The notion of strongly compact set is due to Reinhold Heckmann [1]. A subset *Q* of a space *X* is called *strongly compact* if and only if for every open neighborhood *U* of *Q*, there is a finitary compact set ↑*E* such that *Q* ⊆ ↑*E* ⊆ *U*. (A finitary compact set is the upward closure of a finite set *E*.)

I have already mentioned there that for every sober space *X*, the sobrification of **Q**_{fin}(*X*) of finitary compact subsets of *X*, with the upper Vietoris topology, is the space **Q**_{s}(*X*) of so-called *strongly* compact saturated subsets of *X*, not the whole Smyth hyperspace **Q**(*X*) as one might expect [2, Proposition 7.33]. Let us explore this in a bit more detail.

First, we need to notice that every strongly compact set is compact. If *Q* is strongly compact indeed, then every open cover (*U _{i}*)

_{i ∈ I}of

*Q*is such that

*Q*⊆ ↑

*E*⊆ ∪

_{i ∈ I}

*U*for some finite set

_{i}*E*. Then

*E*is included in the union of finite many of the sets

*U*, which therefore form a finite subcover of

_{i}*Q*.

If *X* is a continuous dcpo in its Scott topology, or more generally a locally finitary compact space, then, conversely, every compact set *Q* is strongly compact. Indeed, for every open neighborhood *U* of *Q*, we use locally finitary compactness in order to find finitary compact neighborhoods ↑*E _{x}* of each point

*x*of

*Q*, and included in

*U*; finitely many cover

*Q*, and the resulting finite union is a finitary compact set that contains

*Q*and is included in

*U*.

But, in more general spaces, there are in general strictly more compact sets than strongly compact sets. For example, in any T_{1} space, it is easy to see that the strongly compact sets are exactly the finite sets, and that is very far from exhausting the collection of compact sets.

## The sobrification of **Q**_{fin}(*X*)

Let **Q**(*X*) be the space of all compact saturated subsets of *X*, with the upper Vietoris topology: its basic open subsets are of the form ☐*U* ≝ {*Q* ∈ **Q**(*X*) | *Q* ⊆ *U*}, where *U* ranges over the open subsets of *X*. **Q**(*X*) is a T_{0} space, whose specialization ordering is *reverse* inclusion.

One can say more. First, there is a monad (**Q**, η, μ) on the category **Top**_{0} of T_{0} spaces, whose unit η* _{X}* :

*X*→

**Q**(

*X*) maps every point

*x*of

*X*to ↑

*x*, and whose multiplication μ

*:*

_{X}**Q**(

**Q**(

*X*)) →

**Q**(

*X*) maps every element

**of**

*Q***Q**(

**Q**(

*X*)) to ∪

**, the union of all the elements of**

*Q***. It is practical to note that η**

*Q*

_{X}^{–1}(☐

*U*)=

*U*and μ

_{X}^{–1}(☐☐

*U*)=☐

*U*for every open subset

*U*of

*X*; this is useful in order to show that the unit and the multiplication are continuous, for example.

We will also need to observe that the operator ☐ commutes with finite intersections and with directed unions of open sets; the latter is a consequence of the compactness of elements of **Q**(*X*).

**Lemma.** For every sober space *X*, **Q**(*X*) is sober.

*Proof.* Let **C** be an irreducible closed subset of **Q**(*X*). We consider the family **F** of all open subsets *U* of *X* such that ☐*U* intersects **C**. **F** is non-empty, upwards-closed and if *U* and *V* are any two elements of **F**, then since **C** is irreducible, ☐*U* ∩ ☐*V* intersects **C**. Since ☐*U* ∩ ☐*V* =☐(*U* ∩ *V*), *U* ∩ *V* is in **F**. Hence, **F** is a filter of open subsets of *X*. Similarly, but using the fact that ☐ commutes with directed unions of open sets, **F** is Scott-open. By the Hofmann-Mislove theorem (Theorem 8.3.2 in the book), **F** is the collection of open neighborhoods of some compact saturated subset *Q*_{0} of *X*.

If *Q*_{0} were not in **C**, then it would be in its complement, which is open, so there would be a basic open set ☐*U* such that *Q*_{0} ∈ ☐*U* and ☐*U* does not intersect **C**. The latter means that *U* is not in **F**, hence does not contain *Q*_{0}; this contradicts *Q*_{0} ∈ ☐*U*. Therefore *Q*_{0} is in **C**.

In order to show that **C** is the closure of *Q*_{0}, it remains to show that every element *Q* of **C** is below, namely contains, *Q*_{0}. It suffices to show that every open neighborhood *U* of *Q* contains *Q*_{0}, and that is easy: since *Q* ⊆ *U*, ☐*U* intersects **C** at *Q*, so *U* is in **F**, and therefore contains *Q*_{0}. ☐

One might guess that **Q**(*X*) would be the sobrification of its subspace **Q**_{fin}(*X*) of finitary compact subsets, but that is wrong. For example, you can check that, if *X* is T_{2}, then **Q**_{fin}(*X*) is already sober, and very different from **Q**(*X*). Since that will follow from the next result, I will not bother to show this.

**Proposition.** For every sober space *X*, the sobrification of **Q**_{fin}(*X*) is **Q**_{s}(*X*), the subspace of **Q**(*X*) consisting of strongly compact saturated subsets of *X*.

*Proof.* Since **Q**(*X*) is sober, the sobrification of **Q**_{fin}(*X*) is its Skula-closure, as we have seen in this post. (Namely, the closure in the Skula topology. The Skula topology is generated by the open *and* the closed sets, or equivalently by the sets *U* ∩ ↓*z*, where *z* ranges over the points and *U* ranges over the open sets in the original topology.)

If *Q* is in the Skula-closure of **Q**_{fin}(*X*), then for every open neighborhood *U* of *Q*, ☐*U* ∩ ↓_{Q}*Q* is a Skula-open neighborhood of *Q* in **Q**(*X*); I am writing ↓** _{Q}** for downward closure in

**Q**(

*X*) (remember that the ordering there is

*reverse*inclusion!). And that set must therefore intersect

**Q**

_{fin}(

*X*). In other words, there is a finitary compact set ↑

*E*in ☐

*U*∩ ↓

_{Q}*Q*. We expand this, and we obtain

*Q*⊆ ↑

*E*⊆

*U*. Therefore

*Q*must be strongly compact saturated.

Conversely, for every strongly compact saturated subset *Q* of *X*, we claim that *Q* is in the Skula-closure of **Q**_{fin}(*X*). In other words, for every open neighborhood **U** of *Q* in **Q**(*X*), we show that **U** contains some finitary compact set containing *Q*. Necessarily, **U** contains a basic open neighborhood ☐*U* of *Q*. Since *Q* is strongly compact, there is a finitary compact set ↑*E* such that *Q* ⊆ ↑*E* ⊆ *U*; then ↑*E* is in ☐*U*, hence in **U**. ☐

One can then show that the monad (**Q**, η, μ) cuts down to a monad (**Q**_{s}, η, μ), and both monads themselves restrict to monads on the category **Sob** of sober spaces. As I have said in this post, the algebras of the monad (**Q**_{s}, η, μ) on **Sob** are exactly the deflationary sober semilattices *with small semilattices* [1, Theorem 7.37], and the proof is a pretty simple extension of what we had done with **Q**_{fin} in that same post.

Instead, I would like to mention a funny hyperspace commutation theorem, which I happened to obtain recently by rereading carefully Matthew de Brecht and Tatsuji Kawai’s paper [3].

## Hyperspace commutation results

Let **H**(*X*) denote the Hoare hyperspace of *X*: that is the set of closed subsets of *X*, with the lower Vietoris topology, given by subbasic open sets ♢*U* ≝ {*C* ∈ **Q**(*X*) | *C* intersects *U*}, where *U* ranges over the open subsets of *X*. As we have seen here, Schalk proved that **H**(*X*) is *always* a sober space. This also defines a monad (**H**, η, μ) [let me reuse the same η and μ as before] whose unit η* _{X}* :

*X*→

**H**(

*X*) maps every point

*x*of

*X*to ↓

*x*, and whose multiplication μ

*:*

_{X}**H**(

**H**(

*X*)) →

**H**(

*X*) maps every element

**of**

*C***H**(

**H**(

*X*)) to cl(∪

**). Also, η**

*C*

_{X}^{–1}(♢

*U*)=

*U*and μ

_{X}^{–1}(♢♢

*U*)=♢

*U*for every open subset

*U*of

*X*.

What de Brecht and Kawai achieved was to show that **QH***X* and **HQ***X* are homeomorphic, provided that *X* is consonant; and that this is actually an if and only if. This is the continuation of a long line of work, showing similar isomorphisms in more restricted categories, mostly of domains. The locale theoretic analogue of this result is due to Steven Vickers and Christopher Townsend [4] shows that the same commutation between the two localic analogues of **Q** and **H** (the so-called upper and lower power locale constructions) holds *without* any assumption on the locale.

Replacing **Q** by **Q**_{s}, I will show that **Q**_{s}**H***X* and **H Q**

_{s}

*X*are homeomorphic for

*every*topological space, consonant or not, sober or not.

Oh, by the way, if you read Heckmann [1], you will see that this commutation requires a property that he calls U_{S}-conformity, and it may puzzle you that I will not require any assumption on *X* at all. The main reason for this is that he considers the Scott topology, not the lower Vietoris topology, on **H***X*.

Let us start the proof. This is essentially the same as de Brecht and Kawai’s proof, replacing **Q** by **Q**_{s}. I will simply reorganize it, and mention how each step adapts when we replace **Q** by **Q**_{s}.

## The first isomorphisms: **QH**=**O**_{p}**O**_{σ}, **Q**_{s}**H**=**O**_{p}**O**_{p}

The first step consists in showing that there is a homeomorphism between **QH***X* and **O**_{p}**O**_{σ}*X*, valid for every topological space *X* [3, Theorem 6.10]. (Well, de Brecht and Kawai only show an order-isomorphism here, not a homeomorphism, but that is not too far.) Here **O***X* is the collection of open subsets of *X*, and we can topologize it in two ways:

- with the Scott topology of the inclusion ordering, yielding a space that I will write as
**O**_{σ}*X*; - or with what I will call the
*pointwise topology*, whose subbasic open sets are [*x*∈] ≝ {*U*∈**O***X*|*x*∈*U*}, where*x*ranges over the points of*X*. I will write the resulting space**O**_{p}*X*.

The homeomorphism is defined as follows:

- φ
maps every_{X}∈*Q***QH***X*to the collection of open subsets*U*of*X*that intersect every*C*∈;*Q* - ψ
maps every_{X}∈*U***O**_{p}**O**_{σ}*X*to the intersection of the sets ♢*U*, where*U*ranges over.*U*

Let us check this.

**Proposition.** φ* _{X}* and ψ

*form a pair of mutually inverse continuous maps between*

_{X}**QH**

*X*and

**O**

_{p}

**O**

_{σ}

*X*. We have φ

_{X}^{–1}([

*U*∈]) = ☐♢

*U*, and ψ

_{X}^{–1}(☐♢

*U*) = [

*U*∈] for every open subset

*U*of

*X*.

*Proof.* For every ** Q** ∈

**QH**

*X*, we first verify that

**≝ φ**

*U**(*

_{X}**) is Scott-open. It is clearly upwards-closed. Given any directed family (**

*Q**U*)

_{i}_{i∈I}of open subsets of

*X*whose union is in

**, by definition of**

*U***, every**

*U**C*∈

**intersects ∪**

*Q*_{i ∈ I}

*U*, hence is in some ♢

_{i}*U*. Therefore the sets ♢

_{i}*U*(

_{i}*i*∈

*I*) form an open cover of

**, which is also directed. Since**

*Q***is compact,**

*Q***is included in some ♢**

*Q**U*. Therefore

_{i}*U*intersects every

_{i}*C*∈

**, so is in φ**

*Q**(*

_{X}**)=**

*Q***.**

*U*We verify that φ_{X}^{–1}([*U* ∈]) = ☐♢*U*. For every ** Q** ∈ φ

_{X}^{–1}([

*U*∈]), by definition, the collection φ

*(*

_{X}**) of all open subsets**

*Q**V*of

*X*that intersect every

*C*∈

**contains**

*Q**U*, namely every

*C*∈

**intersects**

*Q**U*; equivalently,

*Q*∈ ☐♢

*U*.

This shows, in particular, that φ* _{X}* is continuous.

Let us turn to ψ* _{X}* . We first show that: (∗) for every upwards-closed family

**of open subsets of X, for every open subset**

*U**V*of

*X*, ψ

*(*

_{X}**) ⊆ ♢**

*U**V*if and only if

*V*∈

**. The if direction is clear. Conversely, we assume ψ**

*U**(*

_{X}**) ⊆ ♢**

*U**V*and we consider the closed set

*C*≝

*X*–

*V*.

*C*is not in ♢

*V*, hence not in ψ

*(*

_{X}**). Therefore**

*U**C*fails to intersect some

*U*∈

**. By definition of**

*U**C*,

*U*is included in

*V*, and therefore

*V*is in

**, since**

*U***is upwards-closed.**

*U*It follows that ψ* _{X}*(

**) is compact, as we show by using Alexander’s subbase lemma (Theorem 4.4.29 in the book): if ψ**

*U**(*

_{X}**) ⊆ ∪**

*U*_{i ∈ I}♢

*U*, where (

_{i}*U*)

_{i}_{i∈I}is a directed family of open subsets of X, then ψ

*(*

_{X}**) ⊆ ♢**

*U**V*where

*V*≝ ∪

_{i ∈ I}

*U*, since ♢ commutes with arbitrary unions; so

_{i}*V*is in

**by (∗). Since**

*U***is Scott-open, some**

*U**U*is in

_{i}**, and therefore ψ**

*U**(*

_{X}**) ⊆ ♢**

*U**U*, using (∗) once again.

_{i}It also follows from (∗) that ψ_{X}^{–1}(☐♢*U*) = [*U* ∈]. In particular, ψ* _{X}* is continuous.

For every open subset *U* of *X*, (φ* _{X}* ◦ ψ

*)*

_{X}^{−1}(☐♢

*U*) = ψ

*−1([*

_{X}*U*∈]) = ☐♢

*U*. Hence, for every

**∈**

*Q***QH**

*X*,

**and φ**

*Q**(ψ*

_{X}*(*

_{X}**)) belong to the same open sets. Since**

*Q*

**QH***X*

**is T**

_{0}, they are equal. We prove that ψ

*◦ φ*

_{X}*is the identity map in the same manner. ☐*

_{X}The following modification (which does not appear in [3]) is concerned with the space **Q**_{s}**H***X* of *strongly* compact saturated subsets of **H***X*. Correspondingly, we replace **O**_{p}**O**_{σ}*X* with **O**_{p}**O**_{p}*X*. (Mind the change from σ to p in indices!) In order to try and avoid some confusion, we write ☐^{s}*U* for the collection of *strongly* compact saturated subsets of an open set *U*. Notice that **Q**_{s}**H***X* is a topological subspace of **QH***X*, and that **O**_{p}**O**_{p}*X* is a topological subspace of **O**_{p}**O**_{σ}*X*: its elements are the pointwise open subsets of **O***X*, all of them are Scott-open, and the topology is indeed the subspace topology.

**Proposition.** The pair of homeomorphisms φ* _{X}* and ψ

*restrict (and corestrict) to homeomorphisms between*

_{X}

**Q**_{s}

**H***X*and

**O**

_{p}

**O**

_{p}

*X*. We have φ

_{X}^{–1}([

*U*∈]) = ☐

^{s}♢

*U*, and ψ

_{X}^{–1}(☐

^{s}♢

*U*) = [

*U*∈] for every open subset

*U*of

*X*.

*Proof.* We first show that φ* _{X}* maps every

*strongly*compact set

**∈**

*Q*

**Q**_{s}**H***X*to a

*pointwise*open subset

**≝ φ**

*U**(*

_{X}**). To this end, let**

*Q**U*be any element of

**; we will find an open subset**

*U***of**

*V***O**

_{p}

*X*such that

*U*∈

**⊆**

*V***. By definition of φ**

*U**,*

_{X}*U*intersects every

*C*∈

**, so**

*Q***is included in ☐**

*Q*^{s}

*U*. Since

**is strongly compact, there are finitely many closed subsets**

*Q**C*

_{1}, …,

*C*of X such that

_{n}**⊆ ↑**

*Q***{**

_{H}*C*

_{1}, …,

*C*} ⊆ ☐

_{n}^{s}

*U*, where ↑

**denotes upward closure in**

_{H}**H**

*X*. Since ↑

**{**

_{H}*C*

_{1}, …,

*C*} ⊆ ☐

_{n}^{s}

*U*, every

*C*intersects

_{i}*U*, say at

*x*. Then

_{i}*U*is in

**≝ ∩**

*V*_{i=1}

*[*

^{n}*x*∈]. It remains to show that

_{i}**is included in**

*V***= φ**

*U**(*

_{X}**). For every**

*Q**V*∈

**, every point**

*V**x*is in

_{i}*V*, so

*V*intersects every

*C*, showing that ↑

_{i}**{**

_{H}*C*

_{1}, …,

*C*} ⊆ ♢

_{n}*V*. Since

**⊆ ↑**

*Q***{**

_{H}*C*

_{1}, …,

*C*}, every

_{n}*C*∈

**also intersects**

*Q**V*, so

*V*is in φ

*(*

_{X}**).**

*Q*We turn to ψ* _{X}*. For every closed subset

*C*of

*X*, let us write [

*C*∩] for the collection of open subsets of

*X*that intersect

*C*. If

*C*is of the form ↓{

*x*

_{1}, …,

*x*

_{n}}, then [

*C*∩] is the union of the sets [

*x*

_{1}∈], …, [

*x*

_{n}∈]. Every open subset

**of**

*U***O**

_{p}

*X*can be written as ∪

_{i∈I}∩

_{j=1}

*[*

^{ni}*x*∈]. By writing the outer union as a directed union of finite unions, then distributing the inner intersections over the just created finite unions, and finally observing that any finite union of sets [

_{ij}*x*∈] can be written in the form of [

*C*∩] (as I mentioned at the beginning of this paragraph), we can write

**as a directed union ∪**

*U*_{i∈I}∩

_{j=1}

*[*

^{n}*C*∩], where each

_{ij}*C*is closed in

_{ij}*X*.

A few technical but elementary computations show that ψ* _{X}*(

**) is then equal to the filtered intersection ∩**

*U*_{i∈I}↑

**{**

_{H}*C*

_{i1}, …,

*C*}. (Exercise!) Now let us assume that ψ

_{ini}*(*

_{X}**) is included in some open subset**

*U***of**

*V***H**

*X*. Since

**H**

*X*is sober, as I have mentioned earlier, it is well-filtered, so there is an index

*i*in

*I*such that ↑

**{**

_{H}*C*

_{i1}, …,

*C*} is included in

_{ini}**. Since obviously ψ**

*V**(*

_{X}**) is included in ↑**

*U***{**

_{H}*C*

_{i1}, …,

*C*}, we have shown that ψ

_{ini}*(*

_{X}**) is strongly compact. The remaining claims follow from our previous proposition. ☐**

*U*## The de Brecht-Kawai isomorphism: **QH**=**HQ** ⇔ consonance

There are two other maps, defined by de Brecht and Kawai as follows:

- σ
maps every_{X}∈*C***HQ***X*to the collection of closed subsets*C*of*X*that intersect every*Q*∈.*C* - τ
maps every_{X}∈**Q****QH***X*to the collection of compact saturated subsets*Q*of*X*that intersect every*C*∈;*Q*

Yes, the definition is very symmetric. I will break this symmetry, and concentrate on σ* _{X}*.

Let us study them one after the other. Items 2 and 3 below are Lemma 6.4 of [3]. Item 1 is the reason why the statement **QH**=**HQ** is related to consonance, as we will see. Let me write ■*Q*, for every compact saturated subset *Q* of *X*, for the set of open neighborhoods of *Q*.

**Lemma.** We have:

- φ
o σ_{X}maps every_{X}∈*C***HQ***X*to ∪_{Q∈C}■*Q*; - for every open subset
*U*of X, σ_{X}^{−1}(☐♢*U*) = ♢☐*U*; - σ
is a topological embedding._{X}

*Proof.* 1. For every ** C** ∈

**HQ**

*X*, ∪

_{Q∈C}■

*Q*is a Scott-open subset of

**O**

*X*. For every closed subset

*C*of

*X*,

*C*∈ ψ

*(∪*

_{X}_{Q∈C}■

*Q*) if and only if

*C*intersects every open set

*U*that contains some

*Q*∈

**. This is certainly the case if**

*C**C*intersects

*Q*. Conversely, if

*C*does not intersect

*Q*, then

*U*≝

*X*–

*C*is an open set that does not intersect

*C*. Therefore ψ

*(∪*

_{X}_{Q∈C}■

*Q*) = σ

*(*

_{X}**).**

*C*In particular, σ* _{X}*(

**) is in**

*C***QH**

*X*, showing that σ

*is well-defined, and φ*

_{X}*(σ*

_{X}*(*

_{X}**)) = ∪**

*C*_{Q∈C}■

*Q*, since φ

*and ψ*

_{X}*are mutually inverse.*

_{X}For every open subset *U* of *X*, σ_{X}^{−1}(☐♢*U*) = σ_{X}^{−1}(φ_{X}^{−1}([*U* ∈])), and this is is equal to the collection of closed subsets *C* of **Q***X* such that *U* ∈ ∪_{Q∈C} ■*Q*, namely such that *Q* ⊆ *U* for some *Q* ∈ ** C**. Hence σ

_{X}^{−1}(☐♢

*U*) = ♢☐

*U*. In particular, σ

*is continuous, almost-open, and every almost-open map whose domain is a T*

_{X}_{0}space is injective, so σ

*is a topological embedding. ☐*

_{X}Hence the only thing missing for σ* _{X}* to be a homeomorphism is surjectivity. A space

*X*is

*consonant*if and only every Scott-open subset of

**O**

*X*is a union of sets of the form ■

*Q*, where each

*Q*is compact saturated in

*X*. We have already encountered the concept a few times. Item 1 of the previous lemma shows that if σ

*is surjective, namely if φ*

_{X}*o σ*

_{X}*is surjective, then*

_{X}*X*is consonant. Conversely, we have:

**Lemma.** If *X* is consonant, then σ* _{X}* is surjective.

*Proof.* It suffices to show that φ* _{X}* o σ

*is surjective. Let*

_{X}**be a Scott-open subset of**

*U***O**

*X*. Since

*X*is consonant, we can write

**as a union of sets of the form ■**

*U**Q*. Let

**be the collection of compact saturated subsets**

*C**Q*of

*X*such that ■

*Q*is included in

**. We claim that**

*U***is closed, in other words that its complement is open in**

*C***Q**

*X*. Let

*Q*∈

**Q**

*X*be outside

*. We will show that*

**C***Q*is in some basic open set ☐

*U*that is disjoint from

**. Let us assume that this is not the case, and let us consider any open neighborhood**

*C**U*of

*Q*. By assumption, ☐

*U*intersects

**, say at**

*C**Q’*. Since

*Q’*is in

**, ■**

*C**Q’*is included in

**. Since**

*U**Q’*is in ☐

*U*,

*U*is in ■

*Q’*, so

*U*is in

**. Hence, we have shown that all the open neighborhoods**

*U**U*of

*Q*are in

**. In other words, ■**

*U**Q*is included in

**, so**

*U**Q*is in

**: this is impossible, since we have assumed**

*C**Q*∈

**Q**

*X*to be outside

*.*

**C**In summary, we have built an element ** C** of

**HQ**

*X*. By item 1 of the previous lemma, φ

*o σ*

_{X}*maps it to ∪*

_{X}_{Q∈C}■

*Q*, namely to the union of all the sets of the form ■

*Q*(with

*Q*compact saturated in

*X*) included in

**. Because of our assumption of consonance, that union contains**

*U***, and it is clearly included in**

*U***. Therefore (φ**

*U**o σ*

_{X}*) (*

_{X}**) =**

*C***. ☐**

*U*We have therefore obtained the following result, due to de Brecht and Kawai [3, Theorem 6.13].

**Theorem.** For every topological space *X*, the following are equivalent:

*X*is consonant;- σ
is surjective;_{X} - σ
is a homeomorphism of_{X}**HQ***X*onto**QH***X*.

They also show that the map τ* _{X}* is such that τ

_{X}^{−1}(♢☐

*U*) ⊆ ☐♢

*U*, that this inclusion is an equality if and only if any of the items 1–3 above is satisfied; and that, in that case, τ

*is continuous and is the inverse of σ*

_{X}*.*

_{X}## The isomorphism **Q**_{s}**H**=**HQ**_{s}

What happens if we replace **Q** with **Q**_{s} throughout? One may guess—and one would be right—that everything would work the same way, replacing compact saturated sets by strongly compact saturated sets… and also **O**_{p}**O**_{σ}*X* by **O**_{p}**O**_{p}*X*, since now φ* _{X}* and ψ

*restrict to homeomorphisms between*

_{X}

**Q**_{s}

**H***X*(instead of

**QH***X*) and

**O**

_{p}

**O**

_{p}

*X*(instead of

**O**

_{p}

**O**

_{σ}

*X*). The right variant of the notion of consonance is then the following: let us say that

*X*is

*finitarily consonant*if and only if every

*pointwise*open subset

**of**

*U***O**

*X*(namely, every open subset of

**O**

_{p}

*X*, not

**O**

_{σ}

*X*) is a union of sets of the form ■

*Q*, where

*Q*is

*strongly*compact saturated.

We note that every set ■*Q* is a pointwise open subset of **O***X* (namely, an open subset of **O**_{p}*X*, or equivalently an element of **O**_{p}**O**_{p}*X*), for every strongly compact saturated set *Q*. Indeed, for every *U* ∈ ■*Q*, *Q* is included in *U*, and since *Q* is strongly compact, we can find a finitary compact set ↑*E* such that *Q* ⊆ ↑*E* ⊆ *U*. Let us write the finite set *E* as {*x*_{1}, …, *x*_{n}}. Then *U* is in ∩_{i=1}* ^{n}*[

*x*∈], since each

_{i}*x*is in

_{i}*U*, and ∩

_{i=1}

*[*

^{n}*x*∈] is included in ■

_{i}*, since every open set that contains every*

*Q**x*must contain ↑

_{i}*E*, hence also

*Q*. This finishes to show that ■

*Q*is an open subset of

**O**

_{p}

*X*, for every strongly compact saturated set

*Q*. Hence every union of such sets is also open in

**O**

_{p}

*X*. Finitary consonance requires that those are the only open subsets of

**O**

_{p}

*X*.

We now define σ_{X}^{s} just like σ* _{X}*: it maps every

**∈**

*C***HQ**

_{s}

*X*(instead of

**HQ**

*X*) to the collection of closed subsets

*C*of

*X*that intersect every

*Q*∈

**. And we replay the proofs we did above:**

*C***Lemma.** We have:

- φ
o σ_{X}_{X}^{s}maps every∈*C***HQ**_{s}*X*to ∪_{Q∈C}■*Q*; - for every open subset
*U*of X, (σ_{X}^{s})^{−1}(☐^{s}♢*U*) = ♢☐^{s}*U*; - σ
_{X}^{s}is a topological embedding.

*Proof.* Just as before, ψ* _{X}* (∪

_{Q∈C}■

*Q*) = σ

*(*

_{X}**), where now**

*C***is a closed subset of**

*C***Q**

_{s}

*X*. Since every

*Q*∈

**is strongly compact saturated, we have seen that ∪**

*C*_{Q∈C}■

*Q*is in

**O**

_{p}

**O**

_{p}

*X*, so its image by ψ

*is in*

_{X}

**Q**_{s}

**H***X*. This also shows φ

*(σ*

_{X}*(*

_{X}**)) = ∪**

*C*_{Q∈C}■

*Q*, hence item 1. Items 2 and 3 are proved as before. ☐

The following is also proved just like the analogous result we have proved earlier.

**Lemma.** If *X* is finitarily consonant, then σ_{X}^{s} is surjective. ☐

However, the new thing is that the notion of finitary consonance is *vacuous* (I was keeping that for the end!).

**Fact.** Every topological space is finitary consonant.

*Proof.* Let ** U** be an open subset of

**O**

_{p}

*X*. By definition, we can write

**as ∪**

*U*_{i∈I}∩

_{j=1}

*[*

^{ni}*x*∈]. But that is just ∪

_{ij}_{i∈I}■

*Q*, where

_{i}*Q*≝ ↑{

_{i}*x*

_{i1}, …,

*x*}, and each

_{ini}*Q*is finitarily compact, hence trivially strongly compact. ☐

_{i}Hence we have obtained the final theorem of this post.

**Theorem.** For every topological space *X*, the spaces **Q**_{s}**H***X*, **HQ**_{s}*X* and **O**_{p}**O**_{p}*X* are homeomorphic; the first two, through σ_{X}^{s}, and the first and the third through the pair φ* _{X}*, ψ

*.*

_{X}I could say a lot more. For example, the constructions are all natural in *X*; σ_{X}^{s} defines a distributive law, hence **Q**_{s}** H** is really the functor part of a monad; etc. But that’s enough for today!

- Reinhold Heckmann. An upper power domain construction in terms of strongly compact sets. In: Brookes, S., Main, M., Melton, A., Mislove, M., Schmidt, D. (eds) Mathematical Foundations of Programming Semantics. MFPS 1991. Lecture Notes in Computer Science, vol 598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55511-0_14
- Andrea Schalk.
*Algebras for Generalized Power Constructions*. PhD Thesis, TU Darmstadt, 1993. - Matthew de Brecht and Tatsuji Kawai. On the commutativity of the powerspace constructions. Logical Methods in Computer Science, 15(3), 2019.
- Steven J. Vickers and Christopher F. Townsend. A universal characterization of the double powerlocale. Theoretical Computer Science 316(1-3), 2004, pages 297–321. http://dx.doi.org/10.1016/j.tcs.2004.01.034

— Jean Goubault-Larrecq (October 19th, 2022)