I am a bit stubborn. In my first post on ideal domains, I thought I would be able to extend Keye Martin’s result from metric to quasi-metric spaces. I have said I had failed, but now I think I have succeeded. This leads to a notion that I will call a *quasi-ideal domain*.

Our purpose today is to show that, if *X* is a continuous Yoneda-complete quasi-metric space, then it embeds into an algebraic dcpo, and in fact, in a very specific way: as the subspace of limit elements of a quasi-ideal domain.

The elements of that quasi-ideal domain will again be certain formal balls, and will look very much like Keye Martin’s **B*** _{D}*(

*X*,

*d*). If you refer to the first post, the main difference will be that our ordering ⊑ will be such that (

*x*, 0) ⊑ (

*y*, 0) whenever

*x*≤

*y*— that was not the case before.

## Embedding continuous Yoneda-complete spaces into algebraic domains

Let us therefore define (*x*, *r*) ⊏ (*y*, *s*) iff:

- either (
*x*,*r*) ≪ (*y*,*s*) and*r ≥ 2s* - or
*r*=*s*=0 and*x*≤*y*(i.e.,*d*(*x*,*y*)*=*0, or equivalently, (*x*, 0) ≤ (*y*, 0)).

Note that I wrote ⊏, not ⊑: ⊏ is a kind of strict part of ⊑. Now define (*x*, *r*) ⊑ (*y*, *s*) iff (*x*, *r*) ⊏ (*y*, *s*) or (*x*, *r*) = (*y*, *s*). With that correction, ⊑ is reflexive, and a painful case analysis shows that it is transitive. Also, ⊑ implies ≤, so the relation is antisymmetric, hence an ordering. Fine.

Let me reuse our former name, and call **B*** _{D}*(

*X*) the set of formal balls with the ordering ⊑.

**Lemma.** For every Yoneda-complete quasi-metric space *X*, **B*** _{D}*(

*X*) is a dcpo, and in fact one in which directed suprema are computed exactly as in

**B**(

*X*).

*Proof.* There is a similar lemma in Ideal Models I, namely the very first one, although it only applied to complete metric spaces. I had mentioned that the only place where we needed *X* to be metric was the last line, and this misfortune is now repaired by our new definition of ⊑.

We take a family (*x _{i}*,

*r*)

_{i}*in*

_{i ∈ I}**B**

*(*

_{D}*X*) that is directed with respect to ⊑, and we let (

*x*,

*r*) = (

*d*-lim

*x*, inf

_{i}*r*) be its supremum in

_{i}**B**(

*X*). (We are using the Kostanek-Waszkiewicz Theorem here.) We must show that (

*x*,

*r*) is also its supremum with respect to ⊑. If the supremum is reached, that is clear. Otherwise, we still have a few cases to distinguish. If

*r*=0 for

_{i}*i*large enough, then we can consider the subfamily of formal balls (

*x*,

_{i}*r*) with

_{i}*r*=0, and since ⊑ and ≤ coincide on them, we are done.

_{i}The final case is where *r _{i}*>0 for every

*i*. In that case, using the same trick as Ideal Models I, every (

*x*,

_{i}*r*) is ⊏ some (

_{i}*x*,

_{j}*r*). This implies that

_{j}*r*=0, and also that (

*x*,

*r*)=(

*x*, 0) is an upper bound of the family (with respect to ⊑): for every

*i*, (

*x*,

_{i}*r*) ⊏ (

_{i}*x*,

_{j}*r*) ≤ (

_{j}*x*,

*r*) = (

*x*, 0), so (

*x*,

_{i}*r*) ≪ (

_{i}*x*, 0), namely (

*x*,

_{i}*r*) ⊏ (

_{i}*x*,

*r*) by the first clause of the definition of ⊏.

If (*y*, *s*) is another upper bound of (*x _{i}*,

*r*)

_{i}*(with respect to ⊑), then it is also an upper bound with respect to ≤, so (*

_{i ∈ I}*x*,

*r*) ≤ (

*y*,

*s*). In particular,

*s*=0. Since also

*r*=0, using the second clause of the definition of ⊏, (

*x*,

*r*) ⊑ (

*y*,

*s*). ☐

**Lemma.** For every continuous Yoneda-complete quasi-metric space *X*, *d*, the finite elements of **B*** _{D}*(

*X*) are those of the form (

*x*,

*r*),

*r*≠0.

Recall that a quasi-metric space is *continuous* Yoneda-complete if and only if its (ordinary) poset of formal balls is a continuous dcpo.

The proof is the same as for the second and third lemmas of Ideal Models I. For later reference, we recall how we show that no element (*x*, 0) is finite in **B*** _{D}*(

*X*). Since

*X*is continuous Yoneda-complete, (

*x*, 0) is the supremum of a directed family

*D*= (

*x*,

_{i}*r*)

_{i}*in*

_{i ∈ I}**B**(

*X*) with (

*x*,

_{i}*r*) ≪ (

_{i}*x*, 0) for each

*i*. (We shall need to remember that this implies

*r*>0.) The way-below relation ≪ is the one from

_{i}**B**(

*X*). The family

*D*is directed in

**B**(

*X*) (with respect to ≤) but not necessarily with respect to ⊑. We build another family with the same supremum, which is directed with respect to ⊑. This is a family of formal balls (

*x*,

_{I}*r*), where

_{I}*I*ranges over the finite subsets of

*D*. Each is an element of

*D*, and is above every element of

*I*in the ⊑ ordering. It is also above (

*x*,

_{J}*r*) in the ⊑ ordering, for every proper subset

_{J}*J*of

*I*, and that ensures that the family of formal balls (

*x*,

_{I}*r*) is directed with respect to ⊑. Once this is done, if (

_{I}*x*, 0) were finite in

**B**

*(*

_{D}*X*), we would have (

*x*, 0) ⊑ (

*x*,

_{I}*r*) for some

_{I}*I*, and that is impossible since

*r*>0.

_{I}**Theorem.** Let *X*, *d* be a continuous Yoneda-complete quasi-metric space. Then *X*, with the *d*-Scott topology, is homeomorphic to the subspace of non-finite elements of the algebraic dcpo **B*** _{D}*(

*X*).

*Proof.* The previous lemma shows that the non-finite elements of **B*** _{D}*(

*X*) are the formal balls (

*x*, 0), which are in bijection with the points

*x*of

*X*. By the paragraph right before the statement of the theorem, (

*x*, 0) is the supremum of a family of formal balls, with non-zero radii, and which is directed in

**B**

*(*

_{D}*X*), namely with respect to ⊑. That shows that

**B**

*(*

_{D}*X*) is algebraic.

The rest of the proof is exactly the same as in the corresponding theorem in Ideal Models I. Let us call, temporarily, the *D*-Scott topology the topology induced by the inclusion of *X* into **B*** _{D}*(

*X*).

An open subset *U* of *X* in the *d*-Scott topology is the intersection of *X* with a Scott-open subset *V* of **B**(*X*). Since ⊑ implies ≤, and since directed suprema are computed in **B*** _{D}*(

*X*) as in

**B**(

*X*),

*V*is Scott-open in

**B**

*(*

_{D}*X*), hence

*U*is

*D*-Scott open.

Conversely, let *U* be a *D*-Scott open subset of *X*. For every *x* ∈ *U*, there is a formal ball (*y*, *r*) ⊑ (*x*, 0) with *r*≠0 such that every formal ball (*z*, *0*) such that (*y*, *r*) ⊑ (*z*, 0) is in *U*. Since *r*≠0, (*y*, *r*) ≪ (*x*, 0). For every element *z* such that (*y*, *r*) ≪ (*z*, 0), we plainly observe that (*y*, *r*) ⊑ (*z*, 0), so (*z*, 0) is in *U*. It follows that the intersection of *X* with ↟(*y*, *r*) contains (*x*, 0) and is included in *U*, showing that *U* is *d*-Scott-open. ☐

## Quasi-ideal domains

As we shall see, **B*** _{D}*(

*X*) has extra properties that make it look even more like Keye Martin’s ideal domains.

Let me make a parenthesis, which might otherwise act as a motivation for the whole construction.

Daniele Varacca once asked me whether the following notion had a name: algebraic domains where every element below a finite element is itself finite. That seems like it should be a natural notion, but I don’t think I’ve seen that anywhere in the literature. The closest is the notion of a dI-domain, where we require a lot more: in a dI-domain, every finite point must have only finitely many points below it—in particular those points will all be finite; a dI-domain is also required to be bounded-complete algebraic, with the property that binary suprema of bounded pairs distribute over binary meets, and we will certainly not make any of those assumptions.

Let me call **quasi-ideal domain** any algebraic dcpo in which every element below a finite element is finite. Clearly, every ideal domain satisfies that property. In a quasi-ideal domain, the non-finite elements may fail to be maximal, as the following example shows.

**Fact.** For every set *A*, the powerset **P**(*A*) is quasi-ideal domain. If *A* is infinite, then **P**(*A*) is not an ideal domain.

(The finite elements are the finite subsets of *A*, and certainly any subset of *A* that is included in a finite subset is itself finite.)

Anyway, a quasi-ideal domain has just *two* layers: a lower layer of finite elements, and an upper layer of non-finite elements which can all be obtained as directed suprema of elements from the lower layer.

I will call the non-finite elements the *limit elements* of the quasi-ideal domain.

As you can see, **B*** _{D}*(

*X*) is one example of such a quasi-ideal domain: if (

*x*,

*r*) ⊑ (

*y*,

*s*) where (

*y*,

*s*) is finite (i.e.,

*s*≠0), then (

*x*,

*r*) ≤ (

*y*,

*s*), whence

*r*≥

*s*> 0, and therefore (

*x*,

*r*) is finite as well.

We can then rephrase the previous theorem under the following much stronger form.

**Theorem.** Let *X*, *d* be a continuous Yoneda-complete quasi-metric space. Then *X*, with the *d*-Scott topology, is homeomorphic to the subspace of limit elements of the quasi-ideal domain **B*** _{D}*(

*X*).

## ω-continuous Yoneda-complete quasi-metric spaces

Can we refine all that when *X* is second-countable in its *d*-Scott topology? I do not know yet, but we can if we assume a stronger property, which (by analogy with our notion of continuity for Yoneda-complete quasi-metric spaces) I will call *ω-continuity*.

Call a Yoneda-complete quasi-metric space *X*, *d* *ω-continuous* if and only if its dcpo of formal balls is… ω-continuous.

Recall that a dcpo is called ω-continuous if and only if it has a countable basis. By Norberg’s Lemma 7.7.13, this is equivalent to its Scott topology being second-countable. In particular, every ω-continuous Yoneda-complete quasi-metric space is both continuous Yoneda-complete and second-countable in its *d*-Scott topology. (I have not been able to show that, conversely, every second-countable continuous Yoneda-complete quasi-metric space is ω-continuous, and I would be surprised if that were true.)

Now we can refine everything we have done earlier. Fix a basis *B* of **B**(*X*, *d*), which typically will be countable for an ω-continuous Yoneda-complete quasi-metric space *X*, *d*. Define **B**‘* _{D}*(

*X*,

*d*) as the following variant of

**B**

*(*

_{D}*X*,

*d*):

- The elements of
**B**‘(_{D}*X*,*d*) are the formal balls in the basis*B*, plus the formal balls of the form (*x*, 0),*x*∈*X*. - The ordering ⊑ is defined as before: (
*x*,*r*) ⊑ (*y*,*s*) iff (*x*,*r*) ⊏ (*y*,*s*) or (*x*,*r*) = (*y*,*s*), where (*x*,*r*) ⊏ (*y*,*s*) iff either (*x*,*r*) ≪ (*y*,*s*) and*r ≥ 2s*, or*r*=*s*=0 and*x*≤*y*.

**Lemma.** For every ω-continuous Yoneda-complete quasi-metric space *X*, **B**‘* _{D}*(

*X*) is a dcpo, and in fact one in which directed suprema are computed exactly as in

**B**(

*X*).

This is proved exactly as our first lemma on **B*** _{D}*(

*X*). The important new thing to check is that, given a family (

*x*,

_{i}*r*)

_{i}*in*

_{i ∈ I}**B**‘

*(*

_{D}*X*) that is directed with respect to ⊑, then its sup in

**B**(

*X*) will always be in

**B**‘

*(*

_{D}*X*), that is, it will either be in the basis

*B*or have radius 0. This follows from the case analysis we have already done in our previous similar lemma.

**Lemma.** For every ω-continuous Yoneda-complete quasi-metric space *X*, *d*, the finite elements of **B**‘* _{D}*(

*X*) are those of the form (

*x*,

*r*),

*r*≠0, namely those of the basis

*B*.

That is again proved as before.

As a corollary, we obtain the following variant of our first Theorem: Let *X*, *d* be a continuous Yoneda-complete quasi-metric space, and fix a basis *B* of its dcpo of formal balls. Then *X*, with the *d*-Scott topology, is homeomorphic to the subspace of non-finite elements of the algebraic dcpo **B***‘** _{D}*(

*X*), and the finite elements of the latter are exactly the formal balls in

*B*with non-zero radius. That is, again, proved as before.

When *B* is countable, that implies that **B**‘* _{D}*(

*X*) has countably many finite elements:

**Theorem.** Let *X*, *d* be an ω-continuous Yoneda-complete quasi-metric space. Then *X*, with the *d*-Scott topology, is homeomorphic to the subspace of non-finite elements of the ω-algebraic dcpo **B*** _{D}*(

*X*).

Let us rephrase that in the language of ideal completion remainders (see Ideal Models II): Every ω-continuous Yoneda-complete quasi-metric space is the ideal completion remainder of a countable poset.

We have seen that the ideal completion remainders of countable posets are the quasi-Polish spaces (a theorem due to Matthew de Brecht). We therefore obtain:

**Theorem.** The topological spaces underlying the ω-continuous Yoneda-complete quasi-metric spaces (with the *d*-Scott topology) are exactly the same as those underlying the separable Smyth-complete quasi-metric spaces (with the open ball topology — which in fact coincides with the *d*-Scott topology in that case), namely the quasi-Polish spaces.

That finally answers one of the questions I had about the definition of quasi-Polish spaces: why should we consider *Smyth-complete* spaces instead of, say, Yoneda-complete spaces to define them?

The answer is, provided we take ω-continuous Yoneda-complete spaces, and take care of using the *d*-Scott topology (the right topology, in all situations that I know of):

this *does not matter*.

— Jean Goubault-Larrecq (March 10th, 2016)