There is a famous formula in domain theory, called *Scott’s formula*. If *B* is a basis of a continuous poset *X*, and *f* is a monotonic function from *B* to some dcpo *Y*, then one can define a new function *f’* from the whole of *X* to *Y* by:

*f’*(*x*) ≝ sup_{b ∈ B, b ≪ x} *f*(*b*),

and then *f’* will automatically be a continuous function from *X* to *Y*. There is no reason why it would coincide with *f* on *B*, but it is the best continuous approximation to *f* on *B*, in the sense that *f’* is below *f* on *B* (*f’*(*b*)≤*f*(*b*) for every *b* ∈ *B*) and that it is the largest one with that property (for every continuous function *g* below *f* on *B*, *g*≤*f’*).

This is mentioned as Proposition 5.1.60 in the book, although you should definitely read the errata page (important blooper #13).

I would like to explore a few ways that this formula can be extended, in what cases *f’* coincides with *f* on *B* (i.e., in what cases *f’* is a *continuous extension* of *f*), and finally in what cases any algebraic laws satisfied by *f* will still be satisfied by *f’*.

## c-spaces

We can first extend Scott’s formula to the case where *X* is a *c-space*, not just a continuous poset. A c-space is a space *X* in which, for every point *x*, for every open neighborhood *U* of *x*, there is a point *y* in *U* such that *x* is in the interior of ↑*y*.

It will be practical to write *y* ≪ *x* instead of “*x* is in the interior of ↑*y*“, and ↟*y* for the interior of ↑*y* (not just in c-spaces, but in any topological space). You might be concerned about a possible conflict of notations with the way-below relation on posets. But, in the case where *X* is a continuous poset, with its Scott topology, the usual way-below relation ≪ is exactly given by *y* ≪ *x* if and only if *x* is in the interior of ↑*y*, or equivalently, the interior of ↑*y* is exactly ↟*y*, defined as the set of elements *z* such that *y* is way-below *z* (this is Proposition 5.1.35 of the book); and I will never consider the way-below relation on posets that are not continuous in this post.

Now let me define a *basis* of a topological space *X* as a collection *B* of points of *X* such that for every point *x* in *X*, for every open neighborhood *U* of *x*, there is an element *b* of *B* ∩ *U* such that *b* ≪ *x* (i.e., such that *x* is in the interior of ↑*b*). Any space with a basis must be c-space, and conversely, and c-space *X* has a basis, namely *X* itself. When *X* is a continuous poset with its Scott topology, we retrieve the usual definition of a basis.

Scott’s formula has an easy generalization to c-spaces and their bases, as we now see. Note that *X* need not be sober, in particular; if *X* were sober, then it would be a continuous dcpo with its Scott topology (Proposition 8.3.36 of the book). There is no harm in replacing *Y*, with its Scott topology, with a monotone convergence space either. All spaces are (pre)ordered with their specialization preorderings; this is what allows us to make sense of the adjective “monotonic” below.

**Theorem.** Let *B* be a basis of a c-space *X*, and *f* be a monotonic function from *B* to some monotone convergence space *Y*. For every *x* in *X*, let:

*f’*(*x*) ≝ sup_{b ∈ B, b ≪ x} *f*(*b*)

(Scott’s formula).

Then *f’* is a continuous map from *X* to *Y*, and is the largest continuous map on *X* that is below *f* on *B*.

*Proof.* For any given point *x*, let *F _{x}* denote the family of elements

*b*∈

*B*such that

*b*≪

*x*. We claim that

*F*is directed. Once we have proved this, the supremum that defines

_{x}*f’*(

*x*) will be a supremum of a directed family, hence will be well-defined, since

*Y*is a monotone convergence space, in particular a dcpo in its specialization ordering. Now every open neighborhood

*U*of

*x*contains an element of

*F*, since

_{x}*B*is a basis of

*X*. By taking

*U*≝

*X*,

*F*is non-empty, and for any two elements

_{x}*b*

_{1}and

*b*

_{2}of

*F*, there is an element

_{x}*b*of

*F*in int(↑

_{x}*b*

_{1}) ∩ int(↑

*b*

_{2}); in particular,

*b*

_{1},

*b*

_{2}≪

*b*. Hence

*F*is directed, as promised.

_{x}We claim that *f’* is continuous. For every open subset *V* of *Y*, for every *x* ∈ *X*, *f’*(*x*) ∈ *V* if and only if sup_{b ∈ B, b ≪ x} *f*(*b*) is in *V*. Since *Y* is a monotone convergence space, *V* is Scott-open, so sup_{b ∈ B, b ≪ x} *f*(*b*) ∈ *V* implies that *f*(*b*) ∈ *V* for some *b* ∈ *B* such that *b* ≪ *x*. Conversely, if *f*(*b*) ∈ *V* for some *b* ∈ *B* such that *b* ≪ *x*, then *f’*(*x*), which is even larger, is in *V*. Hence *f’*^{−1}(*V*) is the union of the sets ↟*b*, where *b* ranges over the elements of *B* such that *f*(*b*) ∈ *V*. In other words, *f’*^{−1}(*V*) = ∪_{b ∈ f−1 (V)} int(↑*b*). Since that is open, *f’* is continuous.

For every *x* ∈ *B*, we have that every *b* ∈ *B* such that *b* ≪ *x* (i.e., *x* ∈ int(↑*b*)) is such that *x* is in ↑*b*, namely *b* ≤ *x*; so *f*(*b*) ≤ *f*(*x*) since *f* is monotonic. Taking suprema, we obtain that *f’*(*x*)≤*f*(*x*). Hence *f’* is below *f* on *B*.

Finally, we consider any continuous map *g* from *X* to *Y* that is below *f* on *B*. For every point *x* of *X*, for every open neighborhood *V* of *g*(*x*), *g*^{−1}(*V*) is an open neighborhood of *x*, hence contains some element *b* of *B* such that *b* ≪ *x*, since *B* is a basis of *X*. Since *g*(*b*) ≤ *f*(*b*) and *g*(*b*) ∈ *V*, *f*(*b*) is also in *V*, and therefore *f’*(*x*) ≥ *f*(*b*) is in *V*. We have shown that every open neighborhood *V* of *g*(*x*) contains *f’*(*x*), so *g*(*x*) ≤ *f’*(*x*), showing that *f’* is the largest continuous map below *f* on *B*. ☐

## Continuous extensions

Next, we examine when *f’* is an extension of *f*, namely when *f’* coincides with *f* on *B*. This is dealt with in Corollary 5.1.61 of the book; as with Proposition 5.1.60, you need *Y* to be a dcpo, not just a bdcpo (see the errata page, important blooper #13; you may also keep *Y* as a bdcpo, and then you need to require *B* to be cofinal in *X*).

The view I have just given of Scott’s formula on c-spaces gives us a complete characterization of when *f’* is an extension of *f*.

**Proposition.** Let *B* be a basis of a c-space *X*, *f* be a monotonic map from *B* to a monotone convergence space *Y*, and let *f’* be defined by Scott’s formula. Then *f’* extends *f* if and only if *f* is continuous from *B* (with the subspace topology from *X*) to *Y*.

*Proof.* We have seen that *f’* is continuous from *X* to *Y*. Therefore its restriction to the subspace *B* is continuous. In particular, if *f’* coincides with *f* on *B*, *f* must be continuous from *B* to *Y*.

Conversely, let us assume that *f* is continuous from *B* to *Y*. For every open subset *V* of *Y*, *f*^{−1}(*V*) is open, hence equal to *U* ∩ *B* for some open subset *U* of *X*. For every *x* ∈ *f*^{−1}(*V*)= *U* ∩ *B*, by definition of *B* there is an element *b* ∈ *B* such that *b* ≪ *x* and *b* ∈ *U*. Hence *b* is in *f*^{ −1}(*V*). Equivalently, *f*(*b*) ∈ *V*, and this shows that *f’*(*x*) ≥ *f*(*b*) is in *V*. We have shown that for every *x* ∈ *X*, every open neighborhood *V* of *f*(*x*) also contains *f’*(*x*), so *f*(*x*) ≤ *f’*(*x*). Since *f’* is below *f* on *B*, we conclude that *f* and *f’* coincide on *B*. ☐

When *X* is a continuous poset with basis *B*, and *Y* is a dcpo, the condition that *f* be continuous from *B*, with the subspace topology from *X*, to *Y*, is equivalent to the fact that *f* be *relatively Scott-continuous on B*, which I will define next. There is a pretty subtle point here, as any Scott-continuous map from *B* to *Y* will be relatively Scott-continuous on *B*, but the converse may fail.

Let me call *X*–*supremum* of a family *D* of points of *X* the least upper bound of *D* in *X*, if it exists. Let me call *B*–*supremum* of a family *D* of points of *B* the least upper bound of *D* in *B*, if it exists. Now, given any family *D* of points of *B*,

- If
*D*has an*X*-supremum*x*that happens to be in*B*, then*x*must be the*B*-supremum of*D*as well:*x*is an upper bound of*D*, and for every upper bound*b*of*D*in*B*,*b*is larger than any point of*D*, so*b*≤*x*since*x*is an*X*-supremum of*D*. - However, and this is the catch, if
*D*has a*B*-supremum*b*, then*b*will be an upper bound of*D*in*X*as well, but not necessarily the least one; in other words,*b*may well not be the*X*-supremum of*D*.

For an illustration of this, consider the collection of all subsets of a given topological space*Y*for*X*, its subcollection of closed subsets for*B*, both ordered by inclusion. The*B*-supremum of a collection*D*≝(*C*)_{i}_{i ∈ I}of elements of*B*(closed sets) is cl(∪_{i ∈ I}*C*), and that is in general different from the_{i}*X*-supremum ∪_{i ∈ I}*C*._{i}

Hence, and returning to the case where *B* is a basis of a continuous poset *X*, and *Y* is a dcpo, let me say that a function *f* from *B* to *Y* is *relatively Scott-continuous on B* if and only if *f* is monotonic, and for every directed family (*b _{i}*)

_{i ∈ I}in

*B*with an

*X*-supremum (not a

*B*-supremum!)

*b*that happens to be in

*B*,

*f*(

*b*) = sup

_{i ∈ I}

*f*(

*b*).

_{i}**Lemma.** Given a basis *B* of a continuous poset *X*, and a function *f* from *B* to a dcpo *Y*, *f* is continuous from *B*, with the subspace topology induced by the Scott topology on *X*, if and only if *f* is relatively Scott-continuous on *B*.

*Proof.* Let us first assume that *f* is continuous from *B*, with the subspace topology induced by the Scott topology on *X*. Then *f* is monotonic, and given any directed family (*b _{i}*)

_{i ∈ I}in

*B*with an

*X*-supremum

*b*that happens to be in

*B*, we claim that

*f*(

*b*) = sup

_{i ∈ I}

*f*(

*b*). The inequality

_{i}*f*(

*b*) ≥ sup

_{i ∈ I}

*f*(

*b*) is a consequence of monotonicity. In order to show the reverse inequality, we consider any (Scott-)open neighborhood

_{i}*V*of

*f*(

*b*), and we will show that

*f*(

*b*) is in

_{i}*V*for some

*i*∈

*I*. Since

*f*is continuous,

*f*

^{–1}(

*V*) is open in the subspace topology induced on

*B*by the Scott topology on

*X*, hence is of the form

*U*∩

*B*for some Scott-open subset

*U*of

*X*. Now

*f*

^{–1}(

*V*) =

*U*∩

*B*contains

*b*, which is the

*X*-supremum of (

*b*)

_{i}_{i ∈ I}. Since

*U*is Scott-open (in

*X*), some

*b*is in

_{i}*U*. It is also in

*B*, by definition. Hence

*b*is in

_{i}*U*∩

*B*=

*f*

^{–1}(

*V*), so that

*f*(

*b*) is in

_{i}*V*, as promised.

That direction of the proof did not use the fact that *B* is a basis of *X*, and works perfectly well for any subset *B* of an arbitrary, not necessarily continuous poset *X*.

In the converse direction, we will definitely use the fact that *B* is a basis of *X*, and therefore that *X* is a continuous poset. Let us assume that *f* is relatively Scott-continuous on *B*. We consider any (Scott-)open subset *V* of *Y*, and we claim that *f*^{–1}(*V*) can be written as *U* ∩ *B* for some Scott-open subset *U* of *X*. We define *U* as the union of all the sets ↟*b*, where *b* ranges over *f*^{–1}(*V*); to make it clear, ↟*b* is the collection of points *x* in *X* (not *B*) such that *b* ≪ *x*. Then *U* is (Scott-)open in *X*. For every *b’* ∈ *f*^{–1}(*V*), since *X* is a continuous poset with basis *B*, *b’* is the *X*-supremum of the directed family of all elements *b* ≪ *b’* with *b* ∈ *B*. Since *f* is relatively Scott-continuous on *B*, *f*(*b’*), which is in *V*, is the directed supremum of all elements *f*(*b*) with *b* ≪ *b’*, *b* ∈ *B*. Therefore one such element *f*(*b*) is in *V*, since *V* is Scott-continuous. This shows that *b’* ∈ ↟*b*, where *b* ∈ *f*^{–1}(*V*), and hence that *b’* ∈ *U* ∩ *B*. Conversely, for every *b’* ∈ *U* ∩ *B*, there is a *b* ∈ *f*^{–1}(*V*) such that *b* ≪ *b’*. Since *f* is monotonic and *V* is upwards-closed, *b’* ∈ *f*^{–1}(*V*). ☐

It follows that:

**Fact.** A monotonic function *f* from a basis *B* of a continuous poset *X* to a dcpo *Y* has a Scott-continuous extension *f’* from the whole of *X* to *Y* if and only if *f* is relatively Scott-continuous on *B*.

In Corollary 5.1.61 of the book, the following is stated: if *f* is Scott-continuous from *B* to *Y*, then it has a (unique) continuous extension *f’*. We have seen that being Scott-continuous from *B* to *Y* (=monotonic+mapping existing directed *B*-suprema to suprema) is different from being relatively Scott-continuous on *B*. Corollary 5.1.61 of the book is a *consequence* of the fact we have just stated. Since *X*-suprema of elements of *B* that happen to be in *B* are also *B*-suprema (not the other way around!), any Scott-continuous map from *B* to *Y* is also relatively Scott-continuous on *B*. It just so happens that Corollary 5.1.61 of the book is not the most general result that one could state.

Perhaps more annoyingly, I have realized recently that Corollary 5.1.61 of the book is *almost never* what you need in applications. Most of the time, it is much more natural to show that *f* is *relatively* Scott-continuous. The reason is that, in usual applications, we have a firm grasp over what directed *X*-suprema look like in *X*, but it would require quite some work to characterize *B*-suprema—and, as we have seen, that is in fact useless.

## A sketch of an application

A *continuous valuation* on a topological space *X* is a Scott-continuous map ν from the lattice of open sets **O***X* of *X* to **R**_{+} ∪ {∞} that satisfies the following two conditions:

*strictness*: ν(∅)=0;*modularity*: for all*U*,*V*in**O***X*, ν(*U*∪*V*)+ν(*U*∩*V*)=ν(*U*)+ν(*V*).

I will concentrate on *bounded* continuous valuations ν, namely those such that ν(*X*) < ∞.

Continuous valuations are very close to measures: we can integrate with respect to a continuous valuation, and in fact continuous valuations extend to measures on the Borel σ-algebra of *X* when *X* is LCS-complete, and measures restrict to continuous valuations on the open sets when *X* is hereditarily Lindelöf, in particular when *X* is second-countable. For a space *X* that is both (typically any quasi-Polish space), this even defines a one-to-one correspondence between bounded continuous valuations and bounded measures. I will not expand on this here.

In [1], Klaus Keimel and I gave pretty simple generalizations of what has been known as Choquet-Kendall-Mathéron theorems. Let me give one example of this.

We consider the Smyth hyperspace **Q***Z* of a topological space *Z*. That is the set of non-empty compact saturated subsets of *Z*, and we give it the upper Vietoris topology, whose basic open subsets are ☐*U* ≝ {*Q* ∈ **Q***Z* | *Q* ⊆ *U*}, where *U* ranges over the open subsets of *Z*.

Given a continuous valuation ν on **Q***Z*, we may form a function μ : **O***Z* → **R**_{+} ∪ {∞} by letting μ(*U*) ≝ ν(☐*U*) for every open subset *U* of *X*. Such a function μ has the following properties, which make μ what I call a *continuous credibility*, and which has been known under the name of a (continuous) totally monotone capacity, or a (continuous) totally convex capacity. Namely, μ is Scott-continuous, strict, and satisfies the following condition of *total convexity*:

μ(*U*) ≥ ∑* _{I}* (-1)

_{|I|+1}μ(∩

_{i∈I}

*U*

_{i})

for all open sets *U*, *U*_{1}, …, *U*_{n} such that *U* contains *U*_{1} ∪ … ∪ *U _{n}*, and where the summation extends over all non-empty subsets

*I*of {1, …,

*n*}. This is an inequational form of the so-called inclusion-exclusion formula in probability theory, and I have already mentioned this in relation to the baby Groemer theorem.

The point of the (first of three) Choquet-Kendall-Mathéron theorems in [1] is to give a converse to this construction. This way, continuous credibilities are relatively practical *representations* for continuous valuations on the hyperspace **Q***Z* of non-empty compact saturated subsets of *Z*.

For this, we use that *Z* is a locally compact space (although a core-compact space would suffice), and we start from a function μ : **O***Z* → **R**_{+} (not **R**_{+} ∪ {∞}, so μ is, in fact, bounded) that is Scott-continuous, strict, and totally convex in the sense described above. With this as input, we show that there is a (unique, bounded) continuous valuation ν : **O**(**Q***Z*) → **R**_{+} such that μ(*U*) ≝ ν(☐*U*) for every open subset *U* of *X*. Here is how, omitting most of the details, and in the goal of showing where Scott’s formula is used.

Using the baby Groemer theorem, we can show that there is a function ν : **EO**(**Q***Z*) → **R**_{+} such that μ(*U*) ≝ ν(☐*U*) for every open subset *U* of *Z*. Here **EO**(**Q***Z*) denotes the lattice of *elementary open subsets* of **Q***Z*, which we define as the *finite* unions of basic open sets ☐*U*. This does not use the fact that *Z* is locally compact yet. Also, ν is monotonic, strict and modular. The next move is to extend ν to the whole of **O**(**Q***Z*), and for this we use Scott’s formula on the basis *B* ≝ **EO**(**Q***Z*) of **O**(**Q***Z*); that is indeed a basis because *Z* is locally compact, and therefore also **Q***Z*.

Great: omitting all the details, all that is easy. However, there is still something missing. We would like to show that the extension ν’ of ν that we obtained through Scott’s formula is a continuous valuation. It is continuous, but we also need to verify that it is strict and modular, right? In [1], we proved it by hand, but surely there is a general theorem that would allow us to prove it directly.

This is exactly what I will state. We examine under what conditions any algebraic laws satisfied by ν (or, in general, a continuous map *f* from *B* to *Y*) are still satisfied by its unique continuous extension ν’ (or more generally, *f’* : *X* → *Y*). In the sequel, I will even look at cases where we can ensure that algebraic laws satisfied by *f* are also satisfied by *f’*, even when *f’* is not an extension of *f*, just the largest continuous map below *f* on *B*.

## Preserving algebraic laws

Given a c-space *X* with basis *B* and a monotone convergence space *Y*, we consider formal inequalities of the form:

*g* (_(*g*_{1} (** z**)), …, _(

*g*

_{m}(

**))) ≤**

*z**h*(_(

*h*

_{1}(

**)), …, _(**

*z**h*

_{n}(

**)))**

*z*where *g* and *h* are maps from *Y ^{m}*, resp.

*Y*, to

^{n}*Y*, and

*g*

_{1}, …,

*g*

_{m},

*h*

_{1}, …,

*h*

_{n}are maps from

*X*to

^{k}*X*, for some

*k*∈

**N**. The notation

**stands for a tuple (**

*z**z*

_{1}, …,

*z*

_{k}) of

*k*distinct formal variables.

Such a formal inequality is *satisfied* on *X* (resp., on *B*) by a function *f* : *X* → *Y* if and only if

*g* (*f* (*g*_{1} (** u**)), …,

*f*(

*g*

_{m}(

**))) ≤**

*u**h*(

*f*(

*h*

_{1}(

**)), …,**

*u**f*(

*h*

_{n}(

**)))**

*u*for every tuple ** u** = (

*u*

_{1}, …,

*u*

_{k}) of values in

*X*(resp., in

*B*). For example, a continuous valuation ν on a space X is the same thing as a Scott-continuous map ν:

**O**

*X*→

**R**

_{+}∪ {∞} that satisfies the inequalities:

- _(
*g*_{1}()) ≤*h*(), where*g*_{1}is the 0-ary map with value ∅ and*h*is the 0-ary map with value 0 (in short, the inequality _(∅) ≤ 0, expressing strictness); - _(
*z*_{1}∪*z*_{2})+_(*z*_{1}∩*z*_{2}) ≤ _(*z*_{1})+_(*z*_{2}); - _(
*z*_{1})+_(*z*_{2}) ≤ _(*z*_{1}∪*z*_{2})+_(*z*_{1}∩*z*_{2}).

Note in particular how we expressed the *equality* _(*z*_{1} ∪ *z*_{2})+_(*z*_{1} ∩ *z*_{2}) = _(*z*_{1})+_(*z*_{2}) (modularity) as two inequalities in opposite directions.

Now the answer that I will give below requires the functions *g* and *g*_{1}, …, *g*_{m} to be continuous, but the functions *h*_{1}, …, *h*_{n} will need to obey a different assumption. Let me say that a map *h* : *X* → *Y* is *quasi-open* if and only if for every open subset *U* of *X*, ↑*h*[*U*] is open in *Y*. This is the same idea as a notion that Klaus Keimel once used for the addition operation of a semitopological cone, and which he called *almost open*.

**Proposition.** Let *X* be a c-space with basis *B*, *Y* be a monotone convergence space, and *L* be a formal inequality *g* (_(*g*_{1} (** z**)), …, _(

*g*

_{m}(

**))) ≤**

*z**h*(_(

*h*

_{1}(

**)), …, _(**

*z**h*

_{n}(

**))) such that:**

*z**g*_{1}, …,*g*_{m},*h*_{1}, …,*h*_{n}applied to elements of*B*yield values in*B*;*g*is continuous on*Y*and*g*_{1}, …,*g*_{m}are continuous on*X*;^{k}*h*is monotonic on*Y*and*h*_{1}, …,*h*_{n}are monotonic on*X*.^{k}

For every monotonic map *f* : *B* → *Y*, if *f* satisfies *L* on *B*, and if:

- either
*f*is continuous on*B*, with the subspace topology (in which case*f’*extends*f*), - or
*h*_{1}, …,*h*_{n}are quasi-open on*X*(in which case^{k}*f’*does not necessarily extend*f*),

then *f’* satisfies *L* on *X*.

*Proof.* For every tuple ** u** ≝ (

*u*

_{1}, …,

*u*

_{k}) of values in

*X*, we verify that:

*g* (*f’* (*g*_{1} (** u**)), …,

*f’*(

*g*

_{m}(

**))) ≤**

*u**h*(

*f’*(

*h*

_{1}(

**)), …,**

*u**f’*(

*h*

_{n}(

**)))**

*u*by considering any open neighborhood *V* of the left-hand side, and showing that it contains the right-hand side. By assumption, (*f’* (*g*_{1} (** u**)), …,

*f’*(

*g*

_{m}(

**))) is in**

*u**g*

^{−1}(

*V*). Since

*g*is continuous, the latter is open, so we can find open neighborhoods

*U*

_{1}, …,

*U*

_{m}of

*f’*(

*g*

_{1}(

**)), …,**

*u**f’*(

*g*

_{m}(

**)) respectively such that Π**

*u*_{i=1}

^{m}*U*

_{i}⊆

*g*

^{−1}(

*V*).

Since *Y* is a monotone convergence space, each *U*_{i} is Scott-open. By definition of *f’*, it follows that for each *i* there is an element *b _{i}* ∈

*B*such that

*b*≪

_{i}*g*

_{i}(

**) and**

*u**f*(

*b*) ∈

_{i}*U*

_{i}. In particular,

*f*(

*b*

_{1}, …,

*b*) ∈

_{m}*V*.

Since every *g*_{i} is continuous, ∩_{i=1}^{m}*g*_{i}^{–1}(↟*b _{i}*) is an open neighborhood of

**. Since**

*u**X*is a c-space, with basis

^{k}*B*(exercise!), there is a tuple

^{k}**≝ (**

*c**c*

_{1}, …,

*c*

_{k}) of values of

*B*such that

**∈ ∩**

*c*_{i=1}

^{m}*g*

_{i}

^{–1}(↟

*b*) and

_{i}**≪**

*c***(namely, such that**

*u**c*

_{1}≪

*u*

_{1}, …,

*c*

_{k}≪

*u*

_{k}; remember that this means that each

*u*

_{i}is in the interior of ↑

*c*

_{i}). Then

*b*≤

_{i}*g*

_{i}(

**) for every**

*c**i*, and since

*f*is monotonic,

*f*(

*b*

_{1}, …,

*b*) ≤

_{m}*f*(

*g*

_{1}(

**), …,**

*c**g*

_{m}(

**)). This entails that**

*c**f*(

*g*

_{1}(

**), …,**

*c**g*

_{m}(

**)) ∈**

*c**g*

^{−1}(

*V*), hence that

*g*(

*f*(

*g*

_{1}(

**), …,**

*c**g*

_{m}(

**))) ∈**

*c**V*.

All the elements *g*_{1} (** c**), …,

*g*

_{m}(

**),**

*c**h*

_{1}(

**), …,**

*c**h*

_{n}(

**) are in**

*c**B*, by our assumption 1. Since

*f*satisfies

*L*on

*B*,

*g* (*f* (*g*_{1} (** c**)), …,

*f*(

*g*

_{m}(

**))) ≤**

*c**h*(

*f*(

*h*

_{1}(

**)), …,**

*c**f*(

*h*

_{n}(

**))),**

*c*so *h* (*f* (*h*_{1} (** c**)), …,

*f*(

*h*

_{n}(

**))) is in**

*c**V*.

We now consider two cases, depending on whether assumption 4 or 5 is satisfied.

*First case: assumption 4 is satisfied.* In other words, we assume that *f* is continuous on *B*. Then *f’* extends *f*, as we have seen earlier. Then *h* (*f* (*h*_{1} (** c**)), …,

*f*(

*h*

_{n}(

**))) =**

*c**h*(

*f’*(

*h*

_{1}(

**)), …,**

*c**f’*(

*h*

_{n}(

**))). Since each**

*c**u*

_{i}is in the interior of ↑

*c*

_{i}, in particular

*c*

_{i}≤

*u*

_{i}for each

*i*, so, using the fact that

*f’*,

*h*

_{1}, …,

*h*

_{n}are monotonic,

*h*(

*f*(

*h*

_{1}(

**)), …,**

*c**f*(

*h*

_{n}(

**))) ≤**

*c**h*(

*f’*(

*h*

_{1}(

**)), …,**

*u**f’*(

*h*

_{n}(

**))), and therefore**

*u**h*(

*f’*(

*h*

_{1}(

**)), …,**

*u**f’*(

*h*

_{n}(

**))) is in**

*u**V*.

*Second case: assumption 5 is satisfied.* Then *f’* may fail to extend *f*, but assumption 5 tells us that the maps *h*_{1}, …, *h*_{n} are quasi-open. Since each *u*_{i} is in the interior ↟*c*_{i} of ↑*c*_{i}, *h*_{i}(** u**) is in the open set ↑

*h*

_{i}[↟

*c*

_{1}× … × ↟

*c*

_{k}]. The latter is equal to its own interior, and is therefore included in the interior of the larger set ↑

*h*

_{i}[↑

*c*

_{1}× … × ↑

*c*

_{k}] = ↑

*h*

_{i}(

**). Hence**

*c**h*

_{i}(

**) ∈ int(↑**

*u**h*

_{i}(

**)), in other words**

*c**h*

_{i}(

**) ≪**

*c**h*

_{i}(

**). By definition of**

*u**f’*,

*f*(

*h*

_{1}(

**)), …,**

*c**f*(

*h*

_{n}(

**)) ≤**

*c**f’*(

*h*

_{1}(

**)), …,**

*u**f’*(

*h*

_{n}(

**)). We use the fact that**

*u**h*is monotonic, and we obtain that

*h*(

*f*(

*h*

_{1}(

**)), …,**

*c**f*(

*h*

_{n}(

**))) ≤**

*c**h*(

*f’*(

*h*

_{1}(

**)), …,**

*u**f’*(

*h*

_{n}(

**))), so that**

*u**h*(

*f’*(

*h*

_{1}(

**)), …,**

*u**f’*(

*h*

_{n}(

**))) is in**

*u**V*. ☐

Let us see what this yields on our example of inequalities characterizing valuations (strictness, plus two inequalities for modularity):

- _(∅) ≤ 0;
- _(
*z*_{1}∪*z*_{2})+_(*z*_{1}∩*z*_{2}) ≤ _(*z*_{1})+_(*z*_{2}); - _(
*z*_{1})+_(*z*_{2}) ≤ _(*z*_{1}∪*z*_{2})+_(*z*_{1}∩*z*_{2}).

The first inequality is really _(*g*_{1}()) ≤ *h*(), where *g*_{1} is the 0-ary map with value ∅ and *h* is the 0-ary map. Considering that the basis *B* we took in this example was **EO**(**Q***Z*), the set of finite unions of basic open subsets of **Q***Z*, condition 1 (mapping *B ^{k}* to

*B*) is obvious. Condition 2 is that

*g*

_{1}is continuous (and also

*g*, which is the identity map in that case). Condition 3 is that

*h*is monotonic, which is clear as well.

Condition 1 is also clear for the second and third inequalities: namely, **EO**(**Q***Z*) is closed under binary unions, and also under binary intersections (because ☐*U* ∩ ☐*V* = ☐(*U* ∩ *V*)). As far as conditions 2 and 3 are concerned, both *g* and *h* are the + function on **R**_{+} ∪ {∞}, and that is continuous (for *g*; in particular it is monotonic, which is what we want for *h*). By continuous, I mean jointly continuous, by the way, but separate continuity is enough, by the Banaschewski-Lawson-Ershov observation. The maps *g*_{i} and *h*_{i} are: binary union ∪, binary intersection ∩, and first and second projections (this is how you get *z*_{1}, resp., *z*_{2}, from ** z** = (

*z*

_{1},

*z*

_{2})). And they are all Scott-continuous, hence separately continuous, hence jointly continuous by the Banaschewski-Lawson-Ershov observation, since this example took place in the setting of a locally compact space, hence one such that

**O**

*Z*is a continuous dcpo.

The setting Klaus Keimel and I were in was where *f* (the valuation ν on *B*=**EO**(**Q***Z*) such that μ(*U*) ≝ ν(☐*U*) for every open subset *U* of *Z*) was already continuous on *B*, so condition 4 applies, and the extension *f’* is now automatically a continuous valuation on the space *Z*, finishing the proof.

However, let us imagine that we wanted to use condition 5 instead. We would need to examine when binary union, binary intersections, and first and second projections are quasi-open. Let me state how we can simplify the proof of such statements. We will see that satisfying condition 5 would require more from the underlying space *Z*.

## Quasi-open maps and the way-below relation

Condition number 5 (the one that applies in the previous proposition in the case that *f* is not continuous on *B*, and therefore does not extend to *f’*) is about quasi-openness, which may appear as a mysterious property. It is equivalent to the following one, however.

**Proposition.** For every c-space *X*, for every topological space *Y*, for every natural number *k*, a map *h* : *X ^{k}* → X is quasi-open if and only if it

*preserves*≪, namely if and only if for all

*x*

_{1}, …,

*x*

_{k},

*x’*

_{1}, …,

*x’*

_{k}in

*X*such that

*x*

_{1}≪

*x’*

_{1}, …,

*x*

_{k}≪

*x’*

_{k}, we have

*h*(

*x*

_{1}, …,

*x*

_{k}) ≪

*h*(

*x’*

_{1}, …,

*x’*

_{k}).

*Proof.* Let us assume *h* quasi-open. Then ↑*h*[↟*x*_{1} × … × ↟*x*_{k}] is open. If *x*_{1} ≪ *x’*_{1}, …, *x*_{k} ≪ *x’*_{k}, then *h* (*x’*_{1}, …, *x’*_{k}) is in ↑*h*[↟*x*_{1} × … × ↟*x*_{k}]. Also, ↑*h* [↟*x*_{1} × … × ↟*x*_{k}] is included in ↑*h* [↑*x*_{1} × … × ↑*x*_{k}] = ↑*h* (*x*_{1}, …, *x*_{k}), so *h* (*x’*_{1}, …, *x’*_{k}) is in an open set that is included in ↑*h* (*x*_{1}, …, *x*_{k}). It follows that *h* (*x’*_{1}, …, *x’*_{k}) is in the interior of ↑*h* (*x*_{1}, …, *x*_{k}), which means that *h* (*x*_{1}, …, *x*_{k}) ≪ *h* (*x’*_{1}, …, *x’*_{k}), by our definition of ≪ on topological spaces.

Conversely, let us assume that *h* preserves ≪. Let *U* be any open subset of *X ^{k}*. In order to show that ↑

*h*[

*U*] is open, we show that every element

*y*of ↑

*h*[

*U*] belongs to some open subset

*V*of

*Y*that is included in ↑

*h*[

*U*]. Since

*y*∈ ↑

*h*[

*U*], there is a tuple (

*x’*

_{1}, …,

*x’*

_{k}) in

*U*such that

*h*(

*x’*

_{1}, …,

*x’*

_{k}) ≤

*y*. Notably, there is an open rectangle

*U*

_{1}× … ×

*U*

_{k}included in

*U*and containing (

*x’*

_{1}, …,

*x’*

_{k}). Since

*X*is a c-space, we can find elements

*x*

_{1}≪

*x’*

_{1}, …,

*x*

_{k}≪

*x’*

_{k}in

*U*

_{1}, …,

*U*

_{k}respectively. In particular, (

*x*

_{1}, …,

*x*

_{k}) is in

*U*. Since

*h*preserves ≪,

*h*(

*x*

_{1}, …,

*x*

_{k}) ≪

*h*(

*x’*

_{1}, …,

*x’*

_{k}), namely

*h*(

*x’*

_{1}, …,

*x’*

_{k}) is in the interior

*V*of ↑

*h*(

*x*

_{1}, …,

*x*

_{k}). Since

*h*(

*x’*

_{1}, …,

*x’*

_{k}) ≤

*y*,

*y*is also in

*V*. Finally,

*V*is included in ↑

*h*(

*x*

_{1}, …,

*x*

_{k}), which is included in ↑

*h*[

*U*] since (

*x*

_{1}, …,

*x*

_{k}) is in

*U*. ☐

Returning to our example of the inequalities defining modularity, when are binary union, binary intersections, first and second projections quasi-open? We are working on *X* = **O***Z* where *Z* is locally compact, hence *X* is a continuous dcpo and ≪ is the usual notion of way-below relation. It is easy to see that union, first and second projections do preserve ≪. But binary intersection preserves ≪ if and only if *Z* is *core-coherent*: this is Proposition 5.2.19 in the book.

In other words, relying on condition 5 dispenses us to show that *f* is continuous on *B* (condition 4—that can indeed be a chore!), but requires more from the underlying space *X*.

## Other extension results

Scott’s formula is only one of many possible extension results. I should mention that there is another close one, which requires a lot less on *X* and a lot more on *Y*. Given any dense subset *B* of an arbitrary topological space *X*, a bc-domain *Y*, and a monotonic map *f* from *B* to *Y*, there is a largest continuous map *f’* from *X* to *Y* below *f*; and if *f* is continuous, then *f’* is a continuous extension of *f* from *B* to the whole of *X*. That is part of *Scott’s theorem* that the bc-domains are the injective topological spaces (over topological embeddings).

Explicitly, we first extend *f* to a monotonic map from *X* to *Y* by letting *f*(*x*) be equal to the infimum of all *f*(*b*), where *b* ranges over the elements of *B* above *x*, for every *x* in *X*. In a second step, we define *f’*(*x*) as sup {*y* ∈ *Y* | *x* ∈ int(*f*^{−1}(↟*y*))}. I will not prove this here, maybe another time. (This may be wrong, too. I have just written it mostly from the top of my head, without much verification.)

There is a variant of this where *Y* is required to be a continuous complete lattice, and we obtain a similar result, where this time *B* is an arbitrary (not a dense) subset of *X*.

And there are many other extension results where *X* is obtained as the sobrification of a space *B*, or as its well-filterification, or etc. Oh, well.

- Jean Goubault-Larrecq and Klaus Keimel. Choquet-Kendall-Matheron theorems for non-Hausdorff spaces. Mathematical Structures in Computer Science 21(3), 2011, pages 511-561.

— Jean Goubault-Larrecq (September 20th, 2023)