A *uniform space* is a natural generalization of the notion of a metric space, on which completeness still makes sense [1]. The non-Hausdorff variant of this is called a *quasi-uniform space*, and the purpose of this post is to introduce some of the features of those spaces, with a particular stress on a now very classical construction due to William Pervin [3]. It is rather puzzling that I managed to avoid the subject of quasi-uniform spaces in something like the 7 years that this blog existed!

## The definition of quasi-uniform spaces

Let *X* be a set. A binary relation *R* on *X* is a subset of *X* × *X*. Binary relations compose: *R* o *S* is the set of those pairs (*x*,*z*) such that (*x*,*y*) ∈ *R* and (*y*,*z*) ∈ *S* for some *y* in *X*. Note that *R* o *R* ⊆ *R* if and only if *R* is transitive. A relation *R* is reflexive if and only if it contains the diagonal Δ ≝ {(*x*,*x*) | *x* ∈ *X*}.

A *quasi-uniformity* ** U** on

*X*is a filter of reflexive binary relations on

*X*that satisfies the following relaxed transitivity condition: for every relation

*S*in

**, there is a relation**

*U**R*in

**such that**

*U**R*o

*R*⊆

*S*. Explicitly, it is a family of binary relations on

*X*such that:

- (reflexivity) for every
*R*in, for every*U**x*in*X*, (*x*,*x*) is in*R*; - (filter 1)
*X*×*X*is in;*U* - (filter 2) for every
*R*in, every binary relation*U**S*on*X*such that*R*⊆*S*is in;*U* - (filter 3) for all
*R*,*S*in,*U**R*∩*S*is in;*U* - (relaxed transitivity) for every
*S*in, there is a relation*U**R*insuch that, for all points*U**x*,*y*,*z*in*X*such that (*x*,*y*) ∈*R*and (*y*,*z*) ∈*R*, we have (*x*,*z*) ∈*S*.

A *quasi-uniform space* is a set with a quasi-uniformity ** U**. The relations

*R*in

**are called**

*U**entourages*, a French word that means surroundings. (And also environment, neighborhood, relatives, acquaintances. As most French words, the word also exists in English.)

The primary example is given by hemi-metric spaces. Given any hemi-metric *d* on *X*, one defines a quasi-uniformity * U_{d}* as follows. A

*basic entourage*is a relation of the form [<

*r*] ≝ {(

*x*,

*y*) |

*d*(

*x*,

*y*) <

*r*}, where

*r*varies over the positive real numbers (namely,

*r*>0). Then

*is the family of relations that contains some basic entourage [<*

**U**_{d}*r*].

It is interesting to see why relaxed transitivity holds. Let *S* be in * U_{d}*, so

*S*contains some basic entourage [<

*s*], with

*s*>0. Define

*r*≝

*s*/2 and

*R*as [<

*r*]. For all points

*x*,

*y*,

*z*in

*X*such that (

*x*,

*y*) ∈

*R*and (

*y*,

*z*) ∈

*R*, we have

*d*(

*x*,

*y*)<

*r*and

*d*(

*y*,

*z*)<

*r*, so

*d*(

*x*,

*z*)≤

*d*(

*x*,

*y*)+

*d*(

*y*,

*z*)<

*s*, showing that (

*x*,

*z*) is in

*S*. In other words, relaxed transitivity stems from the triangular inequality.

By the way, the way we have built * U_{d}* is a practical way of defining a quasi-uniformity. A

*base (of entourages*) is any non-empty family

**of binary relations on**

*B**X*satisfying:

- (reflexivity) for every
*R*in, for every*B**x*in*X*, (*x*,*x*) is in*R*; - (filter 3) for all
*R*,*S*in, there is a relation*B**T*insuch that*B**T*⊆*R*∩*S*; - (relaxed transitivity) for every
*S*in, there is a relation*B**R*insuch that*B**R*o*R*⊆*S*.

Then the set of binary relations that contain some element of ** B** forms a quasi-uniformity. This is the quasi-uniformity

*generated by*

**. This is what we just did, with**

*B**consisting of the relations [<*

**B***r*],

*r*>0.

Before we go on, I should mention the original notion of uniformity. Given any binary relation *R* on *X*, let *R*^{op} be its converse: (*x*,*y*) is in *R*^{op} if and only if (*y*,*x*) is in *R*. A quasi-uniformity ** U** is a

*uniformity*if and only if it satisfies the following extra condition:

- (symmetry) for every
*R*in,*U**R*^{op}is in.*U*

The prime example of a uniformity is * U_{d}* provided that

*d*is a

*pseudo*-metric now.

## Quasi-uniform spaces and topological spaces

There is a traditional way to extract a topology from a quasi-uniformity * U*.

Given any point *x* in *X*, and any binary relation *R* on *X*, let *R*[*x*] denote the set {*y* ∈ *X* | (*x*,*y*) ∈ *R*}; namely, the *image* of *x* by *R*. We say that a * U-neighborhood *of

*x*is a set of the form

*R*[

*x*] for some entourage

*R*∈

**. Next, we call**

*U**any subset of*

**U**-open*X*that is a

*-neighborhood*

**U***of each of its points. This forms a topology, which I will call the*

*induced topology*, or the

*topology induced by*

**. One also says that the quasi-uniformity**

*U***is**

*U**compatible*with a topology if and only if that topology is the induced topology.

For example, if *d* is a hemi-metric, a * U_{d}*-neighborhood of

*x*is simply a subset of

*X*that contains the open ball

*B*

^{d}_{x}_{,<r}for some

*r*>0. The topology induced by

*is therefore the open ball topology of*

**U**_{d}*d*.

By the way, for every *x* ∈ *X*, the * U–*neighborhoods

*R*[

*x*] (

*R*∈

**) are neighborhoods of**

*U**x*in the topology induced by

**. Indeed, consider the set**

*U**U*of all the points

*y*∈ X such that there is an

*S*∈

**such that**

*U**S*[

*y*] is included in

*R*[

*x*]. This is open in the induced topology underlying

**, by definition. In fact, it is easy to check that**

*U**U*is the interior of

*R*[

*x*] in that topology. Additionally,

*x*is in

*U*, because one can take

*R*for

*S*in that case. Hence

*R*[

*x*] indeed contains an open set

*U*that contains

*x*.

A natural question arises: which topologies arise as topologies induced by quasi-uniformity? The question was solved by Császár [2]: all topologies whatsoever! The simplest known argument is due to Pervin [3], and is as follows. There are in general several quasi-uniformities that induce the same topology, and we will see an example below; another way of stating the following theorem is to say that there is at least one.

**Theorem (Császár [2], Pervin [3]).** Let *X* be a topological space. For every open subset *U* of *X*, let *R _{U}* be the binary relation {(

*x*,

*y*) |

*x*∈

*U*⇒

*y*∈

*U*}. The finite intersections of relations

*R*,

_{U}*U*∈

**O**

*X*, form a base

**of entourages of a quasi-uniformity on**

*B**X*, whose induced topology is exactly the topology of

*X*. This quasi-uniformity is the

*Pervin quasi-uniformity*of the topology of

*X*.

*Proof.* We check the axioms for a base of entourages. First, (reflexivity) and (filter 3) are clear. In order to verify (relaxed transitivity), we verify the much stronger statement that *R _{U}* o

*R*⊆

_{U}*R*for every open subset

_{U}*U*of

*X*. That inclusion just means that if

*x*∈

*U*⇒

*y*∈

*U*and if

*y*∈

*U*⇒

*z*∈

*U*, then

*x*∈

*U*⇒

*z*∈

*U*. The relation

*R*o

_{U}*R*⊆

_{U}*R*will play an important rôle later in this post.

_{U}We now consider an arbitrary element *S* ≝ *R _{U}*

_{1}∩ … ∩

*R*

_{U}_{n}in

**. We look for an element**

*B**R*of

**such that**

*B**R*o

*R*⊆

*S*, and we simply define

*R*as

*S*. Every pair (

*x*,

*z*) in

*R*o

*R*is such that there is a point

*y*such that (

*x*,

*y*) and (

*y*,

*z*) are in

*R*. Hence (

*x*,

*y*) and (

*y*,

*z*) are in

*R*

_{U}_{i}for every

*i*, 1≤

*i*≤

*n*, which shows that (

*x*,

*z*) is in

*R*

_{U}_{i}o

*R*

_{U}_{i}⊆

*R*

_{U}_{i}for every

*i*, 1≤

*i*≤

*n*, hence that (

*x*,

*z*) is in

*R*=

*S*.

Let ** U** be the quasi-uniformity generated by

**.**

*B*Given any point *x* of *X*, and any basic entourage * S* ≝

*R*_{U}_{1}∩ … ∩

*, the image*

*R*_{U}_{n}*S*[

*x*] is equal to the intersection of the opens sets

*that contain*

*U*_{i}*x*. It follows that every

**-open set is a neighborhood of each of its points, hence is open in the original topology on**

*U**X*. Conversely, let

*U*be any open set in the original topology on

*X*. For every

*x*in

*U*,

*[*

*R*_{U}*x*]=

*U*is a

**-neighborhood of**

*U**x*included in

*U*, so

*U*is

**-open. ☐**

*U*You may have noticed the similarity with Wilson’s theorem (Theorem 6.3.13 in the book), which says that every countably-based topological space is hemi-metrizable. You may have noticed that the proof strategy is very similar, too!

Indeed, each open subset *U* of *X* gives rise to a binary relation *R _{U}*, but that relation has many properties. First and foremost, it is a preordering. Indeed,

*R*o

_{U}*R*⊆

_{U}*R*, as we have shown in the proof above, and that means that

_{U}*R*is transitive; and every entourage is reflexive by definition. Being a preordering, it induces a hemi-metric

_{U}*d*, defined by

_{U}*d*(

_{U}*x*,

*y*) ≝ 0 if (

*x*,

*y*) ∈

*R*, and 1 (or ∞) otherwise. The open ball topology of that hemi-metric contains only

_{U}*U*, the empty set, and the whole space.

Now remember that the topology defined by a countable family of hemi-metrics is hemi-metrizable (Lemma 6.3.11). What we have just said shows the following.

**Proposition.** Every topology is defined by a family of hemi-metrics.

*Proof. *Just take the hemi-metrics *d _{U}*, where

*U*ranges over the open subsets. ☐

As an aside, the topologies defined by a family of pseudo-metrics (symmetric hemi-metrics) are exactly the completely regular topologies, and are also exactly the topologies induces by uniformities. Let me allow not to say why this month, and to postpone it to a later post.

## Morphisms

At this point, one may be tempted to think that there is no point in inventing a new notion (quasi-uniform spaces), since quasi-uniform spaces and topological spaces seem to be one and the same thing.

The difference is in *morphisms*. While the morphisms in the category **Top** of topological spaces are the continuous maps, the morphisms from *X* to *Y* in the category **QUnif** of quasi-uniform spaces are the *uniformly continuous maps*, namely the maps *f* : *X* → *Y* such that (*f*×*f*)^{–1} (*S*) is an entourage of *X* for every entourage *S* of *Y*.

Every uniformly continuous map *f* is continuous for the underlying induced topologies. Indeed, let *V* be any open subset of *Y*. For every *x* ∈ *f*^{–1}(*V*), we need to show that there is a neighborhood of *X* whose image by *f* is included in *V*. We use the fact that *f*(*x*) is in *V*, and the definition of the induced topology on *Y*, to obtain an entourage *S* in *Y* such that *S*[*f*(*x*)] ⊆ *V*. Let *R* be the entourage (*f*×*f*)^{–1} (*S*). Then *R*[*x*] is a neighborhood of *x*, by definition of the induced topology on *X*, and every element of its image by *f* is of the form *f*(*y*) with (*x*,*y*) ∈ *R*; so (*f*(*x*),*f*(*y*)) is in *S*, by definition of *R*, showing that *f*(*y*) is in *S*[*f*(*x*)], hence in *V*.

For quasi-metric spaces *X* and *Y*, with respective quasi-metrics *d _{X}* and

*d*, a uniformly continuous map

_{Y}*f*:

*X*→

*Y*between the corresponding quasi-uniform spaces is a map such that for every

*s*>0, (

*f*×

*f*)

^{–1}([<

*s*]) contains a basic entourage [<

*r*] for some

*r*>0. In other words: for every

*s*>0, there is an

*r*>0 such that for all points

*x*,

*y*such that

*d*(

_{X}*x*,

*y*)<

*r*,

*d*(

_{Y}*f*(

*x*),

*f*(

*y*))<

*s*. This is the usual definition of uniform continuity in (quasi-)metric spaces.

It is well-known that there are continuous maps between metric spaces that are not uniformly continuous, for example the map *x* ↦ 1/*x* on the positive real numbers. Hence that also holds in the realm of (quasi-)uniform spaces.

Rather remarkably, uniform continuity and continuity coincide on Pervin quasi-uniformities arising from topologies. Indeed, let *X* and *Y* be two topological spaces, and consider them with their Pervin quasi-uniformities. Let *f* : *X* → *Y* be a continuous map. For every basic entourage *S* ≝ *R _{V}*

_{1}∩ … ∩

*R*

_{V}_{n}in

*Y*, (

*f*×

*f*)

^{–1}(

*S*) is the set of pairs (

*x*,

*y*) of points of

*X*such that

*f*(

*x*) ∈

*V*

_{1}implies

*f*(

*y*) ∈

*V*

_{1}and … and

*f*(

*x*) ∈

*V*

_{n}implies

*f*(

*y*) ∈

*V*

_{n}, in other words (

*f*×

*f*)

^{–1}(

*S*) is the basic entourage

*R*

_{U}_{1}∩ … ∩

*R*

_{U}_{n}where

*U*

_{1}≝

*f*

^{–1}(

*V*

_{1}), …,

*U*

_{n}≝

*f*

^{–1}(

*V*

_{n}).

What all that means, categorically, is:

- There is a (forgetful) functor
*U*:**QUnif**→**Top**that maps every quasi-uniform space to the underlying topological space (with the induced topology), and every uniformly continuous map*f*to itself, seen as a continuous map. - There is a functor
*Perv*:**Top**→**QUnif**that maps every topological space to itself, seen as a quasi-uniform space with the Pervin quasi-uniformity of its topology. It maps every continuous map*f*to itself, since*f*is uniformly continuous as well, as we have just seen. - The composite functor
*U*o*Perv*is the identity functor on**Top**: indeed, that is what the Császár-Pervin theorem above states.

The composite functor *Perv* o *U* does not seem to have any notable feature, and in particular there does not seem to be any adjunction involving *Perv* and *U*. Notably, the Pervin quasi-uniformity of a topology ** O** is in general neither the smallest nor the largest quasi-uniformity that induces

**.**

*O*Here is a counterexample, again due to Pervin [3]. Consider the quasi-metric d_{R} on the real numbers: d_{R}(*a*,*b*)≝max(*a*–*b*,0). That induces a quasi-uniformity, whose basic entourages are [<*r*] = {(*a*,*b*) ∈ **R** × **R** | *a*<*b*+*r*}, *r*>0. The topology induced by this quasi-uniformity is the open ball topology of d_{R}, which is the Scott topology on **R**. The Pervin quasi-uniformity of this Scott topology has basic entourages of the form *R _{U}*

_{1}∩ … ∩

*R*

_{U}_{n}where each

*U*

_{i}is of the form ]

*a*

_{i}, ∞[.

- No such entourage contains any quasi-metric entourage [<
*r*]: any pair (*a*,*b*) where*b*is strictly larger than every*a*_{i}, and*a*≥*b*+*r*is in*R*_{U}_{1}∩ … ∩*R*_{U}_{n}but not in [<*r*]. - In the other direction, we claim that no quasi-metric entourage [<
*r*] contains any Pervin entourage*R*_{U}_{1}∩ … ∩*R*_{U}_{n}(where*U*_{i}= ]*a*_{i}, ∞[). if*n*=0, this is clear, since the empty intersection is the whole of**R**×**R**. Otherwise, let ε>0 be chosen such that ε<*r*. Then the point (*a*_{1}+ε,*a*_{1}) is in [<*r*], but not in*R*_{U}_{1}∩ … ∩*R*_{U}_{n}, in fact not even in*R*_{U}_{1}.

It follows that the two (Pervin and quasi-metric) quasi-uniformities on **R** are incomparable.

One may, at least, attempt to identify what kinds of quasi-uniform spaces can be obtained by applying *Perv* to arbitrary topological spaces. I do not know the answer to that question. However, if we are prepared to replace topological spaces by sets equipped with a lattice of subsets (not necessarily closed under arbitrary unions), *Perv* generalizes naturally, and the answer to the corresponding question is known, as I will now argue. The corresponding quasi-uniform spaces are called *Pervin spaces*.

## Pervin spaces

As far as I know, Pervin spaces were introduced by Mai Gehrke, Serge Grigorieff and Jean-Éric Pin, see [4]. They form a generalization of the Pervin quasi-uniformities introduced in the Császár-Pervin theorem: we now define the Pervin quasi-uniformity of any family of subsets of a set *X* whatsoever, not necessarily a topology. The formula is the same: for any subset *B* of *X*, *R _{B}* is the set of pairs (

*x*,

*y*) such that

*x*∈

*B*implies

*y*∈

*B*. Accordingly, the Császár-Pervin theorem has the following generalization.

**Theorem.** Let ** F** be a family of subsets of a set

*X*. The finite intersections of relations

*R*,

_{B}*B*∈

**, form a base**

*F***of entourages of a quasi-uniformity**

*B***on**

*U**X*, whose induced topology is exactly the topology

**generated by**

*O***on**

*F**X*. This quasi-uniformity is the

*Pervin quasi-uniformity*of the family

**.**

*F*Proof. The proof that ** U** is a quasi-uniformity is as in the Császár-Pervin theorem. Notably, we need to observe that

*R*o

_{B}*R*⊆

_{B}*R*, namely that

_{B}*R*is a transitive binary relation on

_{B}*X*for every

*B*in

**. This implies that every basic entourage**

*F**R*

_{B}_{1}∩ … ∩

*R*

_{B}_{n}is transitive as well.

Given any point *x* of *X*, and any basic entourage * S* ≝

*R*_{B}_{1}∩ … ∩

*, the image*

*R*_{B}_{n}*S*[

*x*] is equal to the intersection of the sets

*that contain*

*B*_{i}*x*. It follows that every

**-open set is a neighborhood of each of its points (relative to the topology**

*U***), hence is open in**

*O***. Conversely, let**

*O**U*be any open set in

**. For every**

*O**x*in

*U*, there is a finite intersection

*B*_{1}∩ … ∩

*B*_{n}of elements of

**that contains**

*F**x*and is included in

*U*. Let

*≝*

*S*

*R*_{B}_{1}∩ … ∩

*. Then*

*R*_{B}_{n}*S*[

*x*] =

*B*_{1}∩ … ∩

*B*_{n}is a

**-neighborhood of**

*U**x*included in

*U*, so

*U*is

**-open. ☐**

*U*In the proof, we have noticed that the Pervin quasi-uniformity ** U** of

**has a base of**

*F**transitive*entourages (namely, those of the form

*R*

_{B}_{1}∩ … ∩

*R*

_{B}_{n}). The quasi-uniformities with that property are called…

*transitive*.

Here is another property of ** U**. Given any entourage

*R*in

**,**

*U**R*contains some basic entourage

*R*

_{B}_{1}∩ … ∩

*R*

_{B}_{n}. For every subset

*I*of {1, …,

*n*}, let

*C*be the set obtained as the intersection of the sets

_{I}*B*

_{i}with

*i*∈

*I*, and of the complements of the sets

*B*

_{i}with

*i*∉

*I*. The family of all the sets

*C*, where

_{I}*I*ranges over the subsets of {1, …,

*n*}, forms a finite cover of the whole space

*X*. Moreover,

*C*×

_{I}*C*is included in

_{I}*R*

_{B}_{1}∩ … ∩

*R*

_{B}_{n}, hence in

*R*. Indeed, for every pair (

*x*,

*y*) in

*C*×

_{I}*C*, for every index

_{I}*i*, if

*i*∈

*I*then (

*x*,

*y*) is in

*R*

_{B}_{i}because

*y*is in

*B*

_{i}, and if

*i*∉

*I*then (

*x*,

*y*) is in

*R*

_{B}_{i}because

*x*is

*not*in

*B*

_{i}.

A quasi-uniformity ** U** on a set

*X*is

*totally bounded*if and only if for every entourage

*R*∈

**, there is a finite cover of**

*U**X*such that, for every element

*C*of that cover,

*C*×

*C*is included in

*R*. We have just argued that the Pervin quasi-uniformity of a family

**of subsets is always transitive and totally bounded.**

*F*It is instructive to see what this notion of ‘totally bounded’ means when restricted to a hemi-metric space *X*, *d*. For every subset *C* of *X*, for every *r*>0, *C* × *C* is included in [<*r*] if and only if, for all *x*, *y* in *C*, *d*(*x*,*y*)<*r*. Every such subset *C* is either empty, or included in the open ball with center *x* and radius *r* under the symmetrized pseudo-metric *d*^{sym}, where *x* is any given point of *C*. Hence, if *X*, *d* is totally bounded qua quasi-uniform space, then for every *r*>0, there is a finite cover of *X* by (sets included in) open balls of radius *r* under *d*^{sym}. In other words, *X*, *d* is totally bounded qua hemi-metric space (compare with Definition 6.7.4 in the book).

Conversely, if *X*, *d* is totally bounded qua hemi-metric space, then for every *r*>0, *X* has a cover by open balls *B* of radius *r*/2 under *d*^{sym}, and each of these open balls *B* satisfies *B* × *B* ⊆ [<*r*]. Therefore *X*, *d* is totally bounded qua quasi-uniform space. In summary, ‘totally bounded’ really means the same thing as the usual notion on hemi-metric spaces.

Let us return to Pervin quasi-uniformities. The following is due to Mai Gehrke, Serge Grigorieff and Jean-Éric Pin (April 2012; no, I will not give any more precise reference, because the document I have says that it is not finished and therefore must remain confidential—it is fairly easy to find it on the Web nonetheless). The proof I will give is different, though.

**Proposition.** A quasi-uniformity ** U** on a set

*X*is the Pervin quasi-uniformity of a family of subsets of

*X*if and only if it is transitive and totally bounded. In that case:

is the Pervin quasi-uniformity of the family*U*of subsets of*F**X*of the form*R*[*E*] ≝ {*y*∈*X*| ∃*x*∈*E*, (*x*,*y*) ∈*R*}, where*R*ranges over the transitive entourages inand*U**E*ranges over the subsets of*X*;is a subbase of the topology**F**induced by*O*;*U*is also the Pervin quasi-uniformity of the lattice*U*of finite unions of finite intersections of elements of*L*, and*F*is a base of*L*.*O*

Proof. We have already proved the ‘only if’ direction. Let us assume that ** U** is transitive and totally bounded.

We first note that every element of ** F** is open (in

**). Indeed, let us pick any such element. It is of the form**

*O**R*[

*E*], where

*R*is a transitive entourage, and

*E*is a subset of

*X*. For every point

*y*of

*R*[

*E*], every point

*z*∈

*R*[

*y*] is such that (

*y*,

*z*) ∈

*R*, and since (

*x*,

*y*) ∈

*R*for some

*x*in

*E*and

*R*is transitive, (

*x*,

*z*) is also in

*R*, showing that

*z*is in

*R*[

*E*]. This shows that

*R*[

*y*] is a neighborhood of

*y*included in

*R*[

*E*]. Since

*y*is arbitrary in

*R*[

*E*],

*R*[

*E*] is open in

**.**

*O*We claim that, given any element *B*≝*R*[*E*] of ** F**,

*R*is in

_{B}**. (Remember that**

*U**R*is assumed to be transitive, and in

**, here.) Every pair (**

*U**y*,

*z*) in

*R*is in

*R*: if

_{B}*y*is in

*B*=

*R*[

*E*], then (

*x*,

*y*) is in

*R*for some

*x*in

*E*, hence (

*x*,

*z*) is also in

*R*, since

*R*is transitive; so

*z*is in

*B*. This shows that

*R*is included in

*R*. By (filter 2),

_{B}*R*is in

_{B}**.**

*U*A similar argument shows that *R _{A}* is in

**for every**

*U**A*in

**. Let us write**

*L**A*as the finite union (over

*i*=1…

*m*) of the finite intersection (over

*j*=1…

*n*) of elements

_{i}*B*≝

_{ij}*[*

*R*_{ij}*E*] of

_{ij}**. Let**

*F**R*be the intersection of

*all*the relations

*, when*

*R*_{ij}*i*and

*j*vary. We claim that

*R*is included in

*R*. As above, this will entail that

_{A}*R*is in

_{A}**. Given any pair (**

*U**y*,

*z*) in

*R*such that

*y*is in

*A*, there is an index

*i*such that for every

*j*(1≤

*j*≤

*n*),

_{i}*y*is in

*B*=

_{ij}*[*

*R*_{ij}*E*]. Hence for every

_{ij}*j*, there is a point

*x*in

_{j}*E*such that (

_{ij}*x*,

_{j}*y*) is in

*. Since (*

*R*_{ij}*y*,

*z*) is in

*R*, hence in

*, and since*

*R*_{ij}*is transitive, (*

*R*_{ij}*x*,

_{j}*z*) is also in

*for every*

*R*_{ij}*j*. This shows that

*z*is in the intersection of the sets

*B*=

_{ij}*[*

*R*_{ij}*E*], 1≤

_{ij}*j*≤

*n*, hence in

_{i}*A*.

In the converse direction, we claim that every entourage *R* (in ** U**) contains some basic Pervin entourage

*R*

_{B}_{1}∩ … ∩

*R*

_{B}_{n}, where each

*B*is in

_{i}**. Since**

*F***is transitive, and up to the replacement of**

*U**R*by a smaller entourage, we may assume that

*R*is transitive. We now use total boundedness. Let {

*C*_{1}, …,

*C*_{n}} be some finite cover of

*X*such that each product

*C*_{i}×

*C*_{i}is included in

*R*. For each

*i*,

*R*[

*C*_{i}] is in

**, by definition. We let**

*F**B*≝

_{i}*R*[

*C*_{i}]. For every pair (

*x*,

*y*) in

*R*

_{B}_{1}∩ … ∩

*R*

_{B}_{n}, we claim that (

*x*,

*y*) is in

*R*. Since {

*C*_{1}, …,

*C*_{n}} is a cover, there is an index

*i*such that

*x*is in

*C*_{i}, hence in the larger set

*B*=

_{i}*R*[

*C*_{i}]. (Note that

*is included in*

*C*_{i}*R*[

*C*_{i}] because

*R*is reflexive.) Now (

*x*,

*y*) is in

*R*

_{B}_{i}, so

*y*is also in

*B*. It follows that there is a point

_{i}*z*in

*C*_{i}such that (

*z*,

*y*) is in

*R*. Therefore (

*x*,

*z*) is in

*C*_{i}×

*C*_{i}, hence in

*R*. It follows that (

*x*,

*y*) is also in

*R*, since

*R*is transitive. This finishes to show that

*R*

_{B}_{1}∩ … ∩

*R*

_{B}_{n}is included in

*R*.

All this establishes item 1, and also item 3. Item 2 is a consequence of the previous Theorem. ☐

**Warning.** The previous proposition does *not* show that * U* is the Pervin topology of the topology

**, only of its subbase**

*O**, or of its base*

**F****. The latter is closed under finite unions, but not under arbitrary unions in general.**

*L*There is much more that one can say about quasi-uniform spaces… enough to fill a book, quite certainly. One should certainly read Hans-Peter Künzi’s introduction to the subject [5], which is already quite encyclopedic! I was sad to hear that he had recently passed away (see this page). I had met him once or twice; the last time in Leicester, UK, in 2016, and I certainly enjoyed the lively breakfast we had there one morning with his many students and postdocs.

- Nicolas Bourbaki. Topologie générale (éléments de mathématique), chapitre 2. Springer, 2007. (Many other, previous editions, too.)
- Ákos Császár. Fondements de la topologie générale. Gauthier-Villars, Paris, 1960.
- William Joseph Pervin. Quasi-Uniformisation of Topological Spaces.
*Mathematische Annalen*147:316–317, 1962. - Jean-Éric Pin. Pervin Spaces. Slides of a talk given in Coimbra, Portugal, September 2016.
- Hans-Peter Albert Künzi. An Introduction to Quasi-Uniform Spaces. In
*Beyond Topology*, pages 239-304, volume 486 of the*Contemporary Mathematics*series, American Mathematical Society, 2009, edited by Frédéric Mynard and Elliott Pearl.