(a,a)$) is asymptotic.

One reason for this is apparently to render the [[category]] of uniform convergence spaces [[cartesian closed category|cartesian closed]].  So presumably there are uniform convergence spaces $X$ and $Y$ satisfying (6) such that the set of [[uniformly continuous maps]] from $X$ to $Y$ can be given only a pointwise uniform convergence structure.

For the purposes of [[constructive mathematics]], we can also consider another condition:

*  (7).  _Comparison_:  If $F$ is an asymptotic filter, then there is an asymptotic filter $G$ with the property that, whenever $R \in G$, there is some $S \in F$ such that $\neg{R} \cup S = X \times X$.  Assuming [[excluded middle]], we may take $G$ to be $F$ itself (and take $S$ to be $R$), rendering this condition trivial in [[classical mathematics]].

A (pointwise) (quasi)-uniform convergence structure/space that satisfies (7) may be called (quasi)-**uniformly regular space|uniformly regular** (although '(quasi)-uniformly [[locally decomposable space|locally decomposable]]' would be more proper, since there is no reason why such a space should be [[regular space|regular]], even in the symmetric case).

It is traditional to consider only _[[proper filter|proper]]_ asymptotic filters (or equivalently to consider only asymptotic [[nets]]), which allows one to leave out (2); but in that case (5) and (6) must be modified to apply only when the generated filters are proper: (5) applies only when each element of $F$ [[intersection|meets]] each element of $G$, and (6), if used, applies only when $X$ is [[inhabited set|inhabited]].  So really, it is simpler to include the improper filter among the asymptotic filters.

The conditions (1,2,3,4,5,6,7) correspond respectively to the conditions (6,4,5,2,3,1,0) at [[uniform space]] (as of 2013-02-22, in case the latter are ever renumbered).  However, (6′) has no direct analogue.

## Terminology warning

Among the terms defined (in boldface) above, I ([[Toby Bartels]]) invented 'asymptotic filter' and those involving 'pointwise' and 'quasi‑', because no appropriate terms appeared in the reference that I used.  Possibly other terms appear in the literature (besides the fact that 'pointwise' may simply be left out).  That said, my use of 'quasi‑' follows [[quasimetric]] and [[quasiuniform space]], so it at least is probably correct.  On the other hand, 'asymptotic' conflicts with the meaning of [[asymptotic function]] in [[complexity theory]] (which is weaker, and in fact says that functions are asymptotic when their [[logarithms]] are asymptotic in our sense), although it matches the meaning of [[asymptote]] in elementary [[analytic geometry]].  Finally, 'pointwise' was 'weak' in earlier versions of this page, until I changed it to have a more transparent meaning and avoid potential conflicts.  (People say 'weak uniform convergence' in various contexts, but nobody ever says 'pointwise uniform convergence'.)

## Subsidiary concepts

Let $X$ and $Y$ be pointwise uniform convergence spaces.  A [[function]] $f$ from $X$ to $Y$ is __[[uniformly continuous function|uniformly continuous]]__ if $f \times f\colon X \to Y$ takes asymptotic filters only to asymptotic filters.  Then $f$ is a __uniform homeomorphism__ if $f$ is a [[bijection]] whose [[inverse function|inverse]] is also uniformly continuous.  We take the uniformly continuous functions to be the [[morphisms]] of a [[category]] $Unif Conv$ whose [[objects]] are the uniform convergence spaces; then the uniform homeomorphisms are the [[isomorphisms]] of this category.

Now let $X$ and $Y$ be pointwise quasiuniform spaces.  The definitions above may be repeated with 'quasiuniform' instead of 'uniform'.  If we ever use 'uniform' with pointwise quasiuniform convergence spaces, this means to use the pointwise uniform convergence structure generated by the given pointwise quasiuniform convergence structure.  (The pointwise uniform convergence structure generated by a quasiuniform convergence structure will be a uniform convergence structure, so we may leave 'pointwise' out of this paragraph entirely if so desired.)

Let $X$ be a pointwise uniform convergence space.  A filter $F$ on $X$ is __[[Cauchy filter|Cauchy]]__ if it is [[proper filter|proper]] and $F \times F$ (which is a filter on $X \times X$) is asymptotic.  Then the Cauchy filters on $X$ form a Cauchy structure that makes $X$ into a [[Cauchy space]].  If $X$ is a pointwise quasiuniform convergence space, then we may use the same definition.  In any case, the result depends only on the generated uniform convergence structure.

Let $X$ be a pointwise uniform convergence space.  A filter $F$ on $X$ __[[convergence|converges]]__ to a point $a$ in $X$ if $F \times U_a$, where $U_a$ is the principal ultrafilter at $a$ (the collection of all subsets of $X$ to which $a$ belongs), is asymptotic; we may use $U_a \times F$ instead and get the same result.  Then this relation of convergence is a convergence structure that makes $X$ into a [[convergence space]], and this convergence space will be [[reciprocal space|reciprocal]] ($R_1$).  If $X$ is a pointwise quasiuniform convergence space, then we may use the same definition, but the result may no longer be reciprocal, and using $U_a \times F$ instead will give a different convergence structure (and where $U_a$ converges to $b$ one way, $U_b$ converges to $a$ the other way).  Nevertheless, the result depends only on the generated quasiuniform convergence structure.

The convergence space structure induced by the Cauchy structure induced by a pointwise uniform convergence space is the same as the convergence structure induced directly by the pointwise uniform convergence space.  This fails for (pointwise) quasiuniform convergence spaces, but it remains true that every convergent filter is Cauchy.  A pointwise qausiuniform convergence space is __[[complete space|complete]]__ if, conversely, every Cauchy filter converges to at least one point.  Also, the space is __[[Hausdorff space|Hausdorff]]__ if every Cauchy filter (and hence every filter) converges to at *most* one point.  The complete Hausdorff uniform convergence spaces form a [[reflective subcategory]] of $Unif Conv$, so every uniform convergence space has a __[[Hausdorff completion]]__.

## Examples

Let $X$ be a [[metric space]] and call an [[infinite sequence]] in $X \times X$ _asymptotic_ if, when viewed as a [[pairing|pair]] $(a,b)$ of infinite sequences in $X$, these sequences satisfy the relation given in the Motivation above:
$$ \lim_{n \to \infty} {\|a_n - b_n\|} = 0 .$$
The [[eventuality filters]] of the asymptotic sequences in $X \times X$ form a base for a uniform convergence structure.  The same holds even if $X$ is only a [[pseudometric space]] or an [[extended metric space]]; but if $X$ is only a [[quasimetric space]], then we get only a base for a quasiuniform convergence structure.

Now let $X$ be a [[uniform space]] and let a filter be _asymptotic_ if every [[entourage]] belongs to it.  Indeed, if $E$ is the collection of entourages, then $E$ is itself a filter on $X \times X$ and $\{E\}$ is already a base for this uniform convergence structure.  If $X$ is the underlying uniform space of a metric space, then this is the same uniform convergence structure as in the previous paragraph.  If $X$ is only a [[quasiuniform space]], then we get only a quasiuniform convergence structure.

Now let $X$ be a [[Cauchy space]] and let a filter $F$ on $X
\times X$ be _subbasic asymptotic_ if $F = G \times G$ for some Cauchy filter $G$ on $X$.  Then the subbasic asymptotic filters form a subbase for a uniform convergence structure, and the Cauchy structure induced by this uniform convergence structure agrees with the original Cauchy structure on $X$.

Now let $X$ be a [[convergence space]] and let a filter $F$ on $X \times X$ be _subbasic asymptotic_ if $F = G \times G$ for some convergent filter $G$ on $X$.  Then the subbasic asymptotic filters form a subbase for a quasiuniform convergence structure, which is a (pointwise) uniform convergence structure iff $X$ is [[reciprocal space|reciprocal]] ($R_1$).  In any case, the convergence structure induced by this quasiuniform convergence structure agrees with the original convergence structure on $X$.

If $X$ is a [[group object]] in the category $Conv$ of convergence spaces (that is, $X$ is both a convergence space and a [[group]] such the group operations are [[continuous function|continuous]]), then there are two other ways to make $X$ into a uniform convergence space (left and right, which agree if $X$ is an [[abelian group]]), analogous to the ways in which a [[topological group]] becomes a [[uniform space]].  Specifically, let a [[net]] in $X \times X$ be _asymptotic_ if, when viewed as a [[pairing|pair]] $(a,b)$ of nets (with common domain) in $X$, these nets satisfy a relation analogous to that for asymptotic sequences in a metric space above:
$$ \lim_{n \to \infty} a_n b_n^{-1} = 1 $$
(on one side) or
$$ \lim_{n \to \infty} a_n^{-1} b_n = 1 $$
(on the other side).  The [[eventuality filters]] of the asymptotic nets in $X \times X$ form a base for a uniform convergence structure; indeed, they are precisely the proper asymptotic filters.  The convergence structure induced by this uniform convergence structure agrees with the original convergence structure on $X$ (which must be reciprocal, for essentially the same reason that the topology of a topological group must be).  Probably one can get a quasiuniform convergence structure or four out of a [[monoid object]] in $Conv$, but I haven\'t thought that through.

## References

Pages 72--76 (in Part I) in

* Jan Harm van der Walt (2009); _Generalized solutions of systems of nonlinear partial differential equations_ (PhD thesis); [web](http://upetd.up.ac.za/thesis/available/etd-05242009-122628/).

(This dissertation is not really about uniform convergence spaces, but it has the advantage of being free online.)

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uniform cover

# Uniform covers
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## Idea

A uniform cover is an [[open cover]] in which enough of the [[subsets]] have a uniform minimal size.  This doesn\'t make sense in (say) an arbitrary [[topological space]], but it does make sense in a [[uniform space]] (or a [[uniform locale]]).  In fact, it\'s possible to *define* a uniform space by specifying which covers are uniform.

## Definitions

The most general definition (that we have here so far) is in a [[uniform space]] or [[uniform locale|locale]], but we can write down the definition in other familiar situations as well.

### In a metric space

Let $X$ be a (pseudo)[[metric space]].  A collection $\mathcal{C}$ of subsets of $X$ is a __uniform cover__ if:

*  For some [[positive number]] $\epsilon$, every [[open ball]] of radius $\epsilon$ is contained in some element of $\mathcal{C}$.

### In a topological group or vector space

Let $X$ be a [[topological vector space]] or even a [[topological abelian group]].  A collection $\mathcal{C}$ of subsets of $X$ is a __uniform cover__ if:

*  For some [[neighbourhood]] $N$ of $0$ (the [[identity element]] of the group), every set of the form $a + N$ is contained in some element of $\mathcal{C}$.

If $X$ is a [[topological group]], then we have both left-uniform and right-uniform covers, depending on whether we use $a N$ or $N a$ in the definition above.  (I\'m not sure what the convention is, if any, on which is left and which is right.)

### In a uniform space

Sometimes the notion of uniform cover is taken as axiomatic in the definition of [[uniform space]].  But if we define a uniform space in terms of entourages, then we have:

Let $X$ be a [[uniform space]].  A collection $\mathcal{C}$ of subsets of $X$ is a __uniform cover__ if:

*  For some [[entourage]] $U$, every set of the form $U[a] \coloneqq \{ b\colon X \;|\; a \approx_U b \}$ is contained in some element of $\mathcal{C}$.

This definition subsumes both (pseudo)metric spaces and topological groups (although a nonabelian topological group has two uniform structures, corresponding to the two notions of uniform cover).

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uniform filter

Given a [[ring]] $R$, for any left [[ideal]] $I\subset R$ and a set $S\subset R$ define
$$ (I : S) = \{ r \in R \,|\, r S \subset I \}. $$
This is clearly a left ideal again. The special case $(I:R)$ is a two-sided ideal, namely the maximal ideal of $R$ contained in $I$. If $r\in R$ then we write $(I:r) := (I:\{r\})$.

A [[filter]] $F$ in the [[lattice]] of left ideals of a ring $R$ is a __uniform filter__ if $I\in F$ implies $(I:r)\in F$ for any $r\in R$.  Equivalently, the [[Gabriel composition of filters]] satisfies $F\subset F\bullet \{R\}$.
The Gabriel composition of uniform filters is a uniform filter.
Uniform filters are also called __topologizing__, because a non-empty set of left ideals of $R$ is a uniform filter iff it is the family of left ideals of $R$ which form an open neighborhood of $0$ in a "linear topology" on $R$.

The uniform filters of ideals in a ring $R$ bijectively correspond to [[kernel functor]]s on $R$-$\mathrm{Mod}$ (left exact subfunctors of the identity functor). The correspondence goes as follows. If $F$ is a uniform filter, and $M$ in $R$-$\mathrm{Mod}$, define $\sigma_F M$ as the set of all $m\in M$ such that $m$ is annihilated by some left ideal $I$ in $F$. Conversely, given a kernel functor $\sigma$, define a uniform filter $F^\sigma$ to be the filter whose members are all left ideals $I$ such that $\sigma(R/I)=R/I$.

The most important class of uniform filters are [[Gabriel filter]]s.

* Garcia Román, Manuel; Márquez Hernández, Mercedes; Jara, Pascual; Verschoren, Alain, _Uniform filters_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 40 no. 2 (1999), p. 82-126 [numdam](http://www.numdam.org/item?id=CTGDC_1999__40_2_82_0)
* Pierre Gabriel, [[Des catégories abéliennes]], Bulletin de la Société Mathématique de France __90__ (1962), p. 323-448 [numdam](http://www.numdam.org/item?id=BSMF_1962__90__323_0)

category: algebra
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uniform locale

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# Uniform Locales
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## Idea

A **uniform locale** is to a [[uniform space]] as a [[locale]] is to a [[topological space]].

## Definition

While an ordinary uniform space is defined directly in terms of subsets, and the underlying topology then constructed secondarily, in the absence of an underlying set it seems more convenient to define a uniform locale as additional structure on a given locale, together with an additional axiom which essentially says "the underlying topology is the same as the one we started with."

### Basic notions

For any $E\in Op(X\times X)$ and any [[sublocale]] $A$ of $X$, with inclusion map $i:A\to X$, we write $E[A]$ for the image of $(i\times 1)^*(E)$ of $A\times X$ under the projection $A\times X \to X$.  If $A$ is [[overt space|overt]] (such as if $A$ is an open part of $X$ and $X$ is overt, or if [[excluded middle]] holds), then $E[A]$ is an open part of $X$, and can equivalently be defined as $\bigvee \{ V\in Op(X) \mid Pos(V\cap A) \}$, where $Pos(W)$ means "$W$ is [[positive element|positive]]".  Otherwise, it is merely a sublocale.

Similarly, if $E,F\in Op(X\times X)$, we write $E\circ F$ for the image of $\pi_{23}^*(E) \cap \pi_{12}^*(F)$ under the projection $\pi_{13}:X\times X\times X\to X\times X$.  If $X$ is overt, this is an open part of $X\times X$; otherwise it is merely a sublocale.  We also write $E^{-1}$ for the pullback of $E$ along the twist map $X\times X \cong X\times X$.

### Covering uniformities

An [open] **[[open cover|cover]]** of a locale $X$ is a collection $C \subseteq Op(X)$ of open parts of $X$ whose [[join]] is $X$.  For covers $C_i$, we define:

* $C_1$ **refines** $C_2$, written $C_1 \prec C_2$, if every element of $C_1$ is $\le$ in some element of $C_2$.

* $C_1 \wedge C_2 \coloneqq \{ A \wedge B \;|\; A \in C_1, B \in C_2 \}$; this is also a cover.

* $E_C = \bigcup \{ U\times U \mid U\in C \}$, and $C[A] = E_C[A]$.

* For $A \in Op(X)$, we write $C[A] = E_C[A]$ (see above definition).  If $A$ is overt (such as if $X$ is overt, such as if [[excluded middle]] holds), this is equivalent to $\bigvee \{ B \in C \;|\; A \cap B \text{ is positive } \}$, where positive is in the sense of [[positive element]].

* $C^* \coloneqq \{ C[A] \;|\; A \in C\}$.

We now define a **covering uniformity** on a locale $X$ to be a collection of covers, called **[[uniform covers]]**, such that

1. If $C$ is a uniform cover and $C \prec C'$, then $C'$ is a uniform cover.

2. The cover $\{X\}$ is a uniform cover
and if $C_1, C_2$ are uniform covers, then so is $C_1 \wedge C_2$.

3. If $C$ is a uniform cover, there exists a uniform cover $C'$ such that $(C')^* \prec C$.

4. The collection of uniform covers is __admissible__:
for any open part $A\in Op(X)$, we have
   $$ A = \bigvee \{ B | \exists \;\text{ a uniform cover }\; C \;\text{ such that }\; C[B] \le A \} $$

The last condition is the one saying that "the induced topology is again the topology of $X$."; the other conditions correspond precisely to the [uniform-cover definition](uniform+space#ucov) of a uniform topological space.

### Entourage uniformities

An __entourage__ ([Picado and Pultr](#PP), Section XII.1.1)
is an open part $E\in Op(X\times X)$
such that $\{u\in Op(X)\mid u\times u\le E\}$
is an open cover of $X$.

An **entourage uniformity** ([Picado and Pultr](#PP), Section XII.2.3) on a locale $X$ is a collection of [[entourages]] such that:

1. If $E$ is an entourage and $E\subseteq F$, then $F$ is also an entourage.

2. The open part $X\times X$ is an entourage
and if $E$ and $F$ are entourages, then so is $E\cap F$.

3. If $E$ is an entourage, then so is $E^{-1}$.

4. For any entourage $E$, there exists an entourage $F$ such that $F\circ F \subseteq E$.  The sublocale $F\circ F$ is not always open, but we can still ask it to be contained in the open sublocale $E$.

5. The collection of uniform entourages is __admissible__:
for any $U\in Op(X)$ we have $U = \bigvee \{ V\in Op(X) \mid E[V] \subseteq U \text{ for some uniform entourage } E\}$.

### From entourage uniformities to covering uniformities

Given an entourage $E$ on a locale $L$, we define an open cover $U_E$ by setting
$$U_E=\{x\in L\mid x\times x\subset E\}.$$

Given an entourage uniformity $\{E_i\}_i$, we construct
a covering uniformity out of it by taking all open covers $V$
that are refined by some $U_{E_i}$.

### From covering uniformities to entourage uniformities

Given an open cover $U$, we construct an entourage $E_U$
by setting $$E_U=\bigvee_{x\in U}x\times x.$$

Given a covering uniformity $\{U^i\}_i$, we construct an entourage uniformity
out of it by taking all entourages $E$ that contain some $E_{U^i}$.

### Equivalence between covering uniformities and entourage uniformities

Theorem XII.3.3.4 in [Picado and Pultr](#PP) shows that the above correspondence is bijective.  Furthermore, the categories of entourage uniform locales
and covering uniform locales are equivalent (Corollary XII.3.4.3 in the cited book).  However, since their book uses [[classical logic]] throughout, it is not entirely clear whether the same equivalence holds constructively.

## Properties

### Uniformizability

A [[locale]] admits a uniformity if and only if it is [[completely regular]]
([Picado-Pultr, Theorem 2.8.2](#PP)).

### Fine uniformity

Uniformities are closed under unions,
so any [[completely regular]] [[locale]] admits a largest uniformity,
the __fine uniformity__.

The fine uniformity consists of all [[normal covers]] (alias [[numerable covers]]),
where a cover $U$ is __normal__
if there is a sequence of covers $\{U_n\}_{n\ge0}$
such that $U=U_0$ and $U_{n+1}U_{n+1}$ refines $U_n$.

### Compact regular locale

On a [[compact]] [[regular]] [[locale]] the system
of all covers forms a uniformity.
Maps out of [[compact regular locales]]
are automatically uniform.

### Complete uniform locales

A uniform locale $L$ is __complete__
if every dense uniform embedding $L\to X$
is an isomorphism of uniform locales.

The category of complete uniform locales
is a [[reflective subcategory]] of uniform locales.
The reflection functor is known as the __completion__ functor.

Concretely, the completion of a uniform locale $L$
with a system of uniform covers $A$
can be described as follows.
Its opens are downward-closed subsets $U$
of the underlying [[frame]] of $L$
that satisfy the following saturation conditions:

* If for some $a\in L$ we have $\{x\in L\mid x\lt_A a\}\subset U$,
then also $a\in U$.
Here $x\lt_A a$ means that there is $C\in A$
such that $C[x]\le a$.

* If for some $C\in A$ and $a\in L$ we have $a\wedge C\subset U$,
then also $a\in U$.

## Criteria for completeness

Every compact regular locale is complete.
The fine uniformity on a [[paracompact locale]] is complete.

Isbell's theorem states that a [[locale]] is [[paracompact]]
if and only if it admits a complete uniformity.

A [[completely regular]] [[locale]] is [[compact]]
if and only if it admits a complete uniformity
that has a basis in which all covers are finite.

## See also

* [[uniform space]]

* [[preuniform locale]], [[uniform locale]]

* [[premetric locale]], [[metric locale]]

## References

A detailed treatment of uniform locales can be fund in

* {#PP} [[Jorge Picado]], [[Aleš Pultr]], _[[Frames and Locales]]_.

The following paper developes covering uniformities constructively, and includes citations to several other papers that do it classically:

* [[Peter Johnstone]], 1989; A constructive theory of uniform locales. I. Uniform covers; General topology and applications, 179--193; Lecture Notes in Pure and Applied Mathematics 134 (1991).
 {#Johnstone89}

These ideas are developed further in

* Graham Manuell, _Uniform locales and their constructive aspects_, [arxiv:2106.00678](https://arxiv.org/abs/2106.00678)

A constructive and predicative theory in the programme of [[formal topology]] can be found here:

* [[Giovanni Curi]], *On the collection of points of a formal space*, Annals of Pure and Applied Logic* **137** 1-3 (2006)  126-146 $[$[doi:10.1016/j.apal.2005.05.019](https://doi.org/10.1016/j.apal.2005.05.019)$]$

* [[Giovanni Curi]]; Constructive metrisability in point-free topology; Theoretical Computer Science 305 (2003) 1--3, 85--109.

[[!redirects uniform locale]]
[[!redirects uniform locales]]

[[!redirects pointless uniformity]]
[[!redirects pointless uniformities]]

[[!redirects complete uniformity]]
[[!redirects complete uniformities]]

[[!redirects fine uniformity]]
[[!redirects fine uniformities]]


uniform module

Given a (possibly noncommutative) unital ring $R$ (left or right) nonzero $R$-module $M$ is __uniform__ if the intersection of any two nonzero submodules of $M$ is nonzero, or, equivalently, such
that every nonzero submodule of $M$ is [[essential submodule|essential]] in $M$.

A nonzero module $M$ is uniform iff its [[injective envelope]] $E(M)$ is indecomposable. In particular, if
$M$ is itself injective, then it is uniform iff it is indecomposable.

An arbitrary $R$-module $M$ has finite rank if and only if it has an essential submodule which is a finite direct sum of uniform submodules.

* K. R. Goodearl, R. B. Warfield Jr. _An introduction to noncommutative Noetherian rings_, London Math. Soc. Student Texts 61, 2nd ed. 2004

category: algebra
[[!redirects uniform modules]]


uniform property

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## Definition

A _uniform property_ is a [[property]]/[[predicate]] of [[uniform spaces]] which is [[invariant]] under [[isomorphisms]] of uniform spaces.

## Related concepts

* [[topological property]]

* [[epsilontic analysis]]

## References

* Wikipedia, _[Uniform property](https://en.wikipedia.org/wiki/Uniform_property)_

[[!redirects uniform properties]]


uniform space

[[!redirects uniform space draft]]
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# Uniform spaces
* table of contents
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## Idea

Uniform spaces were invented by [[André Weil]], to capture a general notion of [[space]] for which it makes sense to speak of [[uniformly continuous map]]s. Such spaces include (pseudo)[[metric spaces]] and [[topological groups]].

A _uniform structure_ or _uniformity_ on a [[set]] $X$ consists of a collection of global binary [[relations]] $\varepsilon$ which allow us to say when one point $x$ is "$\varepsilon$-close" to another $y$. For a metric space, we have for instance relations expressed by the formula $d(x, y) \lt \varepsilon$, where $\varepsilon$ is a [[positive number]]. For a [[topological group]], we have relations $x y^{-1} \in \varepsilon$, where $\varepsilon$ is a [[neighborhood]] of the identity.

## Definitions

The definition described above is based on [[entourages]]; it was the original one published in [[Bourbaki]] and is still most commonly seen today.  There is an equivalent definition based on [[uniform covers]], although it is more difficult to generalize in some ways (e.g. to [[quasiuniform spaces]]).  The relationship with [[gauge spaces]] (defined below) also allows for another definition. The original definition in [[Bourbaki]] [can be seen](#naive_tame_topology) as describing a contravariant functor from the category of finite sets to a category of [[filters]] of subsets.

### Entourage uniformities
{#ent}

A **uniform structure**, or **uniformity**, on a set $X$ consists of a collection of [[binary relations]] $U \subseteq X \times X$ (called __entourages__ or __vicinities__) satisfying some conditions. Write $x \approx_U y$ if $x$ is related to $y$ through $U$; then the conditions are the following:

1. The [[equality relation]] $\Delta = \{(x, x): x \in X\}$ is contained in every entourage. That is,
   $$ \forall U,\; \forall x,\; x \approx_U x .$$
2. For every entourage $U$, there exists an entourage $V$ such that $V \circ V \subseteq U$, where $\circ$ is the ordinary operation of [[relation|relational composition]]. That is,
   $$ \forall U,\; \exists V,\; \forall x, y, z,\; x \approx_V y \approx_V z \;\Rightarrow x \approx_U z .$$
3. For every entourage $U$, there exists an entourage $V$ such that $V^op \subseteq U$, where $V^\op$ is the [[opposite relation]] to $V$. That is,
   $$ \forall U,\; \exists V,\; \forall x, y,\; y \approx_V x \;\Rightarrow\; x \approx_U y .$$
   In light of axiom (6), it follows that $U^op$ itself is an entourage.
4. There exists an entourage; in light of axiom (6), it follows that the universal relation $X \times X$ is an entourage.
5. If $U, V$ are entourages, so is some subset of $U \cap V$. In light of axiom (6), it follows that $U \cap V$ is an entourage.
6. If $U$ is an entourage and $U \subseteq V \subseteq X \times X$, then $V$ is an entourage.

A set equipped with a uniform structure is called a **uniform space**.

A collection of entourages satisfying (1--5) is a **[[base]]** for a uniformity; a base is precisely what generates a uniformity by taking the upward closure.  A collection satisfying (1--3) is a **preuniformity**. Slightly more generally, we can replace each $V$ in the statement of these axioms with a finite [[intersection]] $V_1 \cap \cdots \cap V_n$ of entourages to get the concept of a **[[subbase]]** for a uniformity.  A subbase is precisely what generates a base by closing up under finite intersections and precisely what generates a uniformity by closing up under finite intersections and taking the upward closure.

#### As a ternary relation
A **uniform structure** or **uniformity** could be represented as a family of binary relations on a set $X$ indexed by a set $S$ of **entourages** or **vicinities**, or equivalently by [[currying]], as a ternary relation $(-)\approx_{(-)}(-):X \times S \times X \to \Omega$, where $\Omega$ is the set of [[truth values]], such that

1. for every entourage $U \in S$ and $x \in X$, $x \approx_U x$

2. for every entourage $U \in S$, there exists an entourage $V \in S$ such that for every $x \in X$, $y \in X$, and $z \in X$, if $x \approx_V y$ and $y \approx_V z$, then $x \approx_U z$.

3. For every entourage $U \in S$, there exists an entourage $V \in S$ such that for every $x \in X$ and $y \in X$, if $x \approx_V y$, then $y \approx_U x$.

4. There is an entourage $U \in S$ and elements $x \in X$ and $y \in X$ such that $x \approx_U y$.

5. If there are entourages $U \in S$ and $V \in S$, then there is an entourage $T \in S$ such that for every element $x \in X$ and $y \in X$, if $x \approx_T y$, then $x \approx_U y$ and $x \approx_V y$.

6. If there is an entourage $U \in S$, then there is an entourage $V \in S$ such that for all $x \in X$ and $y \in X$, if $x \approx_U y$, then $x \approx_V y$.

### Covering uniformities {#ucov}

An equivalent way to characterize a uniform space is by its collection of *[[uniform covers]]*.  Here a **cover** of a set $X$ is a collection $C \subseteq P(X)$ with [[union]] $X$.  For covers $C_i$, we define:

* $C_1$ **refines** $C_2$, written $C_1 \prec C_2$, if every element of $C_1$ is a subset of some element of $C_2$.

* $C_1 \wedge C_2 \coloneqq \{ A \cap B \;|\; A \in C_1, B \in C_2 \}$; this is also a cover.

* For $A \subseteq X$, $C[A] \coloneqq \bigcup \{ B \in C \mid \mathrm{isInhabited}(A \cap B)\}$.

* $C^* \coloneqq \{ C[A] \;|\; A \in C\}$.

We now define a **covering uniformity** on $X$ to be a collection of covers, called **uniform covers**, such that

1. If $C$ is a uniform cover, there exists a uniform cover $C'$ such that $(C')^* \prec C$.

2. There exists a uniform cover; in light of axiom (4), it follows that the cover $\{X\}$ is a uniform cover.

3. If $C_1, C_2$ are uniform covers, so is some cover that refines $C_1 \wedge C_2$. In light of axiom (4), it follows that $C_1 \wedge C_2$ is a uniform cover.

4. If $C$ is a uniform cover and $C \prec C'$, then $C'$ is a uniform cover.

The axioms (2--4) here roughly correspond (respectively) to the axioms (4--6) in the entourage definition.  Axiom (1) takes on all of the real work; any collection of covers that satisfies it may be called a __[[subbase]]__ (but not corresponding directly to a subbase in the previous definition), and then anything satisfying (1--3) is a __[[base]]__.

If $X$ is a uniform space defined in terms of entourages, we give it a covering uniformity by declaring a cover to be uniform if it is refined by $\{ U[x] \;|\; x \in X\}$ for some entourage $U$, where $U[x] \coloneqq \{ y \;|\; x \approx_U y \}$.  Note that this does not mean that a uniform cover "consists of $U$-sized sets" but only that it contains a subcover consisting of sets "no smaller than $U$".

Conversely, given a covering uniformity, we define a base of entourages to consist of sets of the form $\bigcup \{ A \times A \;|\; A \in C\}$ for $C$ a uniform cover.  That is, for each cover $C$, we have a basic entourage $\approx_C$ such that $x \approx_C y$ iff $x, y \in A$ for some $A \in C$.  This defines a bijection between entourage uniformities and covering uniformities.

### Further definitions

We give these in terms of entourages, but they could also be given directly in terms of uniform covers if desired.

The **uniform topology** induced by a uniformity is defined by taking the [[neighborhoods]] of a point $x$ to be sets of the form
$$ U[x] \coloneqq \{ y \;|\; x \approx_U y \} $$
where $U$ is an entourage. (Recall that a subset is [[open subset|open]] iff it is a neighborhood of every point that it contains.) We point out that different uniformities may give rise to the same topology (just as different metrics, even uniformly inequivalent ones, may give rise to the same topology).

In [[classical mathematics]], the uniform topology is always [[regular space|regular]] and indeed [[completely regular space|completely regular]].  In [[constructive mathematics]] this is not necessarily true, and to reproduce classical results it is often useful or necessary to assume that uniform spaces are regular, completely regular, or satisfy an intermediate condition called [[uniformly regular space|uniform regularity]].

Given uniform spaces $X$ and $Y$, a [[function]] $f\colon X \to Y$ is said to be **[[uniformly continuous map|uniformly continuous]]** if for every entourage $V$ of $Y$, $(f \times f)^{-1}(V)$ is an entourage of $X$. Clearly, uniformly continuous functions are [[continuous map|continuous]] with respect to the corresponding uniform topologies, but the converse is false (although see below the discussion in the case where $X$ is [[compact space|compact]]). There is an obvious [[concrete category]] $Unif$ of uniform spaces and uniformly continuous maps.

One feature of uniform space theory which is not available for general topological spaces is the possibility of taking (Cauchy) completions. The relevant definitions are straightforward:

* A **[[Cauchy net]]** in a uniform space $X$ consists of a [[directed set]] $D$ and a function $f\colon D \to X$ such that for every entourage $U$, there exists $N \in D$ such that $f(m) \approx_U f(n)$ whenever $m, n \geq N$. (One can similarly define a **Cauchy filter**.  This definition makes a uniform space into a [[Cauchy space]].)

* A Cauchy net $f\colon D \to X$ **[[convergence|converges]]** to $x\colon X$ if for every entourage $U$, there exists $N \in D$ such that $n \geq N \;\Rightarrow\; x \approx_U f(n)$.  (This makes sense for arbitrary [[nets]] or [[filters]], but it can be proved that any convergent net is Cauchy.  This definition makes a uniform space into a [[convergence space]].)

* A uniform space $X$ is **[[complete space|complete]]** if every Cauchy net/filter in $X$ converges (not necessarily to a unique point).

* A uniform space $X$ is **[[Hausdorff space|Hausdorff]]** or **separated** if every convergent net/filter converges to a unique point, or equivalently if $x = y$ whenever $x \approx_U y$ for every entourage $U$.  (This is a purely topological concept.)

Every uniform space $X$ admits a Hausdorff completion $\overline{X}$, i.e., there is a uniformly continuous map $X \to \overline{X}$ (an [[embedding]] if $X$ is Hausdorff, [[dense map|dense]] in any case), characterized by the following universal property:

* Every uniformly continuous map $f\colon X \to Y$ to a complete Hausdorff uniform space $Y$ extends uniquely to a uniformly continuous map $\overline{f}\colon \overline{X} \to Y$.

In short, the category of complete Hausdorff uniform spaces is a [[reflective subcategory]] of $Unif$.

One can also define a (not necessarily Hausdorff) completion of $X$ by replacing the image of $X$ in $\overline{X}$ by $X$ itself, but this does not have as nice properties; in particular, complete uniform spaces do not form a reflective subcategory of $Unif$.

Every uniform space also has an underlying [[proximity]] (defined there), and the resulting functor $Unif \to Prox$ has a fully faithful right adjoint which identifies proximity spaces with uniform spaces that are [[totally bounded space|totally bounded]].

Uniform spaces can also be identified with [[syntopogenous spaces]] that are both *perfect* and *symmetric*; see [[syntopogenous space]].

Every uniform space $X$ has a [[inequality relation]] (an [[irreflexive]] [[symmetric relation]]) where "$x # y$" means that there exists an entourage $U$ such that $x \mathbin{≉}_U y$.  If $X$ is [[uniformly regular space|uniformly regular]], then this is an [[apartness relation]], i.e. it is also a [[comparison]].    This inequality is tight exactly when the uniform space is Hausdorff.

## The category of uniform spaces

Uniform spaces and uniform maps form a [[topological category]].

In particular, [[limits]] and [[colimits]] of uniform spaces are computed similarly to the case of [[topological spaces]]: take the limit/colimit of underlying sets and equip it with the final/initial uniform structure.

See Chapter 21 in [[Joy of Cats]] for more information.

## Examples

Every [[metric space]] (or more generally any pseudometric space) is a uniform space, with a base of uniformities indexed by positive numbers $\epsilon$.  (You can even get a countable base, for example by using only those $\epsilon$ equal to $1/n$ for some integer $n$.)  Define $x \approx_\epsilon y$ to mean that $d(x,y) \lt \epsilon$ (or $d(x,y) \leq \epsilon$ if you prefer).  Then axiom (2) may be proved by using $\epsilon/2$; similarly, every metric space is [[uniformly regular space|uniformly regular]] in constructive mathematics, which may be proved by using any positive number less than $\epsilon$ and applying the [[comparison]] law.  (The other axioms are easy.)  Every quasi(pseudo)metric space is a quasiuniform space in the same way.  We can also generalise from metric spaces to [[gauge spaces]]; see under Variations below.

Every [[topological group]] is a uniform space, with a base of uniformities indexed by neighbourhoods $U$ of the identity element, in two ways: left and right.  (These two ways agree for [[abelian groups]], of course; they also agree for [[compact group]]s, by the general theorem below for uniformities on [[compact spaces]].  I wonder if that has anything to do with [[Haar measure]]?)  In particular, any [[Banach space]] or [[Lie group]] is a uniform space.  Define $x \approx _U y$ to mean that $x \in y U$ (or $y \in x U$ for the other way).  Then axiom (2) may be proved by invoking the continuity of multiplication; constructively, we cannot prove that every topological group is uniformly regular, although this can be proved for the classical examples of Lie groups and [[topological vector spaces]] (TVSs).

These are in a way the motivating examples.  The theory of uniformly continuous maps was developed first for metric spaces, but it was noticed that, for a [[metrisable]] TVS, it could be described entirely in terms of the topology and the addition, which immediately generalised to non-metrisable TVSs.  The theory of uniform spaces covers both of these (and their generalisations to pseudometric spaces and topological groups) at once.

We can also form certain uniformities on function spaces.  For instance, if $Y$ is a uniform space and $X$ is any set, then the set of functions $Y^X$ has a "uniformity of [[uniform convergence]]"; this is generated by a base of entourages consisting of, for each entourage $U$ of $Y$, the generating entourage
$$ \tilde{U} = \{ (f,g) \mid \forall x\in X, (f(x),g(x))\in U \}. $$
This can be defined for any set $X$, but it is best-behaved when $X$ has some [[compact space|compactness]] properties.  For instance, in [[constructive mathematics]], to prove uniform regularity of $X$ from uniform regularity of $Y$ we seem to need a condition such as that $X$ is a [[covert set]].

## Basic Results

A first wave of results concerns [[separation axioms]]:

* Considered as topological spaces, uniform spaces are [[regular space|regular]] and (more profoundly) [[completely regular space|completely regular]].

As a matter of fact, a significant theorem is that a topology (on a given set) is the uniform topology for some uniformity if and only if it is completely regular.  See also the discussion below on the relation with metric and pseudometric spaces.

* For a uniform space, the following separation conditions are equivalent: $T_0$ (the topology distinguishes points), $T_1$ (points are closed), $T_2$ ([[Hausdorff space|Hausdorff]]), $T_3$ (regular Hausdorff), $T_{3\frac1{2}}$ (completely regular Hausdorff). Of course, this follows from the fact that it is completely regular.

* The category $Unif$ of uniform spaces admits arbitrary small [[products]] (which are preserved by the forgetful functors to [[Top]] and to [[Set]]). Hence it is not generally true that uniform spaces are [[normal space|normal]] (so that separated ones would be $T_4$), because for instance an uncountable power of the [[real line]] (with its usual topology) is not a normal space.

A second wave of results relates uniform spaces to [[pseudometric spaces]] (like metric spaces, but dropping the separation axiom that $d(x, y) = 0$ implies $x = y$):

* Every uniform space embeds uniformly in a product of pseudometric spaces. A uniform space whose uniformity admits a countable subbase is uniformly isomorphic to a pseudometric space (and hence to a metric space if the uniform space is separated, in which case the uniform space is called __metrisable__).

By this result, the results mentioned above on completions of uniform spaces may be proved by appeal to similar results for (pseudo)metric spaces.

We also mention the special case of compact spaces:

* A [[compact Hausdorff space]] $X$ admits a unique uniformity whose corresponding topology is the topology of $X$. The entourages of this uniformity are precisely the neighborhoods of the diagonal in $X \times X$.

It follows then that if $X, Y$ are uniform spaces with $X$ compact Hausdorff and if $f\colon X \to Y$ is continuous, then $f$ is uniformly continuous.

Finally, uniform spaces and uniformly continuous maps form a [[category]] $Unif$.  This category admits a [[closed monoidal category|closed monoidal structure]] in which the function-space $[X,Y]$ is the set of uniformly continuous maps $X\to Y$ with the uniformity of [[uniform convergence]] (mentioned above).  The corresponding tensor product is called the "semi-uniform product", and is *not* [[symmetric monoidal category|symmetric]], because the evaluation map $X \to [[X,Y],Y]$ is not generally uniformly continuous.  See [Isbell, Chapter III, particularly Theorem 26 p46](#Isbell)

On the other hand, $Unif$ also has a different closed *symmetric* monoidal structure in which the internal-hom $\{X,Y\}$ is the space of uniformly continuous functions with the uniformity of *pointwise* convergence, i.e. the subspace uniformity induced by the product uniformity on $Y^X$.  See [Isbell, exercise III.7, p53](#Isbell)

## Variations

Some authors insist that a uniform space must be separated; this can be arranged directly in the definition (that is without reference to the uniform topology or to the concept of convergent net/filter) by adding the following axiom, a sort of converse axiom (1):

* If $x \approx_U y$ for every entourage $U$, then $x = y$.

This makes the discussion of completions slightly simpler.

If the symmetry axiom (3) in the entourage definition is dropped, then the result is a __quasiuniform space__.  Quasiuniform spaces are related to quasi(pseudo)metrics in the same way as uniform spaces are related to (psuedo)metrics.  Perhaps surprisingly, *every* topological space is quasiuniformisable.  (It is rather triciker to define quasiuniform spaces in terms of covers, but technically possible using covers by pairs of sets; see [Gantner and Steinlage](#GS72).)

A set with a preuniformity or subbase (only axioms 1-3) is a **preuniform space**.

A __[[gauge space]]__ consists of a set $X$ and a collection $\mathcal{D}$ of pseudometrics on $X$; one usually requires $\mathcal{D}$ to be a [[filter]].  A gauge space defines a uniform space (necessarily uniformly regular) by taking one basic entourage for each pseudometric in $\mathcal{D}$ and each positive number $\epsilon$; conversely, every uniform space arises in this way, with the pseudometrics in the gauge being those that are uniformly continuous as maps on the product space.  However, gauge spaces form a category with a stricter notion of morphism, in which the categories $Met$ (of [[metric spaces]] and short maps) and $Unif$ (of uniform spaces and uniformly continuous maps) are both [[full subcategories]].  A __quasigauge space__ consists of a set and a collection of quasipseudometrics; every quasiuniform space arises from a quasigauge space.

In weak [[foundations]] of mathematics, the theorems above may not be provable.  In particular, the theorem that every uniform space arises from a gauge space is equivalent (internal to an arbitrary [[topos]] with a [[natural numbers object]]) to [[dependent choice]] (plus [[excluded middle]] if you don\'t require the uniform space to be uniformly regular).  If the concept is to be applied to analysis, then it may be best to define a uniform space as a gauge space satisfying a saturation condition.

There is also a "pointless" notion of uniform space, called a [[uniform locale]].

## Motivation for the axioms

The really critical axioms are (1--3): a collection of binary relations which satisfies those three axioms is a subbase (although not every subbase takes this form), from we can generate a uniform structure straightforwardly. Indeed, (4--6) simply state that the entourages form a [[filter]], so generating a uniformity from a base or subbase is simply the usual generation of a filter from a base or subbase.

We draw particular attention to axiom (2), which may be called an "$\frac{\varepsilon}{2}$" principle. It generalizes a principle familiar from analysis in metric spaces, where one establishes $d(x, z) \lt \varepsilon$ by showing there exists $y$ such that $d(x, y) \lt \frac{\varepsilon}{2}$ and $d(y, z) \lt \frac{\varepsilon}{2}$, and applying the triangle inequality. The utility of this principle for metric spaces, extrapolated in this way, gives uniform spaces much of their power.

For full power in [[constructive mathematics]], we also need [[uniformly regular space|uniform regularity]], which may similarly be called a "something less than $\varepsilon$" principle.  (That is, for any $\varepsilon$ there is an $\varepsilon'$ such that any two points are either $\varepsilon$-close or $\varepsilon'$-far; classically we may take $\varepsilon'$ to be $\varepsilon$, but constructively it\'s better to think of $\varepsilon' \lt \varepsilon$.)  This can actually be combined with axiom (2) into a single statement, as you might expect since $\frac{\varepsilon}{2} \lt \varepsilon$, but that makes the intuition less clear.

Axiom (1) is a nullary version of axiom (2); together they prove that, given any entourage $U$ and any integer $n \geq 0$, there exists an entourage $V$ whose $n$-fold composite is contained in $U$. The symmetry axiom (3) then allows one to take the opposite of $V$ at any point in the composite as well.

Altogether, these may be seen as axiomatising the notion of *approximate* [[equivalence relation|equivalence]].  If $\approx$ is an approximate equivalence relation, then we might expect it to be

*  [[reflexive relation|reflexive]]: $x \approx x$ and
*  [[symmetric relation|symmetric]]: $x \approx y$ iff $y \approx x$, but
*  NOT [[transitive relation|transitive]]: $x \approx y \approx z$ does not necessarily mean that $x \approx z$.

One could stop there, but this is not a very useful notion of approximation.  Instead we generalise to a *family* of approximate equivalence relations and impose the $\frac{\varepsilon}{2}$ principle to allow them to be used.  This is nearly the definition of uniform space; in particular, the axiom (1) states precisely that each entourage is reflexive.  The symmetry axiom (3) in the standard definition is weaker than requiring each individual entourage to be symmetric, but that is not an essential change; every uniformity has a base consisting of its symmetric entourages.  The final three axioms have already been explained as a closure condition; they force equivalent uniformities on a given set (in the sense that the [[identity function]] on the set is uniformly continuous either way) to be equal.

## Categorical interpretation

The idea that a uniformity is an "approximate equivalence relation" can be made precise as follows.  A [[preorder]] is the same as a category [[enriched category|enriched]] over the poset $\mathbb{2}$ of [[truth values]], and it is an equivalence relation if and only if this category is symmetric.  In fancier language, a preorder is a [[monoid]] (or [[monad]]) in the [[bicategory]] $Rel = \mathbb{2} Mat$.  A quasi-uniform space can then be identified with a monoid in the bicategory $Pro Rel$, whose hom-categories are the categories of [[pro-objects]] in the hom-categories of $Rel$, aka [[filters]].  Of course, it is a uniform space just when it is also symmetric.  See also [[prometric space]].

In all these cases, in order to recover the correct notion of morphism abstractly, we must consider monoids in a [[double category]] or [[equipment]] rather than merely a bicategory.

With a uniform structure on a set $X$ associate a contravariant functor from the
 category of finite sets to a category of filters of subsets as follows.
Call a subset $P$ of $X^S$ _big_ iff there is an entourage $U$ such that $x\in X
^S$ provided $(x(s'),x(s''))\in U$ for each $s',s''\in S$. For a metric space, this is the filter of $\epsilon$-neighbourhoods of the diagonal.
This defines a contravariant functor $F_X$ from the category of finite sets to the category of filters of subsets. Axiom 1 says that there is a unique big subset in $F_X(\{\bullet\})$. Axiom 2 says that the filter on $F_X(\{1,2,3\})$ is the pullback of the obvious maps $F_X(\{1,2,3\})\rightarrow F_X(\{1,2\})\rightarrow F_X(\{2\})\leftarrow F_X(\{2,3\})\leftarrow F_X(\{1,2,3\})$. To give a uniformly continuous map (morphism) $f:X\rightarrow Y$ of uniform structures is the same as to give a natural transformation $F_X \Rightarrow F_Y$. Indeed, such a natural transformation is determined by the function $f_{\{\bullet\}}:F_X(\{\bullet\})\rightarrow F_Y(\{\bullet\})$, and continuity of $F_X(\{1,2\})\rightarrow F_Y(\{1,2\})$ mean uniform continuity.

With a filter $F$ on $X$ associate the functor $E_F$ as follows: $E_F(S)$ is $F^S$.
A filter $F$ on $X$ is Cauchy iff the obvious map $E_F\rightarrow F_X$ is well-defined.

## Related concepts

* [[uniform locale]]

* [[equicontinuous family of functions]]

* [[complete uniform space]] and uniform completion

* [[uniform convergence]]

* [[uniformly continuous map]]

[[!include generalized uniform structures - table]]

## References

*  [[André Weil]], _Sur les espaces à structure uniforme et sur la topologie générale_, Actualités Sci. Ind. 551, Paris, (1937).

* [[John Kelley]], _General Topology_, GTM 27, 1955.

* [[Nicolas Bourbaki]], *Uniform Structures*, Chapter II in: *General topology*, Elements of Mathematics, Springer (1971, 1990, 1995) [[doi:10.1007/978-3-642-61701-0](https://doi.org/10.1007/978-3-642-61701-0)]

* Eric Schechter, _[[Handbook of Analysis and its Foundations]]_ (1996)

* Warren Page, _Topological Uniform Structures_, Dover

* I.M. James, _Topologies and Uniformities_, Springer

* Norman Howes, _Modern analysis and topology_, Springer

* P. Samuel, _Ultrafilters and compactifications of uniform spaces_, Trans. Amer. Math. Soc. __64__ (1948) 100--132

* {#Isbell} J. R. Isbell, _Uniform spaces_, Math. Surveys __12__, Amer. Math. Soc. 1964

* T. E. Gantner and R. C. Steinlage, _Characterizations of quasi-uniformities_, J. London Math Soc (2) 5 (1972) (defines quasi-uniformities using covers)
 {#GS72}

* {#naive_tame_topology} M. Gavrilovich, [_A naive diagram-chasing approach to formalisation of tame topology_](http://mishap.sdf.org/by:gavrilovich/mintsGE.pdf#3.1.6), arXiv:[1807.06986](https://arxiv.org/abs/1807.06986), Sections 3.1.2, 3.1.6.

category: topology

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uniformly continuous map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
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#### Topology
+--{: .hide}
[[!include topology - contents]]
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# Uniformly continuous maps
* table of contents
{: toc}

## Idea

Recall that a [[continuous map]] $f$ between [[spaces]] $X$ and $Y$ has the property that $f$ maps nearby points to nearby points, which may be formalised by first picking one point, then considering how nearby you want the points to be, then picking another point sufficiently nearby.

The concept of _uniformly continuous map_ $f$ is based on the same intuition but a different formalisation: first you pick how nearby you want the points to be, then you pick two points sufficiently nearby.  This results in a stronger criterion, definable in a less general context.

Traditionally this is formalized

* for [[Archimedean fields]] (def. \ref{BetweenArchimedeanFields} below)

* for [[metric spaces]] (def. \ref{BetweenMetricSpaces} below)

but the definition makes sense more generally

* for [[uniform spaces]] (def. \ref{BetweenUniformSpaces} below)

and in fact

* for [[quasi-uniform spaces]] (def. \ref{BetweenQuasiuniformSpaces} below).

## Definitions

+-- {: .num_defn #BetweenUniformSpaces}
###### Definition
**(uniform continuous map between [[uniform spaces]])

Let $X$ and $Y$ be [[uniform spaces]], each defined as a [[set]] equipped with a collection of [[entourages]].  A __uniformly continuous map__ from $X$ to $Y$ is a [[function]] between their [[underlying set]]s such that, given any [[entourage]] $E$ on $Y$, there is an entourage $D$ on $X$ such that $f(a)$ and $f(b)$ are $E$-close in $Y$ whenever $a$ and $b$ are $D$-close in $X$:
$$ \forall\, E\colon \mathcal{U}Y,\; \exists\, D\colon \mathcal{U}X,\; \forall\, a, b\colon X,\; a \approx_D b \;\Rightarrow\; f(a) \approx_E f(b) .$$

=--

A definition may also be given in terms of [[uniform covers]].

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

The definition \ref{BetweenUniformSpaces} is exactly like the definition of _[[continuous map]]_ between uniform spaces, except for the order of the [[quantifiers]] $\exists\, D$ and $\forall\, a$.

=--

+-- {: .num_defn #BetweenQuasiuniformSpaces}
###### Definition
**(uniformly continuous map between [[quasiuniform spaces]])**

The definition \ref{BetweenUniformSpaces} in terms of [[entourages]] extends immediately to [[quasiuniform spaces]], in which case we may speak of __quasiuniformly continuous maps__ since some authors use 'uniformly continuous' for a map which is uniformly continuous between the spaces\' symmetrisations.

=--

+-- {: .num_defn #AntiuniformlyContinuousMaps}
###### Definition
**(antiuniformly continuous map)**

An __antiuniformly continuous map__, is defined as a uniform map, but such that the order in which the points are compared is reversed:
$$ \forall\, E\colon \mathcal{U}Y,\; \exists\, D\colon \mathcal{U}X,\; \forall\, a, b\colon X,\; a \approx_D b \;\Rightarrow\; f(b) \approx_E f(a) .$$

=--

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

Between uniform spaces viewed as symmetric quasiuniformly continuous spaces, quasiuniformly continuous maps (def. \ref{BetweenQuasiuniformSpaces}), antiuniformly continuous maps (def. \ref{AntiuniformlyContinuousMaps}), and uniformly continuous maps (def. \ref{BetweenUniformSpaces}) are the same.

=--

In the particular case of [[metric spaces]], it is common to see this definition in elementary form:

+-- {: .num_defn #BetweenMetricSpaces}
###### Definition
**(uniformly continuous map between [[metric spaces]])**

Given [[metric spaces]] $X$ and $Y$, a __uniformly continuous map__ from $X$ to $Y$ is a function $f\colon X\to Y$ between their underlying sets such that, given any [[positive number|positive]] [[real number]] $\epsilon$, there is a positive number $\delta$ such that the distance in $Y$ between $f(a)$ and $f(b)$ is less than $\epsilon$ whenever the distance in $X$ between $a$ and $b$ is less than $\delta$:
$$ \forall\, \epsilon \gt 0,\; \exists\, \delta \gt 0,\; \forall\, a, b\colon X,\; d_X(a, b) \lt \delta \;\Rightarrow\; d_Y(f(a), f(b)) \lt \epsilon .$$
Again, this is exactly like the definition of [[continuous map]] between metric spaces, except for the order of the quantifiers $\exists\, \delta$ and $\forall\, a$.

=--

+-- {: .num_defn #BetweenArchimedeanFields}
###### Definition
**(uniformly continuous map between [[Archimedean fields]])**

Given [[Archimedean fields]] $X$ and $Y$, a __uniformly continuous map__ from $X$ to $Y$ is a function between their underlying sets such that, given any positive element $\epsilon \gt 0$, there is a positive element $\delta \gt 0$ such that the maximum of $f(a) - f(b)$ and $f(b) - f(a)$ is less than $\epsilon$ whenever the maximum of $a - b$ and $b - a$ is less than $\delta$:
$$ \forall\, \epsilon \gt 0,\; \exists\, \delta \gt 0,\; \forall\, a, b\colon X,\; \max(a - b, b - a) \lt \delta \;\Rightarrow\; \max(f(a) - f(b), f(b) - f(a)) \lt \epsilon .$$
Again, this is exactly like the definition of [[continuous map]] between Archimedean fields, except for the order of the quantifiers $\exists\, \delta$ and $\forall\, a$.

=--

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

A __uniform [[homeomorphism]]__ is a uniformly continuous [[bijection]] whose [[inverse]] is also uniformly continuous (which is *not* automatic).  Two (quasi)uniform spaces are __uniformly homeomorphic__ if there exists a uniform homeomorphism between them.  We may also speak of __antiuniform homeomorphisms__ between __antiuniformly homeomorphic__ quasiuniform spaces.

=--

### As structure

In [[dependent type theory]], one could change the [[universal quantifiers]] and [[existential quantifiers]] in the definition of uniformly continuous function into [[dependent product types]] and [[dependent sum types]].

\begin{definition}
Let $\mathrm{R}_+ \coloneqq \sum_{x:\mathbb{R}} \epsilon \gt 0$ denote the positive real numbers. Given [[metric spaces]] $(X, d_X)$ and $(Y, d_Y)$, a **uniformly continuous function** between $X$ and $Y$ is a function $f:X \to Y$ between their underlying sets with a dependent function which says:

> Given any positive real number $\epsilon \gt 0$, there is as structure a positive real number $\delta \gt 0$ such that for all elements $a:X$ and $b:X$, $\delta_Y(f(a), f(b))$ is less than $\epsilon$ whenever $\delta_X(a, b)$ is less than $\delta$

$$\prod_{\epsilon:\mathrm{R}_+} \sum_{\delta:\mathbb{R}_+} \prod_{a:X} \prod_{b:X} (\delta_X(a, b) \lt \delta) \to (\delta_Y(f(a), f(b)) \lt \epsilon)$$

By the [[type theoretic axiom of choice]], which is simply the distributivity of [[dependent function types]] over [[dependent sum types]], this is the same as saying that

> There exists as structure a function on the positive real numbers $\omega:\mathrm{R}_+ \to \mathrm{R}_+$ such that for all positive real numbers $\epsilon \gt 0$ and for all elements $a:X$ and $b:X$, $\delta_Y(f(a), f(b))$ is less than $\epsilon$ whenever $\delta_X(a, b)$ is less than $\omega(\epsilon)$

$$\sum_{\omega:\mathrm{R}_+ \to \mathrm{R}_+} \prod_{\epsilon:\mathrm{R}_+} \prod_{a:X} \prod_{b:X} (\delta_X(a, b) \lt \omega(\epsilon)) \to (\delta_Y(f(a), f(b)) \lt \epsilon)$$
\end{definition}

There exists a similar definition for [[uniform spaces]]:

\begin{definition}
Given uniform spaces $(X, \mathcal{U}(X), \approx)$ and $(Y, \mathcal{U}(Y), \approx)$, a **uniformly continuous function** between $X$ and $Y$ is a function $f:X \to Y$ with a dependent function which says:

> Given any [[entourage]] $E:\mathcal{U}(Y)$, there is as structure an entourage $D:\mathcal{U}(X)$ such that for all elements $a:X$ and $b:X$, $f(a) \approx_{E} f(b)$ whenever $a \approx_{D} b$

$$\prod_{E:\mathcal{U}(Y)} \sum_{D:\mathcal{U}(X)} \prod_{a:X} \prod_{b:X} (a \approx_{D} b) \to (f(a) \approx_{E} f(b))$$

By the [[type theoretic axiom of choice]], which is simply the distributivity of [[dependent function types]] over [[dependent sum types]], this is the same as saying that

> There exists as structure a function $\omega:\mathcal{U}(Y) \to \mathcal{U}(X)$ between the sets of entourages such that for all entourages $E:\mathcal{U}(Y)$ and for all elements $a:X$ and $b:X$, $f(a) \approx_{E} f(b)$ whenever $a \approx_{\omega(E)} b$

$$\sum_{\omega:\mathcal{U}(Y) \to \mathcal{U}(X)} \prod_{E:\mathcal{U}(Y)} \prod_{a:X} \prod_{b:X} (a \approx_{\omega(E)} b) \to (f(a) \approx_{E} f(b))$$
\end{definition}

## Properties

Every uniformly continuous map between uniform spaces is [[continuous map|continuous]] (between the underlying [[topological spaces]]) and in fact [[Cauchy map|Cauchy continuous]] (between the underlying [[Cauchy spaces]]).  Also, every uniformly continuous or antiuniformly continuous map between quasiuniform spaces is Cauchy continuous.  Conversely, every [[short map|short]] or even [[Lipschitz map|Lipschitz]] map between metric spaces (or Lipschitz [[manifolds]]) is uniformly continuous.

A [[composite]] of uniformly continuous maps is uniformly continuous, as is any [[identity function]] between (quasi)uniform spaces.  The composite of two antiuniformly continuous maps is uniformly continuous.  Thus uniform spaces are the [[objects]] of a [[category]] whose morphisms are the uniformly continuous maps as [[morphisms]], and quasiuniform spaces are the objects of two categories: one with uniformly continuous maps as morphisms and one with both uniformly continuous maps and antiuniformly continuous maps as morphisms (so that quasiuniform spaces are the objects of an $\mathcal{M}$-[[M-category|category]]).

## Examples

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

## Related concepts

* [[uniform convergence]]

* [[uniform property]]

* [[modulus of continuity]]

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uniformly hyperfinite algebra

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

#Contents#
* table of contents
{:toc}

## Definition

A [[C-star algebra]] is called a _uniformaly hyperfinite algebra_ or a _Glimm algbebra_ (after [Glimm](#Glimm)) if it arises as the [[sequential colimit]] of an increasing sequence of [[von Neumann algebra factor|type In]]-algebras (finite [[matrix]] algebras), hence if it is the [[norm]]-closure of the [[union]] $\cup_{i \in \mathbb{N}} A_i$ of a sequence $A_0 \hookrightarrow A_1 \hookrightarrow \cdots$ of an increasing sequence of $I_{n_i}$-factors.

## Properties

All type III [[von Neumann algebra]] [[von Neumann algebra factor|factors]] can be constructed from uniformly hyperfinite algebras ([Powers](#Powers)).

## Related concepts

These algebras naturally appear as [[algebras of observables]] in [[quantum lattice systems]].

## References

The notion was introduced in

* J. Glimm, _On a certain class of operator algebras_, Transactions of the AMS 95 (1960)
  {#Glimm}

See also

* R. Powers, _Representations of uniformly hyperfinite algebras and their associated von Neumann rings_, Ann. Math. (2) 86 (1967) ([Euclid](http://projecteuclid.org/euclid.bams/1183528960))
 {#Powers}

[[!redirects uniformly hyperfinite algebras]]


uniformly regular space

# Uniformly regular spaces
* table of contents
{: toc}

## Idea

In [[classical mathematics]], every [[uniform space]] is a [[regular space]], and indeed a [[completely regular space]].  In [[constructive mathematics]], however, this fails to be true.  The notion of *uniformly regular* uniform space is an intermediate notion in between regularity and complete regularity for uniform spaces, which is frequently useful in the constructive theory of uniform spaces; classically every uniform space is uniformly regular.  (Indeed, some authors include uniform regularity in the constructive *definition* of a uniform space.)

## Definition

Let $X$ be a [[uniform space]].  When the uniformity of $X$ is defined in terms of entourages, then $X$ is **uniformly regular** if it satisfies the following property:

* For every entourage $U$, there exists an entourage $V$ such that $\neg{V} \cup U = X \times X$, where $\neg{V}$ is the [[complement]] of $V$. That is,
  $$ \forall U,\; \exists V,\; \forall x, y,\; x \approx_U y \;\vee\; x \mathbin{≉}_V y .$$

With [[excluded middle]], we can take $V$ to be $U$ itself, so every uniform structure is uniformly regular in classical mathematics.  Note that any uniformity induced by a [[gauge space|gauge]] is uniformly regular (as long as the gauging distances take values in [[located real numbers]] as is usually understood).

When the uniformity of $X$ is defined in terms of uniform covers, then the equivalent condition is:

*  If $C$ is a uniform cover, there exists a uniform cover $C'$ such that, for any two points $x, y$, either $x, y \in A$ for some $A \in C$ or $x, y \in B$ for no $B \in C'$.

With [[excluded middle]], we can take $C'$ to be $C$.

Note that uniform regularity is not literally a "uniformization" of [[regular space|regularity]] but rather of [[locally decomposable space|local decomposability]]; so it could also be called **uniform local decomposability**.  In the case of uniform spaces, it implies regularity (see below), so we generally call it uniform regularity; however, for [[quasi-uniform spaces]] (and [[uniform convergence spaces]]) it would be more proper to say 'uniformly locally decomposable' instead.

## Examples

* Every [[metric space]], and indeed any [[gauge space]], is uniformly regular, as long as the metric(s) take values in [[located real numbers]].  It suffices to consider a basic entourage $U_{d,\varepsilon} = \{ (x,y) \mid d(x,y)\lt\varepsilon \}$, in which case we can take $V = \{ (x,y) \mid d(x,y)\lt\varepsilon/2 \}$ and use the fact that every located real number is either $\lt \varepsilon$ or $\gt \varepsilon/2$.

* A [[discrete space|discrete]] uniform space is uniformly regular if and only if the underlying set has [[decidable equality]].  (Indeed, a discrete topological space is [[regular space|regular]] if and only if it has decidable equality.)  Thus, the statement "every uniform space is (uniformly) regular" is equivalent to [[excluded middle]].

* In particular, not every [[topological group]] is uniformly regular: any group without decidable equality can be equipped with the discrete topology.  However, the classical examples of Lie groups and [[topological vector spaces]] (TVSs) are uniformly regular.

* However, if a topological group is a [[regular space]], then it is uniformly regular.  For if $U$ is any neighborhood of the identity $e$ in a topological group $X$, by regularity we have a neighborhood $V$ of $e$ and an open set $G$ with $V\cap G = \emptyset$ and $G\cup U = X$, and in particular $\neg V\cup U = X$, whence the entourages defined by $U$ and $V$ satisfy the desired condition.

* Every [[completely regular space|completely regular]] uniform space should be uniformly regular, although the correct constructive definition of "completely regular" is not entirely clear.

### Compact spaces

The examples above show that uniform regularity coincides with regularity for discrete uniform spaces and topological groups.  Another class of spaces for which this holds are compact ones.

+--{: .un_theorem}
###### Theorem
Every [[compact space|compact]] [[regular space]] admits a unique compatible uniformity that is uniformly regular.
=--

+--{: .proof}
###### Proof
Let $X$ be compact regular.  Since compactness and regularity are properties of the open-set lattice, $X$ is still compact regular when regarded as a locale, and hence also Hausdorff as a locale.  Moreover, it is locally compact, and hence the locale and spatial products $X\times X$ coincide.  Furthermore, $X\times X$ is also compact regular.

Define the entourages to be all the neighborhoods of the diagonal in $X\times X$; this obviously satisfies all the axioms of a uniformity except possibly the square-root (and uniform regularity).

Let $U$ be a neighborhood of the diagonal in $X\times X$.  Since $X\times X$ is regular, and using the definition of the product topology, for each $x\in X$ there is a square neighborhood $V_x\times V_x$ and an open set $G_x\subseteq X\times X$ such that $(V_x\times V_x)\cap G_x = \emptyset$ and $G_x \cup U = X\times X$.  The open sets $V_x$ cover the compact space $X$, hence finitely many of them suffice.  Let $V = (V_{x_1}\times V_{x_1}) \cup \cdots \cup (V_{x_n}\times V_{x_n})$, and $G = G_{x_1} \cap \cdots \cap G_{x_n}$.  Then $V$ is a neighborhood of the diagonal, and $G$ an open set such that $G\cup U = X\times X$ and $V \cap G = \emptyset$.  In particular, $\neg V \cup U \subseteq G\cup G = X\times X$, so this uniformity (if it exists) is uniformly regular.

Now, $G$ is the union of rectangular open neighborhoods $A\times B$.  Note that since $G$ is disjoint from the diagonal, any $A\times B \subseteq G$ satisfies $A\cap B = \emptyset$.  Moreover, since $X$ is regular, each $A$ and $B$ is the union of open sets that are well-inside it; thus $G$ is the union of rectangular open neighborhoods $C\times D$ such that there are opens $A,B$ with $C\triangleleft A$ and $D\triangleleft B$ and $A\cap B = \emptyset$.  Since $G\cup U = X\times X$ and $X\times X$ is compact, we can cover it by $U$ together with finitely many such rectangles $C\times D$, say $X\times X = U \cup (C_1\times D_1) \cup\cdots \cup (C_m \times D_m)$.

For each of these rectangles $C_i\times D_i$, let $E_i$ be an open set such that $E_i \cup A_i \cup B_i = X$ and $E_i \cap (C_i \cup D_i) = \emptyset$, and let $W_i = (A_i\times A_i) \cup (B_i \times B_i) \cup (E_i \times E_I)$.  Then $W_i$ is a neighborhood of the diagonal; let $W = W_1 \cap \cdots \cap W_m$.

We claim $W\circ W \subseteq U$.  Let $(x,z) \in W\circ W$, so that there is a $y$ with $(x,y)\in W$ and $(y,z) \in W$.  Now we have either $(x,z)\in U$ (which is what we want) or $(x,z) \in C_i\times D_i$ for some $i$, so it suffices to rule out the latter possibility.  Since $(x,y)\in W$, we have $(x,y) \in W_i$, hence $(x,y)\in A_i\times A_i$ or $(x,y)\in B_i\times B_i$ or $(x,y)\in E_i\times E_i$.  But $x\in C_i$, which is disjoint from $B_i$ and $E_i$, so it must be that $(x,y)\in A_i\times A_i$ and hence $y\in A_i$.  Similarly, we conclude that $(y,z)\in B_i\times B_i$ so that $y\in B_i$.  But $A_i\cap B_i = \emptyset$, a contradiction.

It remains to show that the uniform topology on $X$ is the original one.  Clearly if $U$ is an entourage then each $U[x]$ is a neighborhood of $x$.  Conversely, if $V$ is a neighborhood of $x$, then by regularity we have a neighborhood $W$ of $x$ and an open $G$ with $W\cap G = \emptyset$ and $G\cup V = X$.  Then $Z = (V\times V) \cup (G\times G)$ is a neighborhood of the diagonal, i.e. an entourage, such that $Z[x] =V$.

Finally, we show this is the unique uniformity compatible with a compact regular topology.  Suppose $X$ is any compact regular uniform space, and $U$ any neighborhood of the diagonal; we want to show $U$ is an entourage.  For any $x\in X$, there is an entourage $V$ such that $V_x[x] \times V_x[x] \subseteq U$.  Let $W_x$ be an entourage such that $W_x \circ W_x \subseteq V_x$.  Since $X$ is compact, it is covered by finitely many of the neighborhoods $W_x[x]$, say $X = W_{x_1}[x_1] \cup \cdots\cup W_{x_k}[x_k]$.  Let $W = W_1 \cap \cdots \cap W_k$; we claim $W\subseteq U$.  For let $(y,z)\in W$; then we have $y\in W_{x_k}[x_k]$ for some $k$, whence also $z\in V_{x_k}[x_k]$, and thus $(y,z) \in V_{x_k}[x_k]\times V_{x_k}[x_k] \subseteq U$.
=--

## Properties

### Regularity

Every uniformly regular uniform space is a [[regular topological space]] in its uniform topology.  For if $U[a]$ is a uniform neighborhood of $a$, let $V$ be an entourage such that $\neg V\cup U = X\times X$, and $W$ an entourage such that $W\circ W \subseteq V$.  Then $W[a]$ is a neighborhood of $a$, and the interior of $W[\neg V[a]]$ is an open set $G$, such that $G\cap W[a] = \emptyset$ and $G\cup U[a] = X$.

It appears to be unknown whether there exist regular uniform spaces that are not uniformly regular.  Regularity implies uniform regularity for at least three large classes of uniform spaces: discrete ones, compact ones, and topological groups (plus, trivially, metric and gauge spaces, which are automatically uniformly regular).

### Apartness

Recall that every uniform space $X$ has a [[inequality relation]] (an [[irreflexive]] [[symmetric relation]]) where "$x # y$" means that there exists an entourage $U$ such that $x \mathbin{≉}_U y$.  If $X$ is uniformly regular, then this is an [[apartness relation]], i.e. it is also a [[comparison]].  For if $x \mathbin{≉}_U z$, let $V \circ V \subseteq U$ and $\neg{W} \cup V = X \times X$.  Then for any $y$, we cannot have both $x \approx_V y$ and $y\approx_V z$, so we must have either $x \mathbin{≉}_W y$ or $y \mathbin{≉}_W z$, whence either $x # y$ or $y # z$.  (In fact, this is a special case of the fact that any [[regular space]] admits an apartness relation, discussed at [[regular space]]).

### Locatedness

Uniform regularity has some nice consequences for the existence of (almost/uniformly) [[located subsets]]: see there for details.

### Uniform apartness spaces

A uniformly regular uniform space can be equivalently described in terms of the *complements* of entourages, i.e. by giving notions of when two points are "at least $\varepsilon$ apart" rather than "at least $\varepsilon$ close".

+-- {: .un_defn}
###### Definition
A **uniform apartness space** is a set $X$ equipped with a collection of binary relations $A\subseteq X\times X$ called *anti-entourages* such that

1. $(x,x)\notin A$ for $x\in X$ and anti-entourages $A$, i.e. each anti-entourage is [[irreflexive relation|irreflexive]].
2. For every anti-entourage $A$, there exists an anti-entourage $B$ such that whenever $(x,z)\in A$ and $(y,z)\notin B$ we have $(x,y)\in B$.
3. For every anti-entourage $A$, there exists an anti-entourage $B$ such that $A\subseteq B^{op}$ (hence, by axiom (6), $A^{op}$ is an anti-entourage).
4. There exists an anti-entourage (hence, by axiom (6), the empty set is an anti-entourage).
5. If $A$ and $B$ are anti-entourages, so is some superset of $A\cup B$ (hence, by axiom (6), $A\cup B$ is itself).
6. If $A$ is an anti-entourage and $B\subseteq A$, then $B$ is an anti-entourage.
=--

If $X$ is a uniform space, we can make it a uniform apartness space by declaring the anti-entourages to be the sets contained in the complement of some entourage, i.e. $A$ is an anti-entourage if $A \subseteq \neg U = (X\setminus U)$ for some entourage $U$.  Most of the axioms are obvious.  For axiom (2), we use transitivity of a uniformity: if $V\circ V \subseteq U$, then if $(x,z)\notin U$ and $\neg((y,z)\notin V)$, then if $(x,y)\in V$ we can derive a contradiction by assuming $(y,z)\in V$ (double negations can be removed constructively when the goal is a contradiction) and obtaining $(x,y)\in U$ from $V\circ V = U$; thus $(x,y)\notin V$.  Note also that axiom (5) needs only the direction of de Morgan's law that is true constructively, $\neg U \cup \neg V \subseteq \neg (U\cap V)$.)

Thus, any uniform space gives rise to a uniform apartness space.  If our original uniform space was uniformly regular, then this uniform apartness space is again "uniformly regular" in the sense that for any anti-entourage $A$ there is an anti-entourage $B$ such that $\neg A\cup B = X\times X$: if $A\subseteq \neg U$, let $B = \neg V$ where $U\cup \neg V = X\times X$.

(We note in passing that a uniformly regular uniform apartness space also satisfies a stronger version of axiom (2), namely that for every anti-entourage $A$, there exists an anti-entourage $B$ such that whenever $(x,z)\in A$, for any $y$ we have either $(x,y)\in B$ or $(y,z)\in B$.  To see this, first let $C$ be for $A$ as in the standard axiom (2), then let $B$ be for $C$ as in uniform regularity.  Now given $(x,z)\in A$, we have either $(y,z)\in B$ or $(y,z)\notin C$; but the latter implies $(x,y)\in C$ and hence $(x,y)\in B$.)

In the converse direction, any uniform apartness space gives rise to a uniform space whose entourages are the supersets of the complements of the anti-entourages, and uniform regularity is again preserved.  These constructions are not inverse in general, but they are in the uniformly regular case.  For this it suffices to show that any entourage $U$ in a uniformly regular uniform space contains the double-complement $\neg\neg V$ of some entourage $V$, and dually for any anti-entourage $A$ in a uniformly regular uniform apartness space there is an anti-entourage $B$ containing $\neg\neg A$.  But for the first it suffices to have $U\cup \neg V = X\times X$, and for the second it suffices to have $\neg A \cup B = X\times X$.  Thus, there is a bijection between uniformly regular uniformities and uniformly regular uniform apartnesses on any set $X$.

We define a function $f:X\to Y$ between uniform apartness spaces to be *uniformly continuous* if for every anti-entourage $B$ in $Y$, there is an anti-entourage $A$ in $X$ such that if $(f(x),f(y))\in B$ then $(x,y)\in A$.  Since a conditional implies its contrapositive, the constructions above become functors $Unif \to AUnif$ and $AUnif \to Unif$.  Moreover, we have:

+-- {: .un_theorem}
###### Theorem
The above functors form an [[idempotent adjunction]] $Unif \rightleftarrows AUnif$ over $Set$, whose fixed points contain the uniformly regular spaces on each side.  Thus, in particular, the categories of uniformly regular uniform spaces and uniformly regular uniform apartness spaces are isomorphic over $Set$.
=--
+-- {: .proof}
###### Proof
The unit and counit are the identity functions, which are uniformly continuous since a statement implies its double negation, and obviously satisfy the triangle identities.  The adjunction is idempotent since a negated statement is equivalent to its double negation, and we have seen above that uniformly regular spaces are among the fixed points.
=--

## Related pages

* [[uniform space]]
* [[regular topological space]]

category: topology

[[!redirects uniformly regular space]]
[[!redirects uniformly regular spaces]]
[[!redirects uniformly regular uniform space]]
[[!redirects uniformly regular uniform spaces]]
[[!redirects uniformly regular]]
[[!redirects uniform regularity]]
[[!redirects located uniform space]]
[[!redirects located uniform spaces]]
[[!redirects uniform apartness space]]
[[!redirects uniform apartness spaces]]
[[!redirects uniformly locally decomposable space]]
[[!redirects uniformly locally decomposable spaces]]

[[!redirects quasiuniformly locally decomposable space]]
[[!redirects quasiuniformly locally decomposable spaces]]


unimath > history

see *[[UniMath project]]*


UniMath project

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### Context
#### Constructivism, Computability
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[[!include constructivism - contents]]
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#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_UniMath_ is a library of [[formal logic|formalized]] [[mathematics]] ([[formal proof]]) based on the [[univalent foundations of mathematics]] ([[homotopy type theory]] with the [[univalence axiom]]) and currently developed with the [[Coq]] [[proof assistants]].

Current "coordinating committee" (according to the [home page](http://unimath.github.io/UniMath/)):

* [[Benedikt Ahrens]]

* [[Daniel Grayson]]

* [[Michael Lindgren]]

* [[Peter LeFanu Lumsdaine]]

* [[Ralph Matthes]]

* [[Niels van der Weide]]

Web resources:

* *Univalent Mathematics in [[Coq]]*

  [unimath.github.io/UniMath](http://unimath.github.io/UniMath)

  * [table of contents](https://unimath.github.io/UniMath/UniMath/CONTENTS.html)

  * [index](https://unimath.github.io/doc/UniMath/d4de26f/)

* *Univalent Mathematics in [[Agda]]*

  [unimath.github.io/agda-unimath](https://unimath.github.io/agda-unimath)

## Further references

* [[Benedikt Ahrens]], _UniMath: its origins, present, and future_ (2017) [[pdf](https://unimath.github.io/bham2017/UniMath_origins-present-future.pdf), [[Ahrens-UniMathHistory.pdf:file]]]

## Related projects

[[!include proof assistants and formalization projects -- list]]

[[!redirects UniMath]]
[[!redirects unimath]]

category: reference


unimodular integral lattice

#Contents#
* table of contents
{:toc}

## Idea

An [[integral lattice]] whose _discriminant_ is +1 or -1.

## Examples

* [[E8 lattice]]

* [[Leech lattice]]

## Related concepts

* [[modular integral lattice]]

## References

* {#Borcherds99} [[Richard Borcherds]], _The Leech lattice and other lattices_ ([arXiv:math.NT/9911195](http://arxiv.org/abs/math.NT/9911195))

* Wikipedia, _[Unimodular lattice](http://en.wikipedia.org/wiki/Unimodular_lattice)_

[[!redirects unimodular lattice]]
[[!redirects unimodular lattices]]
[[!redirects unimodular integral lattices]]


unintentional type theory

# Unintentional type theory
* table of contents
{: toc}

## Idea

In contrast to [[intensional type theory]], which is a [[type theory]] containing an [[identity type]] $x=y$ whose elements are identifications of $x$ and $y$, **unintentional type theory** has a [[identity type|mistaken identity type]] $x \overset{?}{\approx} y$ whose elements are inadvertent conflations of $x$ and $y$.  The mistaken identity type

> greatly increases the expressive power of UTT by internalizing many proofs which previously required metametatheoretic techniques (e.g. user error on a blackboard).  [(Angiuli, p1)](#Angiuli)

## Definition

The formation rule for the mistaken identity type says that any two [[terms]], there is a type of inadvertent conflations of them.

$$\frac{\Gamma\vdash M:A \qquad \Gamma\vdash N:B}{\Gamma \vdash (M:A \overset{?}{\approx} N:B) \,type}$$

Note that the terms being (potentially) conflated do not have to belong to the same type.

The introduction rule says that from any reason to conflate two terms we can obtain an inhabitant of the mistaken identity type.  It is important to note that the original reason for conflation of the terms is *not* remembered by the proof term.

$$\frac{\Gamma\vdash M:A \qquad \Gamma\vdash N:B \qquad \Gamma\vdash\text{FIXME: are these equal?}}{\Gamma\vdash confl(M,N) :(M:A \overset{?}{\approx} N:B)}$$

Finally, the elimination rule says that if we have a mistaken conflation of $M$ and $N$, we may replace $M$ by $N$ anywhere inside another term $P$ whose type depends on $M$.  The precise syntax of this rule is somewhat complicated; see [(Angiuli, p2)](#Angiuli) for details.

## Semantics

* In the [[homotopy type theory|homotopical interpretatation]] of unintentional type theory, every type has a tower of iterated mistaken identity types, representing the compounding nature of errors.  This is captured semantically by Angiuli's [[(∞,1)-topos|(∞,1)-accidentopos]] model, in which [[types]] are interpreted by [[globular sets]] that could be confused with an [[∞-groupoid]].

* One might expect that unintentional type theory also admits a model in the [[effective topos|ineffective topos]].  According to Awodey, the disassemblies are too immodest for this to work, but a related construction may be possible.

* [[philosophy|Philosophically]], the "unintended [[semantics]]" of unintentional type theory is described by the [[meaning explanation|meaningless explanation]].

## Computer implementations

Unintentional type theory has not yet been implemented in computer proof assistants.  However, Bauer has proposed that such implementation would be very useful pedagogically.  This is because UTT can formalize the way children are taught to solve algebraic equations in school, by starting out pretending that two things are equal.  Using a proof assistant with UTT could therefore help ensure that students are not doing anything right.

## References

* [[Carlo Angiuli]], *The $(\infty,1)$-accidentopos model of unintentional type theory*.  [Sigbovik '13](http://sigbovik.org/2013/); April 1, 2013. [PDF](https://carloangiuli.com/papers/unintentional.pdf)
 {#Angiuli}

category: joke

[[!redirects mistaken identity type]]
[[!redirects (∞,1)-accidentopos]]


union

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###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **union** is a [[join]] of [[subset]]s or (more generally) [[subobject]]s.  This includes the traditional [[set theory|set-theoretic]] union of subsets of some ambient set.

The [[duality|dual]] notion is that of [[intersection]]/[[meet]].

Unions of completely arbitrary sets make sense only in [[material set theory]], where their existence is guaranteed by the [[axiom of union]].  In [[structural set theory]], unions of arbitrary sets can generally be replaced by [[disjoint unions]], and unions of arbitrary [[subsets]] are represented by [[wide pushouts]] of the corresponding [[injections]] from the [[intersection]] of the subsets.

A [[coherent category]] is one having well-behaved finite unions of subobjects, and an [[infinitary coherent category]] is one having all well-behaved small unions of subobjects.

## Properties

* For [[subsets]]: [[images preserve unions]], [[pre-images preserve unions and intersections]]. See at _[[interactions of images and pre-images with unions and intersections]]_.

## Related concepts

* [[intersection]]

* [[disjoint union]]

* [[regular cardinal]]

[[!redirects union]]
[[!redirects unions]]


union structure

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### Context
#### Set theory
+-- {: .hide}
[[!include set theory - contents]]
=--
=--
=--

\tableofcontents

## Definition

Given a set $V$ with an [[extensional relation]] $\prec$, a **proto-union structure** is a [[function]] $U:V \to V$ such that

* for all elements $a \in V$, $b \in V$, and $c \in V$, if $a \prec b$ and $b \prec c$, then $a \prec U(c)$.

Given a set $V$ with an [[extensional relation]] $\prec$, a **union structure** is a proto-union structure where additionally

* for all elements $a \in V$ and $c \in V$, if $a \prec U(c)$, then there exists $b \in V$ such that $a \prec b$ and $b \prec c$.

Uniqueness of $U(c)$ follows from $\prec$ being an [[extensional relation]].

## Foundational concerns

In any [[material set theory]], instead of postulating the mere existence of a set $U$ in which $a \in b$ and $b \in c$ implies that $a \in U$, one could add a primitive unary operation $U(c)$ which takes material sets $c$ and returns a material set $U(c)$ such that for all $a$ and $b$, $a \in b$ and $b \in c$ implies that $a \in U(c)$.

## See also

* [[axiom of union]]

* [[pairing structure]]

* [[powerset structure]]

* [[natural numbers structure]]

[[!redirects union structure]]
[[!redirects union structures]]


unipotent group scheme

[[!redirects unipotent group schemes]]

## Idea

An element $r$ of a ring with multiplicative unit is called *unipotent element* if $r-1$ is nilpotent.

(...)

+-- {: .num_theorem}
###### Theorem and Definition
Let $G$ be an [[affine scheme|affine]] [[group scheme|k-group]]. Then the following conditions are equivalent.

1. The [[completion]] of the [[Cartier duality|Cartier dual]] $\hat D(G)$ of $G$ is a connected formal group.

1. Any [[multiplicative group scheme|multiplicative subgroup]] of $G$ is zero.

1. For any subgroup $H$ of $G$ with $H\neq 0$ we have $Gr_k(H,\alpha_k)\neq 0$.

1. Any algebraic quotient of $G$ is an extension of subgroups of $\alpha_k$.

1. (If $p\neq 0)$, $\cap Im V^n_G =e$.

An affine group scheme satisfying these conditions is called *unipotent group scheme*.
=--