Let us continue our exploration of quasi-uniform spaces. I was recently reading a paper by Jimmie Lawson [2] about stably compact spaces, and which, I must confess, I did not know about until a few months ago. It contains a wealth of information on stably compact spaces, giving a lucid and comprehensive view of the area. If I could rewrite Chapter 9 of the book, I would definitely take example on this paper.

And, in Section 3.5 of [2], surprise! Jimmie mentions a neat result about the quasi-uniformities that induce the topology of a stably compact space, which he attributes to Künzi and Brümmer [3]. I immediately decided to talk about it here.

Let me say what this is about. In the setting of uniform spaces, it has been well-known for a long time that the topology of a compact Hausdorff space is induced by a unique uniformity [1, §4, 1, théorème 1]. Jimmie says that this result extends to stably compact spaces, and I tried to understand [3] in order to explain why in a hopefully simpler way. Unfortunately, the expected result, which would be that there is a unique quasi-uniformity that induces any given stably compact topology, is wrong, as I will demonstrate below. We will see what goes wrong, and progressively make our way towards a theorem that indeed generalizes the situation on compact Hausdorff spaces.

## Is the topology of stably compact space induced by a unique quasi-uniformity?

That is the natural conjecture we might have that would generalize the aforementioned theorem on compact Hausdorff spaces. A quick reading of [2] or [3] might convince you that this what Künzi and Brümmer somehow proved, but that would be wrong.

Here is a simple counter-example—in fact almost the same counter-example as last time. Consider the interval [0, 1] with the Scott topology of ≤, whose open intervals are the empty set, the whole interval, and the half-open intervals ]*a*,1], with 0<*a*<1. This is a stably compact space. But there are at least two quasi-uniformities that induce the Scott topology:

- the Pervin quasi-uniformity, generated by the entourages
*R*_{U}≝ {(*x*,*y*) ∈ [0,1] × [0,1] |*x*∈*U*implies*y*∈*U*}, where*U*ranges over the Scott-open subsets of [0, 1], and - the
*quasi-metric quasi-uniformity*, namely the quasi-uniformity generated by the entourages [<*r*] ≝ {(*x*,*y*) ∈ [0,1] × [0,1] |*x*<*y*+*r*}, where*r*is a (non-zero) positive real number, in other words the quasi-uniformity induced by the quasi-metric d_{R}on [0,1].

Now consider *R*_{U}, where *U* ≝ ]1/2, 1]. That cannot contain any finite intersection [<*r*_{1}] ∩ … ∩ [<*r*_{n}] of entourages of the second quasi-uniformity. We argue as follows. First, if *R*_{U} contained such an intersection, then it would included the smaller entourage [<*r*], where *r* ≝ min (*r*_{1}, …, *r*_{n}, 3/2). Note that *r* is non zero. Second, the pair (1/2+*r*/3, 1/2–*r*/3) is in [<*r*], but not in *R*_{U}. (The reason of the “3/2” term in the definition of *r* is to make sure that the two components of this pair are in the interval [0,1].) Hence *R*_{U} is not in the quasi-metric quasi-uniformity on [0, 1]. This shows that the Pervin quasi-uniformity differs from the quasi-metric quasi-uniformity. (It in fact contains it strictly.)

## The specialization preordering

Given any quasi-uniformity ** U** on

*X*, there is a binary relation ∩

**, which is simply the intersection of all the entourages in**

*U***: (**

*U**x*,

*y*) is in ∩

**if and only if, for every entourage**

*U**R*in

**, (**

*U**x*,

*y*) is in

*R*. It turns out that this is a preordering, and a familiar one.

**Lemma A.** For every quasi-uniformity ** U** on

*X*, ∩

**is the specialization preordering of the topology induced by**

*U***.**

*U*Proof. Let ≤ be the specialization preordering of the topology induced by ** U**. If

*x*≤

*y*, then by definition every neighborhood of

*x*contains

*y*. This applies to every neighborhood of the form

*R*[

*x*] with

*R*an entourage in

**, so**

*U**y*is in

*R*[

*x*] for every

*R*∈

**. In other words, (**

*U**x*,

*y*) is in

*R*for every

*R*∈

**, namely, (**

*U**x*,

*y*) is in ∩

**. Conversely, if (**

*U**x*,

*y*) is in ∩

**, then for every open neighborhood**

*U**U*of

*x*in the induced topology, by definition there is an entourage

*R*such that

*R*[

*x*] is included in

*U*. Since (

*x*,

*y*) is in ∩

**, in particular (**

*U**x*,

*y*) is in

*R*, so

*y*is in

*R*[

*x*], and therefore in

*U*. It follows that

*x*≤

*y*. ☐

## The dual quasi-uniformity

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

*X*, there is a

*dual*quasi-uniformity

**U**^{–1}, whose entourages are the opposites

*R*

^{–1}(also written as

*R*

^{op}) of entourages

*R*in

**—by definition, (**

*U**x*,

*y*) is in

*R*

^{–1}if and only if (

*y*,

*x*) is in

*R*.

The quasi-uniformity ** U** induces a topology on

*X*, and from now on I will consider

*X*as a topological space with that topology.

The dual quasi-uniformity **U**^{–1} also induces a topology on *X*, but what can we say about it? Without any further assumption, we can say at least two things. The first one has to do with specialization preorderings. Let me write *X*^{–1} for *X* with the topology induced by **U**^{–1}.

**Lemma B.** Let *X* be a set with a quasi-uniformity ** U**, and let ≤ be the specialization preordering of

*X*. The specialization preordering of

*X*

^{–1}is

*≤*

^{–1}(which I will also write as ≥).

*Proof.* By Lemma A, the specialization preordering of *X*^{–1} is ∩(**U**^{–1}), the intersection of all the relations *R*^{–1}, when *R* ranges over the entourages of ** U**. This is also equal to (∩

*)*

**U**^{–1}, namely ≥, by Lemma A again. ☐

The second thing has to do with compact saturated subsets.

**Lemma C.** Let *X* be a set with a quasi-uniformity ** U**. Every compact saturated subset of

*X*is closed in

*X*

^{–1}.

*Proof.* Let *Q* be a compact saturated subset of *X*, and *O* be its complement. We wish to show that *O* is open in *X*^{–1}, and that means that for every point *x* outside *Q*, we would like to find an entourage *R* ∈ ** U** such that

*R*

^{–1}[

*x*] is included in

*O*, in other words does not intersect

*Q*.

We fix such a point *x* outside *Q*. Since *Q* is saturated, it is the intersection of its open neighborhoods. Hence there is an open neighborhood *U* of *Q* that does not contain *x*. By definition of the topology of *X*, for every point *y* of *Q* (which is in *U*), there is an entourage *S _{y}* ∈

**such that**

*U**S*[

_{y}*y*] ⊆

*U*.

Using the property of quasi-uniformities which we called (relaxed transitivity) last time, for each such *y* ∈ *Q*, there is a further entourage *R _{y}* ∈

**such that**

*U**R*o

_{y}*R*⊆

_{y}*S*.

_{y}Each set *R _{y}*[

*y*] is a neighborhood of

*y*. Hence the collection of all interiors of sets

*R*[

_{y}*y*],

*y*∈

*Q*, forms an open cover of

*Q*. Since

*Q*is compact, we can extract a finite subcover. In other words, there is a finite subset

*E*of

*Q*such that

*Q*is included in ∪

_{y ∈ E}

*R*[

_{y}*y*]. Let

*R*be the intersection of the entourages

*R*,

_{y}*y*∈

*E*. Since

*E*is finite, this is again an entourage, that is, it is again in

**. We claim that**

*U**R*is the desired entourage, namely that

*R*

^{–1}[

*x*] does not intersect

*Q*.

In order to prove this, we will show that *R*^{–1}[*x*] does not intersect the larger set ∪_{y ∈ E} *R _{y}*[

*y*]. If it did, then

*R*

^{–1}[

*x*] would intersect

*R*[

_{y}*y*] for some

*y*∈

*E*, say at

*z*. Since

*z*∈

*R*[

_{y}*y*], we would have (

*y*,

*z*) ∈

*R*. Since

_{y}*z*∈

*R*

^{–1}[

*x*], we would have (

*z*,

*x*) ∈

*R*, hence (

*z*,

*x*) ∈

*R*. Therefore (

_{y}*y*,

*x*) would be in

*R*o

_{y}*R*, hence in

_{y}*S*. In turn, this would mean that

_{y}*x*would be in

*S*[

_{y}*y*], and therefore in

*U*. But

*U*does not contain

*x*by definition. We have reached a contradiction, and that terminates the proof. ☐

This is promising! If *X* is stably compact, then we would expect its de Groot dual *X*^{d} to coincide with *X*^{–1}, and the results above are natural steps in that direction.

I do no think that we can say much more without requiring some further properties from *X*. We now look at the case where *X* is core-compact, then we will specialize to the locally compact case, and then to the stably compact case.

## Core-compact induced topologies

Last time, we had noticed that the Pervin quasi-uniformity was in general neither the smallest nor the largest quasi-uniformity inducing a given topology. If that topology is core-compact, the situation changes, as we see now.

Let therefore *X* be a core-compact space, and let us write ⋐ for the way-below relation on **O***X*. For every pair of open subsets *U*, *V* such that *U* ⋐ *V*, let us write [*U* ⇒ *V*] for the binary relation consisting of all the pairs of points (*x*,*y*) such that *x* ∈ *U* implies *y* ∈ *V* (namely, such that *x* is not in *U*, or *y* is in *V*). We have:

**Proposition D.** Let *X* be a core-compact space. The finite intersections of relations [*U* ⇒ *V*], where *U*, *V* range over the pairs of open subsets such that *U* ⋐ *V*, form a base for a quasi-uniformity *U*_{0}. The quasi-uniformity *U*_{0} is the smallest compatible quasi-uniformity, namely the smallest uniformity that induces the given topology on *X*.

(Note added on December 12th, 2020: Proposition D is essentially Lemma 5 of [3].)

*Proof.* In order to verify that *U*_{0} is a quasi-uniformity, or rather that the given finite intersections form a base of a quasi-uniformity, let me recall that we need to verify the following, where **B_{0} **is the collection of intersections [

*U*

_{1}⇒

*V*

_{1}] ∩ … ∩ [

*U*

_{n}⇒

*V*

_{n}],

*n*∈

**N**, where for each

*i*,

*U*

_{i}and

*V*

_{i}are open and

*U*

_{i}⋐

*V*

_{i}:

- (reflexivity) for every relation
*R*in, for every*B*_{0}*x*in*X*, (*x*,*x*) is in*R*; - (filter 3) for all
*R*,*S*in, there is a relation*B*_{0}*T*insuch that*B*_{0}*T*⊆*R*∩*S*; - (relaxed transitivity) for every
*S*in, there is a relation*B*_{0}*R*insuch that*B*_{0}*R*o*R*⊆*S*.

Only the latter requires an argument. Given any *S* ≝ [*U*_{1} ⇒ *V*_{1}] ∩ … ∩ [*U*_{n} ⇒ *V*_{n}] in **B_{0}**, we use interpolation (remember that

*X*core-compact means that

**O**

*X*is a continuous dcpo) and we find open subsets

*W*

_{i}such that

*U*

_{i}⋐

*W*

_{i}⋐

*V*

_{i}for each

*i*. Let

*R*be the intersection of the 2

*n*relations [

*U*

_{i}⇒

*W*

_{i}] and [

*W*

_{i}⇒

*V*

_{i}], 1≤

*i*≤

*n*. Every pair of points (

*x*,

*y*) in

*R*o

*R*is such that there is a point

*z*with (

*x*,

*z*) and (

*z*,

*y*) in

*R*. In particular, if

*x*is in

*U*

_{i}then

*z*is in

*W*

_{i}; then

*y*is in

*V*

_{i}. As this holds for every

*i*, (

*x*,

*y*) is in

*S*.

In order to show that *U*_{0} is the smallest compatible quasi-uniformity, let ** U** be an arbitrary compatible quasi-uniformity. For every pair of open subsets

*U*and

*V*of

*X*such that

*U*⋐

*V*, we claim that [

*U*⇒

*V*] is in

**. Since**

*U***is closed under finite intersections, this will show that**

*U*

*U*_{0}is included in

**.**

*U*For every point *x* of *V*, there is an entourage *R*_{x} in ** U** such that

*R*

_{x}[

*x*] is included in

*V*, since the topology of

*X*is induced by

**. Using (relaxed reflexivity), we find an entourage**

*U**S*

_{x}in

**such that**

*U**S*o

_{x}*S*⊆

_{x}*R*

_{x}.

The interiors of the sets *S*_{x}[*x*], when *x* varies in *V*, form an open cover of *V*. Since *U* ⋐ *V*, we can extract a finite subcover of *U*. In other words, there is a finite subset *E* of *V* such that *U* is included in the union of the interiors of the sets *S*_{x}[*x*], *x* ∈ *E*. Let *R* be the intersection of the finitely many entourages *S*_{x}, *x* ∈ *E*. This is again in ** U**. Now every pair of points (

*y*,

*z*) in

*R*is in [

*U*⇒

*V*]: if

*y*is in

*U*, then

*y*is in

*S*

_{x}[

*x*] for some

*x*∈

*E*; since (

*y*,

*z*) in

*R*, it is also in

*S*

_{x}, so

*z*is in

*S*[

_{x}*S*[

_{x}*x*]], hence in

*R*

_{x}[

*x*] (since

*S*o

_{x}*S*⊆

_{x}*R*

_{x}), and therefore in

*V*.

We have shown that *R* is included in [*U* ⇒ *V*]. Since *R* is in ** U**, so is [

*U*⇒

*V*]. This shows that every subbasic entourage [

*U*⇒

*V*] of

*U*_{0}is in

**, hence that**

*U*

*U*_{0}is included in

**. ☐**

*U*## Locally compact induced topologies

We can prove a similar result on locally compact spaces, using a slightly different subbase of entourages in order to define *U*_{0}. For every compact saturated subset *Q* of *X* and every open neighborhood *V* of *Q*, let us write [*Q* ⇒ *V*] for the binary relation consisting of all the pairs of points (*x*,*y*) such that *x* ∈ *Q* implies *y* ∈ *V*.

If *X* is locally compact, then we claim that those entourages form a subbase for the same quasi-uniformity *U*_{0}. In one direction, given any pair of open sets *U* ⋐ *V*, there must be a compact saturated subset *Q* such that *U* ⊆ *Q* ⊆ *V* (Theorem 5.2.9 in the book). Then [*U* ⇒ *V*] contains [*Q* ⇒ *V*]. In the other direction, given any compact saturated subset *Q* of *X* and every open neighborhood *V* of *Q*, then by interpolation (Proposition 4.8.14), there is a compact saturated set *Q’* included in *V* and whose interior *U* contains *Q*. We see that [*Q* ⇒ *V*] contains [*U* ⇒ *V*]. Notice that we also have *U* ⋐ *V*. We can rephrase all this as follows.

**Proposition D’.** Let *X* be a locally compact space. The finite intersections of relations [*Q* ⇒ *V*], where *Q* ranges over the compact saturated subsets of *X* and *V* ranges over the open neighborhoods of *Q*, form a base for the smallest compatible quasi-uniformity *U*_{0}.

To sum up, and still assuming *X* locally compact, a base of entourages of *U*_{0} is given by the finite intersections [*Q*_{1} ⇒ *U*_{1}] ∩ … ∩ [*Q*_{n} ⇒ *U*_{n}], *n* ∈ **N**, where each *Q*_{i} is compact saturated, each *U*_{i} is open, and *Q*_{i} ⊆ *U*_{i}. Those basic entourages can be rewritten in another form.

We rely on a bit of Boolean gymnastics. A pair (*x*,*y*) is in [*Q*_{1} ⇒ *U*_{1}] ∩ … ∩ [*Q*_{n} ⇒ *U*_{n}] if and only if the conjunction (over all *i*, 1≤*i*≤*n*) of the formulae ‘*x* ∈ *X*–*Q*_{i} or *y* ∈ *U*_{i}‘ holds. Distributing ors over ands, that is equivalent to the disjunction, over all subsets *I* of {1, …, *n*}, of ‘*x* ∈ *X*–*Q*_{I} and *y* ∈ *U*_{I}‘, where *U*_{I} is the intersection of the sets *U*_{i}, *i* ∈ *I*, and *Q*_{I} is the *union* of the sets *Q*_{i}, *i* ∈ *I*. (If *I* is empty, *U*_{I} is equal to the whole of *X*.)

Hence we have shown that if *X* is locally compact, any basic entourage [*Q*_{1} ⇒ *U*_{1}] ∩ … ∩ [*Q*_{n} ⇒ *U*_{n}] of *U*_{0} is equal to a finite union of products (*X*–*Q*_{I}) × *U*_{I}, where each *Q*_{I} is compact saturated and each *U*_{I} is open.

Moreover, that union contains (≤) ≝ {(*x*,*y*) ∈ *X* × *X* | *x*≤*y*}, the graph of the specialization preordering ≤. Indeed, for any pair of points *x* and *y* such that *x*≤*y*, by Lemma A, (*x*,*y*) is in every entourage of *U*_{0}, in particular in [*Q*_{1} ⇒ *U*_{1}] ∩ … ∩ [*Q*_{n} ⇒ *U*_{n}] = ∪_{I ⊆ {1, …, n}} (*X*–*Q*_{I}) × *U*_{I}.

In other words, if *X* is locally compact, then any basic entourage [*Q*_{1} ⇒ *U*_{1}] ∩ … ∩ [*Q*_{n} ⇒ *U*_{n}] of *U*_{0} is an open neighborhood of (≤) in *X*^{d} × *X*. It follows that *every* entourage of *U*_{0}, which contains such a basic entourage by definition, is also a neighborhood of (≤) in *X*^{d} × *X*.

Oh yes, I am using the notation *X*^{d} for the de Groot dual of *X*, even when *X* is not stably compact. Its topology, the *cocompact topology*, is the topology *generated* by the complements of the compact saturated subsets of *X*. The closed subsets of *X*^{d} are the intersections of compact saturated subsets of *X*, and with compactness and coherence, those are also the filtered intersections of compact saturated subsets of *X*. When *X* is well-filtered, those filtered intersections are themselves compact saturated, and we retrieve the definition of the de Groot dual of the book.

## Stably compact spaces

**Proposition E.** If *X* is stably compact, then *U*_{0} is exactly the collection of neighborhoods of (≤) in *X*^{d} × *X*.

*Proof.* Given what we have just argued, it is enough to show that every neighborhood *R* of (≤) in *X*^{d} × *X* is in *U*_{0}. We will use the fact that (*X*^{patch},≤) is a compact pospace. (See Section 9.1 of the book for the relation between compact pospaces and stably compact spaces.)

For every *x* ∈ *X*, (*x*,*x*) is in (≤), so there is an open neighborhood *X*–*Q*_{x} of *x* in *X*^{d} (namely, *Q*_{x} is compact saturated in *X* and *x* is not in *Q*_{x}) and an open neighborhood *U*_{x} of *x* in *X* such that (*X*–*Q*_{x}) × *U*_{x} ⊆ *R*. The set (*X*–*Q*_{x}) ∩ *U*_{x} = *U*_{x} – *Q*_{x} is open in the patch topology of *X*, which is by definition the coarsest topology containing both the open subsets of *X* and the complements of compact saturated subsets of *X*. Therefore the sets *U*_{x} – *Q*_{x}, *x* ∈ *E*, forms an open cover of *X*^{patch}.

Since *X*^{patch} is compact, there is a finite subset *E* of *X* such that the sets *U*_{x} – *Q*_{x}, *x* ∈ *E*, already forms an open cover of *X*^{patch}. Let us consider *S* ≝ ∪_{x ∈ E} (*X*–*Q _{x}*) ×

*U*

_{x}. We observe that

*S*is included in

*R*: this is because (

*X*–

*Q*

_{x}) ×

*U*

_{x}⊆

*R*for every

*x*∈

*X*.

For each subset *I* of *E*, let *Q _{I}* be the intersection of the sets

*Q*

_{x}with

*x*∈

*I*, and let

*U*be the union of the sets

_{I}*U*

_{x}with

*x*∈

*I*. The pairs (

*y*,

*z*) that are in

*S*are exactly those that satisfy the disjunction (over all

*x*in

*E*) of the formulae ‘

*y*∉

*Q*

_{x}and

*z*∈

*U*

_{x}‘. By distributing ands over ors, they are those that satisfy the conjunction, over all subsets

*I*of

*E*, of the formulae ‘ for some

*x*in

*I*,

*y*is not in

*Q*

_{x}, or for some

*x*in

*E*–

*I*,

*z*is in

*U*

_{x}‘, equivalently, ‘

*y*∉

*Q*

_{I}or

*z*∈

*U*

_{E–I}‘. In other words,

*S*is equal to ∩

_{I ⊆}

_{E}[

*Q*

_{I}⇒

*U*

_{E–I}].

This (almost) proves that *S* is in *U*_{0}, and therefore that the larger relation *R* is in *U*_{0}. We still need to check that *Q*_{I} is included in *U*_{E–I} for every subset *I* of *E*! And that holds because the union of the sets *U*_{x} – *Q*_{x}, *x* ∈ *E*, is the whole of *X*. Indeed, the latter implies that the union of the larger sets *U*_{x} (instead of *U*_{x} – *Q*_{x}) with *x* ∈ *E*–*I*, and of the larger sets *X* – *Q*_{x} (instead of *U*_{x} – *Q*_{x}) with *x* ∈ *E*–*I*, is also the whole of *X*. But that new union is *U*_{E–I} ∪ (*X* – *Q*_{I}), and the fact that it is equal to *X* is exactly equivalent to the inclusion *Q*_{I} ⊆ *U*_{E–I}. ☐

We can also rewrite this as follows. The diagonal is the set of pairs (*x*,*x*). This is the graph (=) of the equality relation =.

**Proposition E’.** If *X* is stably compact, then *U*_{0} is exactly the collection of neighborhoods of the diagonal (=) in *X*^{d} × *X*.

Indeed, any neighborhood *R* of (=) in *X*^{d} × *X* must contain ∩*U*_{0}=(≤), by Lemma A.

## Duality

I have already said that, given a set *X* with a quasi-uniformity ** U**, seen with the induced topology, every compact saturated subset of

*X*is closed in

*X*

^{–1}. This means that the cocompact topology on

*X*is coarser than the topology of

*X*

^{–1}. When

**is**

*U*

*U*_{0}, the minimal compatible quasi-uniformity (see Proposition D), those two topologies

*coincide*, as we now argue.

**Proposition F.** Let *X* be a locally compact topological space, and ** U** be the minimal compatible quasi-uniformity

*U*_{0}. The topology induced by the dual quasi-uniformity

**U**^{–1}on

*X*coincides with the cocompact topology.

*Proof.* Let *O* be an open subset of *X* in the topology induced by **U**^{–1}. By definition, for every *x* ∈ *O*, there is a basic entourage *R*≝[*Q*_{1} ⇒ *U*_{1}] ∩ … ∩ [*Q*_{n} ⇒ *U*_{n}] (where each *Q*_{i} is compact saturated, each *U*_{i} is open, and *Q*_{i} ⊆ *U*_{i}) such that *R*^{–1}[*x*] ⊆ *O*.

We expand the definition of *R*: *R*^{–1}[*x*] is the set of points *y* such that for every *i*, if *y* ∈ *Q*_{i} then *x* ∈ *U*_{i}; equivalently, such that for every *i*, if *x* ∉ *U*_{i} then *x* ∉ *Q*_{i}; in other words, it is the complement of *Q*_{I}, where *Q*_{I} is the union of the sets *Q*_{i}, *i* ∈ *I*, and *I* is the collection of indices *i* such that *x* ∉ *U*_{i}. *Q*_{I} is compact saturated, so its complement *R*^{–1}[*x*] is open in the cocompact topology. This complement contains *x* (*R*^{–1}[*x*] always contains *x*), and is included in *O*. This shows that *O* is an open neighborhood, in the cocompact topology, of each of its points, so that *O* is open in the cocompact topology.

Conversely, let *O* be any open subset of *X* in the cocompact topology. Its complement *Q* is compact saturated in *X*. By Lemma C, *O* is open in the topology induced by **U**^{–1}. ☐

We finally reach the result promised at the beginning of this post.

**Theorem.** Let *X* be a stably compact topological space. There is a unique quasi-uniformity ** U** that induces the topology of

*X*and such that the dual quasi-uniformity

**U**^{–1}induces the cocompact topology, and this is the minimal compatible quasi-uniformity

*U*_{0}.

*Proof.* Existence is by Proposition F. In order to show uniqueness, we fix a quasi-uniformity ** U** that induces the topology of

*X*and such that the dual quasi-uniformity

**U**^{–1}induces the cocompact topology. By Proposition D’,

**contains**

*U*

*U*_{0}, so we concentrate on showing the reverse inclusion.

Let *R* be any entourage of ** U**. There is an entourage

*S*in

**such that**

*U**S*o

*S*⊆

*R*. For every

*x*∈

*X*, it follows that

*S*

^{–1}[

*x*] ×

*S*[

*x*] is included in

*R*: every pair (

*y*,

*z*) in

*S*

^{–1}[

*x*] ×

*S*[

*x*] is such that (

*y*,

*x*) ∈

*S*and (

*x*,

*z*) ∈

*S*, so (

*y*,

*z*) ∈

*R*.

*S*

^{–1}[

*x*] is an open neighborhood of

*x*in

*X*

^{d}since

**U**^{–1}induces the cocompact topology on

*X*, and

*S*[

*x*] is an open neighborhood of

*x*in

*X*since

**induces the original topology on**

*U**X*. Therefore

*R*is an open neighborhood of (

*x*,

*x*) in

*X*

^{d}×

*X*, for every

*x*∈

*X*. In other words,

*R*is an open neighborhood of (=) in

*. By Proposition E’,*

*X*^{d}×*X**R*in

*U*_{0}, and this finishes the proof. ☐

## An extremely short glimpse at the bitopological view

The proper way of stating the latter theorem is by appealing to the notion of a *bitopology*. (Yet another concept I have never talked about, at least until now!) A bitopology on a set *X* is simply a pair of topologies on *X*, and a bitopological space is simply a set with a bitopology. Most concepts in the theory of bitopological space rely on the interaction of the two topologies, and I really should say more about this some time. (Tomáš Jakl, notably, drew my attention to this subject again pretty recently.)

Every stably compact space has a canonical bitopology, which consists of its original topology, and its cocompact topology. If, as above, we define the cocompact topology as being *generated* by the complements of compact saturated sets, this even makes sense for arbitrary spaces. I do not know of any standard name for this bitopology; let me call it the *cocompact-open* bitopology on *X*.

Every quasi-uniformity ** U** on a space

*X*induces a bitopology on

*X*, too, namely the one formed by the topology induced by

**and the topology induced by**

*U*

**U**^{–1}. Lemma C states that this bitopology is pairwise finer than the cocompact-open bitopology (with the obvious meaning of “pairwise finer”). Proposition F states that the bitopology induced by the minimal quasi-uniformity

*U*_{0}on a locally compact space

*coincides*with the cocompact-open bitopology.

Using this bit of vocabulary, we can rephrase our findings as follows, when *X* is stably compact:

- there is no unique quasi-uniformity that induces the topology of
*X*, - but there is a minimal one
*U*_{0}, *U*_{0}is also the unique quasi-uniformity that induces the cocompact-open bitopology on*X*,- and is simply the set of open neighborhoods of (≤), or equivalently of the diagonal (=), in
*X*^{d}×*X*.

Let me conclude by saying how this generalizes the result mentioned at the very beginning of this post: that the topology of a compact Hausdorff space is induced by a unique uniformity [1, §4, 1, théorème 1]. We argue as follows. If *X* is compact Hausdorff, then its cocompact and its open topologies coincide. By what we have seen, there is a unique quasi-uniformity ** U** on

*X*such that both

*U*and

**U**^{–1}induce the topology of

*X*. That is, of course,

*U*_{0}. Since

*X*

^{d}=

*X*when

*X*is compact Hausdorff,

*U*_{0}is simply the set of open neighborhoods of (=), and that is a

*symmetric*quasi-uniformity (i.e.,

**=**

*U*

**U**^{–1}), in other words it is a uniformity.

- Nicolas Bourbaki. Topologie générale (éléments de mathématique), chapitre 2. Springer, 2007. (Many other, previous editions, too.)
- Jimmie Lawson. Stably compact spaces. Mathematical Structures in Computer Science 21(1):125-169, Feb. 2011.
- Hans-Peter Albert Künzi and Guillaume C. L. Brümmer. Sobrification and bicompletion of totally bounded quasi-uniform spaces, Mathematical Proceedings of the Cambridge Philosophical Society 101(2):237–247, 1987.

— Jean Goubault-Larrecq (November 21st, 2020)