Last time, we had started to study the following poset *L*_{fan}, introduced independently in [3, 1, 4]: *L*_{fan} is **N** × **N**, with equality as ordering on the first component and the usual ordering on the second component, plus an extra bottom element ⊥, plus an extra top element ⊤:

We had seen that *L*_{fan} is a complete lattice with the Chen-Kou-Lyu property—in particular, (*L*_{fan} × *L*_{fan})_{σ} = (*L*_{fan})_{σ} × (*L*_{fan})_{σ}, where the σ subscript means “with the Scott topology”—although it is neither core-compact nor first-countable (in its Scott topology).

The Scott topology on *L*_{fan} can be described explicitly. For every function *f* : **N** → **N**, let *U _{f}* be the subset of those pairs (

*m*,

*n*) of

**N**×

**N**such that

*n*≥

*f*(

*m*), plus ⊤.

Then the Scott-open subsets of *L*_{fan} are exactly the sets *U _{f}*, where

*f*ranges over the functions from

**N**to

**N**, plus the empty set, plus the whole space

*L*

_{fan}. Additionally,

*U*⊆

_{f}*U*if and only if

_{g}*f*≥

*g*, so that

**O**(

*L*

_{fan}) is order-isomorphic to the poset (

**N**→

**N**)

^{op}of functions from

**N**to

**N**, ordered with the

*opposite*ordering ≥ (not ≤), with an additional bottom element (∅) and an additional top element (

*L*

_{fan}).

## Sobriety

We start this month’s post with a nice piece of news about *L*_{fan}.

**Lemma A.** *L*_{fan} is sober in its Scott topology.

*Proof.* Let *C* be an irreducible subset of *L*_{fan}. It is non-empty, and downwards-closed, hence contains ⊥. If it only contains ⊥, then *C* is the downward closure of ⊥. If *C* contains ⊤, then *C* = ↓⊤. Henceforth, let us assume that *C* is neither of those. But *C* is closed, the closed sets are the complements of open sets, and we know what the open sets are: *C* is the complement of *U _{f}*, for some function

*f*from

**N**to

**N**. Also,

*f*is not the constant 0 function, since otherwise

*C*would be {⊥}.

For every *m* ∈ **N**, let us say that *C* has a *non-trivial column*, and precisely that it has *non-trivial column number* *m* if and only if *C* contains a point (*m*, *n*) for some *n*, equivalently if and only if *f*(*m*)≠0. If so, there is a largest point (*m*, *n*) in *C’*s column *m*, namely (*m*, *f*(*m*)–1). Since *f* is not the constant 0 function, *C* has at least one non-trivial column. We claim that it has exactly one.

Otherwise, let us say that columns *m* and *m’* are non-trivial, with *m*≠*m’*. Let *g* : **N** → **N** be as *f*, except that *g* maps *m* to *f*(*m*)–1, and let *g’* be as *f* except that it maps *m’* to *f*(*m’*)–1. Then *C* intersects *U _{g}* at (

*m*,

*f*(

*m*)–1),

*U*at (

_{g’}*m’*,

*f*(

*m’*)–1), but

*C*does not intersect

*U*∩

_{g}*U*, since the latter is equal to

_{g’}*U*

_{sup(}

_{g}_{,}

_{g’}_{)}=

*U*. This contradicts the fact that

_{f}*C*is irreducible (see Trick 8.2.3 in the book).

Therefore *C* has exactly one non-trivial column, say with number *m*. Then *C* = ↓(*m*, *f*(*m*)–1). In all cases, *C* is the downward-closure of a (unique) point, so *L*_{fan} is sober. ☐

This was proved by Chen, Kou and Lyu [1, Example 4.13].

## Weak Hausdorffness, and being a maximal limit space

The following properties were not considered by Chen, Kou and Lyu [1], or by Hertling [4], or by Xu, Shen, Xi and Zhao [3], but I think they deserve to be mentioned, specially considering that they were the topic of some of my recent posts.

A space *X* is *weakly Hausdorff* in the sense of Klaus Keimel and Jimmie Lawson [5] if and only if for any two points *x* and *y* in *X*, for every open neighborhood *W* of ↑*x* ∩ ↑*y*, there is an open neighborhood *U* of *x* and there is an open neighborhood *V* of *y* such that *U* ∩ *V* ⊆ *W*. I have already written a few times about that notion.

**Lemma B.** *L*_{fan} is weakly Hausdorff in its Scott topology.

*Proof.* We take two points *x* and *y* and an open neighborhood *W* of ↑*x* ∩ ↑*y*. The existence of *U* and *V* is obvious if *x*≤*y* (or symmetrically if *y*≤*x*): simply take the whole space for *U* and *W* for *V*. Hence let us assume that *x* and *y* are incomparable. Then none of *x* or *y* is equal to ⊤ or to ⊥, so we can write *x* as (*m*, *n*) and *y* as (*m’*, *n’*), where additionally *m*≠*m’*.

The intersection ↑*x* ∩ ↑*y* is then equal to {⊤}. If *W* is the whole space *L*_{fan}, then we can simply take *U* and *V* equal to the whole space as well. Henceforth, let us assume that *W* is a proper open subset of *L*_{fan}. Since *W* contains {⊤}, it is non-empty, hence of the form *U _{f}*, for some function

*f*:

**N**→

**N**, by Lemma E. We let

*g*:

**N**→

**N**be defined as

*f*, except that it maps

*m*to 0; and

*g’*be defined as

*f*except that it maps

*m’*to 0. Then

*x*= (

*m*,

*n*) is in

*U*,

_{g}*y*= (

*m’*,

*n’*) is in

*U*, and

_{g’}*U*∩

_{g}*U*=

_{g’}*U*

_{sup(}

_{g}_{,}

_{g’}_{)}=

*U*=

_{f}*W*, so we can take

*U*≝

*U*and

_{g}*V*≝

*U*. ☐

_{g’}In this post (March 2024) on Hoover’s maximal limit spaces [6], namely on spaces in which every convergent filter has a unique largest limit, we have shown (Theorem I there) that a maximal limit space is the same thing as a weakly Hausdorff, binary-bounded-sup-complete, monotone convergence space. By Lemma A, *L*_{fan} is sober, hence a monotone convergence space (see Proposition 8.2.34 in the book), it is weakly Hausdorff by Lemma B, and it is certainly binary-bounded-sup-complete, since it is a complete lattice, as we have seen last time. Therefore:

**Proposition C.** *L*_{fan} is a maximal limit space.

In particular, it is coherent, too. We will retrieve this below, when we study the compact saturated subsets of *L*_{fan}.

## The dcpo **O**(*L*_{fan}) of Scott-open subsets of *L*_{fan}

The complete lattice **O**(*L*_{fan}) of Scott-open subsets of *L*_{fan} consist of the sets *U _{f}*, with

*f*:

**N**→

**N**, ordered with the opposite of the pointwise ordering on functions

*f*, plus a top element (the whole space

*L*

_{fan}), plus a bottom element (the empty set).

Let us write **N**_{ω} for the set of natural numbers **N**, with an extra element ω above all natural numbers. We order it in the usual way: 0 ≤ 1 ≤ … ≤ *n* ≤ … ≤ ω.

For every function *g* from **N** to **N**_{ω}, let dom *g* be the set of numbers *m* ∈ **N** such that *g*(*m*)≠ω. This is the *domain* of *g*, which I will write as dom *g*. We let *FF* be the set of functions *f* from **N** to **N**_{ω} whose domain is finite, ordered pointwise. Hertling [4, before Lemma 4.13] uses actual total functions from a finite subset of **N** to **N** (not **N**_{ω}), but I think this definition of *FF* will be more practical.

For every *g* ∈ *FF*, let *V _{g}* be the collection of Scott-open subsets of

*L*

_{fan}of the form

*U*with

_{f}*f*≤

*g*, plus

*L*

_{fan}itself. The following is due to Hertling [4, Lemma 4.14].

**Proposition D.** The sets *V _{g}*,

*g*∈

*FF*, plus {

*L*

_{fan}} and

**O**(

*L*

_{fan}), form a base of the Scott topology on

**O**(

*L*

_{fan}).

*Proof.* We first check that *V _{g}* is Scott-open. It is clear that

*V*is upwards-closed; notably, if

_{g}*U*is in

_{f}*V*, then

_{g}*f*≤

*g*, so every

*U*containing

_{f’}*U*must be such that

_{f}*f’*≤

*f*(that was Fact F of last time), whence

*U*is in

_{f’}*V*, too. We consider any directed family

_{g}*D*of Scott-open subsets of

*L*

_{fan}whose supremum is in

*V*, and we wish to show that

_{g}*D*intersects

*V*. If

_{g}*L*

_{fan}∈

*D*, then some element of

*D*(

*L*

_{fan}itself) is in

*V*, and we are done. Hence we assume that

_{g}*L*

_{fan}is not in

*D*. It cannot be the case that

*D*only contains the empty set, since sup

*D*would be the empty set, and ∅ is not in

*V*. Then we may assume that

_{g}*D*does not contain the empty set, without loss of generality: removing the empty set yields a cofinal subfamily, which will in particular be directed and have the same supremum. This way, we may assume that

*D*is a directed family of sets of the form

*U*,

_{fi}*i*∈

*I*. Using the order-isomorphism with (

**N**→

**N**)

^{op}plus extra top and bottom elements (Fact F of last time), its supremum is

*U*where

_{f}*f*is the pointwise infimum of (

*f*)

_{i}_{i ∈ I}.

By assumption, *U _{f}* is in

*V*, so

_{g}*f*≤

*g*. For each one of the elements

*m*of the domain of

*g*, we can then find an index

*i*∈

*I*such that

*f*(

_{i}*m*)≤

*g*(

*m*); indeed, the family (

*f*(

_{i}*m*))

_{i ∈ I}is well-founded, namely it reaches its least element. Since dom

*g*is finite and (

*f*)

_{i}_{i ∈ I}is directed (with respect to ≥), we can take the same

*i*∈

*I*for every

*m*∈ dom

*g*; namely, for every

*m*∈ dom

*g*,

*f*(

_{i}*m*)≤

*g*(

*m*). For every natural number

*m*outside dom

*g*, we have

*g*(

*m*)=ω, so

*f*(

_{i}*m*)≤

*g*(

*m*) as well. Therefore

*f*≤

_{i}*g*. This terminates our argument that

*V*is Scott-open.

_{g}Second, **O**(*L*_{fan}) is trivially Scott-open, and {*L*_{fan}} is Scott-open, too: given any directed family of Scott-open subsets (*U _{i}*)

_{i ∈ I}of

*L*

_{fan}whose union is in {

*L*

_{fan}}, namely whose union is equal to

*L*

_{fan}, or equivalently, whose union contains ⊥, some

*U*must contain ⊥ already, namely

_{i}*U*=

_{i}*L*

_{fan}.

Finally, we need to show that the set *V _{g}*, {

*L*

_{fan}} and

**O**(

*L*

_{fan}) form a base of of the Scott topology on

**O**(

*L*

_{fan}). Let

*U*∈

**O**(

*L*

_{fan}), and let

**be a Scott-open subset of**

*W***O**(

*L*

_{fan}) such that

*U*∈

**. If**

*W**U*=

*L*

_{fan}, then

*U*∈ {

*L*

_{fan}} ⊆

**. If**

*W**U*=∅, then

*=*

**W****O**(

*L*

_{fan}). It remains to see what happens when

*U*=

*U*for some function

_{f}*f*from

**N**to

**N**. For every

*k*∈

**N**, let

*g*:

_{k}**N**→

**N**

_{ω}map every natural number

*m*<

*k*to

*f*(

*m*), and all others to ω. Clearly,

*U*=

*U*∈

_{f}*V*for every

_{gk}*k*∈

**N**(since

*f*≤

*g*), and we claim that

_{k}*V*⊆

_{gk}**for some**

*W**k*∈

**N**; this will finish the proof.

Let us assume that *V _{gk}* is not included in

**for any**

*W**k*∈

**N**. For every

*k*∈

**N**, there is an element of

*V*that is not in

_{gk}**. That element cannot be**

*W**L*

_{fan}, since

*L*

_{fan}is in

**(**

*W***is non-empty since it contains**

*W**U*, and upwards-closed, hence contains the largest element of

**O**(

*L*

_{fan})). It cannot be the empty set either, since the empty set is not in

*V*. Hence it must be equal to

_{gk}*U*for some function

_{fk}*f*:

_{k}**N**→

**N**. For the record, we will to remember that: (a)

*U*∈

_{fk}*V*and: (b)

_{gk}*U*∉

_{fk}**.**

*W*Let *f’* : **N** → **N** map every number *m* to max(*f*(*m*), *f*_{0}(*m*), …, *f _{m}*(

*m*)). Then

*f*≤

*f’*, and: (c)

*f*(

_{k}*m*)≤

*f’*(

*m*) for every

*m*≥

*k*.

For every *k* ∈ **N**, we now define *f’ _{k}* :

**N**→

**N**as mapping every number

*m*<

*k*to

*f*(

*m*) and every number

*m*≥

*k*to

*f’*(

*m*). Then

*f’*=

*f’*

_{0}≥

*f’*

_{1}≥ … ≥

*f’*≥ …, and inf

_{k}_{k ∈ N}

*f’*=

_{k}*f*. Using the order-isomorphism with (

**N**→

**N**)

^{op}plus extra top and bottom elements (Fact F of last time),

*U*=

_{f’}*U*

_{f’}_{0}⊆

*U*

_{f’}_{1}⊆ … ⊆

*U*

_{f’}_{k}⊆ …, and sup

_{k ∈ N}

*=*

*U*_{f’}_{k}*U*. Since the latter is in

_{f}**, which is Scott-open, some**

*W**is in*

*U*_{f’}_{k}**.**

*W*We claim that *f’ _{k}* ≥

*f*. For every

_{k}*m*≥

*k*,

*f’*(

_{k}*m*) =

*f’*(

*m*) ≥

*f*(

_{k}*m*), by (c) (see a few lines back). For every

*m*<

*k*,

*f’*(

_{k}*m*) =

*f*(

*m*); since

*U*is in

_{fk}*V*(by (a)),

_{gk}*f*≤

_{k}*g*, so

_{k}*f*(

_{k}*m*)≤

*g*(

_{k}*m*), but since

*m*<

*k*and by definition of

*g*,

_{k}*g*(

_{k}*m*)=

*f*(

*m*); so

*f’*(

_{k}*m*) =

*f*(

*m*) =

*g*(

_{k}*m*) ≥

*f*(

_{k}*m*).

We can now conclude: *f’ _{k}* ≥

*f*, so by the usual isomorphism (Fact F of last time),

_{k}*is included in*

*U*_{f’}_{k}*; since*

*U*_{f}_{k}*is in*

*U*_{f’}_{k}**and**

*W***is upwards-closed,**

*W**is in*

*U*_{f}_{k}**, too. But that contradicts the definition of**

*W**, namely (b). Therefore*

*U*_{f}_{k}*V*⊆

_{gk}**for some**

*W**k*∈

**N**, as promised. ☐

As a side remark, the base given in Proposition D is countable. Therefore, and while *L*_{fan} is not first-countable hence not second-countable, **O**(*L*_{fan}) *is* second-countable in its own Scott topology, as was noted by Hertling.

*L*_{fan} is consonant

A space *X* is *consonant* if and only if for every open subset *U* of *X*, for every Scott-open neighborhood ** W** of

*U*in

**O**

*X*, there is a compact saturated subset

*Q*of

*X*such that

*Q*⊆

*U*and every open neighborhood of

*Q*in

*X*belongs to

**. I have talked quite a few times about that property. Let me recall that every locally compact space is consonant (Exercise 5.4.12 in the book), and that every regular Čech-complete space is consonant (the Dolecki-Greco-Lechicki theorem, see Exercise 8.3.4 in the book).**

*W***Lemma E.** Every sober space *X* such that **O***X* has a base of Scott-open subsets ** U** that are filters (viz., [upwards-closed, non-empty and] such that the intersection of any two elements of

**is in**

*U***) is consonant.**

*U**Proof.* Let *U* be an open subset of *X*, and ** W** be a Scott-open neighborhood of

*U*in

**O**

*X*. By assumption, there is a Scott-open neighborhood

**of**

*U**U*included in

**that is a filter. By the Hofmann-Mislove theorem (Theorem 8.3.2 in the book), since**

*W**X*is sober,

**is the collection of open neighborhoods of some compact saturated subset**

*U**Q*of

*X*. This is the desired set

*Q*. ☐

The sets *V _{g}*, with

*g*∈

*FF*, are filters. Given any two open subsets in

*V*, if one is equal to the whole of

_{g}*L*

_{fan}, then their intersection is the other one, which is in

*V*; otherwise, they are of the form

_{g}*and*

*U*_{f}*for some functions*

*U*_{f‘}*f*,

*f’*:

**N**→

**N**such that

*f*,

*f’*≤

*g*, and by the order-isomorphism with (

**N**→

**N**)

^{op}plus a top and a bottom element (Fact F of last time), their intersection is

*U*_{max(}

_{f}_{,}

_{f’}_{)}; since max(

*f*,

*f’*)≤

*g*,

*U*_{max(}

_{f}_{,}

_{f’}_{)}is also in

*V*. It is easy to see that {

_{g}*L*

_{fan}} and

**O**(

*L*

_{fan}) are filters, too. Those form a base of the Scott topology on

**O**(

*L*

_{fan}), by Proposition D. Since

*L*

_{fan}is sober (Lemma A), we can apply Lemma E, and we obtain the following.

**Proposition F.** *L*_{fan} is consonant in its Scott topology.

## The compact saturated subsets of *L*_{fan}

Given any function *g* ∈ *FF*, let *Q _{g}* be the subset of

*L*

_{fan}consisting of ⊤ plus all pairs (

*m*,

*n*) where

*m*∈ dom

*g*and

*n*≥

*g*(

*m*). Hence

*Q*is defined just like the Scott-open set

_{g}*U*, except that

_{f}*g*is a slightly different kind of function; also, while both

*Q*and

_{g}*V*are indexed by the same kind of functions

_{g}*g*,

*V*was a Scott-open subset of

_{g}**O**(

*L*

_{fan}), while

*Q*will be a compact saturated subset of

_{g}*L*

_{fan}. There is a link, though:

*V*is the collection of open neighborhoods of

_{g}*Q*in

_{g}*L*

_{fan}.

Here is an illustration of what *Q _{g}* looks like, with a function

*g*of domain {0, 2}:

**Lemma G.** The compact saturated subsets of *L*_{fan} are the empty set, the whole space *L*_{fan} and the sets of the form *Q _{g}*, where

*g*ranges over

*FF*. They are all finitary compact, namely, upward closures of finite sets.

*Proof.* All those sets—the empty set, *L*_{fan}, *Q _{g}*—are finitary compact; notably,

*Q*is the upward closure of the finite set of points (

_{g}*m*,

*g*(

*m*)),

*m*∈ dom

*g*if dom

*g*is non-empty, and is the upward closure of ⊤ if dom

*g*is empty. Therefore they are all upwards-closed and compact.

Conversely, let *Q* be a non-empty proper compact saturated subset of *L*_{fan}. Being proper, it does not contain ⊥. Since it is non-empty, it contains ⊤. For every *m* ∈ **N**, let us say that column *m* is *empty* if and only if *Q* contains no pair (*m*, *n*) for any *n* (but for this particular value of *m*); it is *non-empty* otherwise.

Let *I* be the set of indices *m* ∈ **N** such that column *m* is non-empty. There is a smallest natural number *n* such that (*m*, *n*) is in *Q*, and we call it *g*(*m*). For every number *m* outside *I*, we let *g*(*m*) ≝ ω. This defines a function *g* from **N** to **N**_{ω}, whose domain is *I*.

By way of contradiction, let us assume that *I* is infinite. We build a function *f _{m}* :

**N**→

**N**, one for each

*m*∈

*I*, which maps

*m*to

*g*(

*m*), every

*m’*∈

*I*–{

*m*} to

*g*(

*m’*)+1, and all other indices to some arbitrary value, say 0. The Scott-open sets

*U*,

_{fm}*m*∈

*I*, form an open cover of

*Q*: every point of

*Q*(other than ⊤, which is in any

*U*anyway—and there is such a

_{fm}*U*, since

_{fm}*I*is non-empty) is of the form (

*m*,

*n*) with

*n*≥

*g*(

*m*)=

*f*(

_{m}*m*), hence is in

*U*. That open cover has no strict subcover: if we remove any set

_{fm}*U*(

_{fm}*m*∈

*I*) from the cover, the point (

*m*,

*f*(

*m*)) of

*Q*fails to be in any

*U*,

_{fm’}*m’*∈

*I*–{

*m*}. Since

*I*is infinite, this contradicts the fact that

*Q*is compact.

Hence *I* is finite. Therefore *g* is in *FF*, and it is easy to see that *Q*=*Q _{g}*. ☐

**Fact H.** For all *g*, *g’* ∈ *FF*, *Q _{g}* ⊇

*Q*if and only if

_{g’}*g*≤

*g’*.

With this characterization, we retrieve some results we have already obtained, in a more concrete fashion:

*L*_{fan}is not core-compact. Indeed,*L*_{fan}is sober (Lemma A) and in that case, core-compactness is equivalent to local compactness (see Theorem 8.3.10 in the book). But the only compact saturated subset of*L*_{fan}with non-empty interior is*L*_{fan}itself. Indeed, the proper compact saturated subsets are of the form*Q*, and none contains any Scott-open subset of the form_{g}*U*. Explicitly,_{f}*U*⊆_{f}*Q*if and only if_{g}*f*≥*g*, and that cannot happen for an**N**-valued function*f*and a function*g*in*FF*.*L*_{fan}is coherent. This is a consequence of the fact that*L*_{fan}is a maximal limit space, which we have seen in Proposition C. More directly, the intersection of*Q*and of_{g}*Q*is equal to_{g’}*Q*, where_{h}*h*maps every*m*∈**N**to max(*g*(*m*),*g’*(*m*)).

## The Smyth hyperspace of *L*_{fan}

Last time (Corollary C), we had mentioned that, because *L*_{fan} has the Chen-Kou-Lyu property, the Scott topology coincides with the lower Vietoris topology on the Hoare hyperspace **H**(*L*_{fan}). Having explored the shape of compact saturated subsets of *L*_{fan}, we can now investigate the other classical hyperspace, the Smyth hyperspace.

The Smyth hyperspace **Q***X* of a space *X* is 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. With what we know about the compact saturated subsets of *L*_{fan}, we can show the following.

**Proposition I.** The upper Vietoris topology coincides with the Scott topology of ⊇ on **Q**(*L*_{fan}) (where *L*_{fan} is given its Scott topology).

*Proof.* First, every upper Vietoris open subsets of **Q**(*L*_{fan}) is Scott-open. Indeed, since *L*_{fan} is sober (Lemma A), hence well-filtered (Proposition 8.3.5 in the book), the supremum of any directed family (*Q _{i}*)

_{i∈I}in

**Q**(

*L*

_{fan}) is its intersection ∩

_{i∈I}

*Q*(Proposition 8.3.25 in the book), and ∩

_{i}_{i∈I}

*Q*is in a basic open set ☐

_{i}*U*if and only if some

*Q*is in ☐

_{i}*U*already (this is well-filteredness).

Second, let ** U** be any Scott-open subset of

**Q**(

*L*

_{fan}). For every

*Q*∈

**, we will find a Scott-open subset**

*U**V*of

*L*

_{fan}such that

*Q*∈ ☐

*V*⊆

**. By Lemma G,**

*U**Q*may have one of three possible forms. If

*Q*=

*L*

_{fan}, then

**is the whole of**

*U***Q**(

*L*

_{fan}), since

**is upwards-closed and**

*U**Q*is the least element of

**Q**(

*L*

_{fan}); then we may define

*V*as

*L*

_{fan}. If

*Q*is empty, then we may take

*V*≝ ∅. In the case that remains,

*Q*=

*Q*for some unique

_{g}*g*∈

*FF*, and we build a sequence of functions

*g*in

_{n}*FF*by induction on

*n*∈

**N**, as follows. The invariants we will maintain are:

- (A)
*Q*is in_{gn};*U* - (B) the domain of
*g*_{n}_{+1}is equal to the domain of*g*plus the one-element set {_{n}*k*}, where (_{n}*k*)_{n}_{n∈N}is a fixed enumeration without repetition of those natural numbers that lie outside the domain of*g*.

We start by defining *g*_{0} as *g*. Given that we have defined *g _{n}*, we observe that the functions

*g*[

_{n}*k*↦

_{n}*m*], defined just as

*g*except that

_{n}*k*(which is outside the domain of

_{n}*g*) is mapped to the natural number

_{n}*m*, form a monotone sequence when

*m*varies over

**N**, and that their supremum is

*g*. Through the order-isomorphism stated in Fact H,

_{n}*Q*is the supremum of the monotone sequence of compact saturated sets

_{gn}*Q*

_{gn}_{[}

_{kn}_{ ↦ }

_{m}_{]},

*m*∈

**N**. Since

*Q*is in

_{gn}**and**

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

*U**Q*

_{gn}_{[}

_{kn}_{ ↦ }

_{m}_{]}is in

**for some**

*U**m*∈

**N**. We pick one such

*m*, for example the least one, and we define

*g*

_{n}_{+1}as

*g*[

_{n}*k*↦

_{n}*m*]. Hence

*Q*

_{gn}_{+1}is in

**(see (A)) and (B) is true.**

*U*Now that we have defined the functions *g _{n}*, we define

*f*:

**N**→

**N**as the pointwise infimum of

*g*,

_{n}*n*∈

**N**. There is a simpler way of describing

*f*: for every

*m*∈

**N**,

*g*(

_{n}*m*) may be equal to ω for small values of

*n*, but for any

*n*large enough,

*g*(

_{n}*m*) will be the same natural number, independently of

*n*; and that number is

*f*(

*m*). This is a consequence of (B).

We let *V* ≝ *U _{f}*. This is Scott-open in

*L*

_{fan}. Also,

*Q*=

*Q*is included in

_{g}*U*, hence is an element of ☐

_{f}*V*. Indeed, ⊤ is in

*U*, and all the other elements of

_{f}*Q*are pairs (

_{g}*m*,

*n*) of natural numbers with

*n*≥

*g*(

*m*); since

*n*is a natural number,

*g*(

*m*)≠ω, so

*m*is in the domain of

*g*=

*g*

_{0}, and therefore

*g*(

*m*)=

*f*(

*m*), whence

*n*≥

*f*(

*m*).

Finally, we claim that ☐*V* ⊆ ** U**. Let

*Q’*be any element of ☐

*V*.

*Q’*cannot be the whole of

*L*

_{fan}, since otherwise ⊥ would be in

*Q’*, hence in

*V*=

*U*, and that is impossible. If

_{f}*Q’*is empty, then certainly it is in

**, because**

*U***is upwards-closed, non-empty (**

*U***contains**

*U**Q*) and therefore contains the top element of

**Q**(

*L*

_{fan}), which is the empty set. Finally, and using Lemma G, the only case that remains is that

*Q’*would be of the form

*Q*for some function

_{g’}*g’*in

*FF*. Since

*Q’*=

*Q*⊆

_{g’}*V*=

*U*, every pair (

_{f}*m*,

*n*) such that

*n*≥

*g’*(

*m*) satisfies

*n*≥

*f*(

*m*), so

*g’*≥

*f*. Since the domain dom

*g’*of

*g’*is finite, and since the domains of the functions

*g*form a monotone sequence of sets dom

_{n}*g*∪ {

*k*

_{0}, …,

*k*

_{n}_{–1}} as

*n*varies, and whose union is the whole of

**N**, it must be the case that dom

*g’*⊆ dom

*g*for some

_{n}*n*large enough. But

*f*coincides with

*g*on dom

_{n}*g*, so

_{n}*g’*≥

*f*entails that

*g’*≥

*g*: for every

_{n}*m*in dom

*g’*⊆ dom

*g*,

_{n}*g’*(

*m*)≥

*f*(

*m*)=

*g*(

_{n}*m*), and for every

*m*outside dom

*g’*,

*g’*(

*m*)=ω. By Fact H,

*Q*lies above (namely, is included in)

_{g’}*Q*. But, by (A),

_{gn}*Q*is in

_{gn}**. Since**

*U***is upwards-closed,**

*U**Q*(=

_{g’}*Q’*) is in

**. Since**

*U**Q’*is arbitrary in ☐

*V*, this shows that ☐

*V*⊆

**. ☐**

*U*Xiaoyong Xi calls *conformal* any well-filtered topological space *X* such that the upper Vietoris and Scott topologies of ⊇ coincide on **Q***X*. (I am not sure where this comes from. He gave me some names for the inventors of the notion, but I was unable to find a corresponding reference.)

One can read Lemma 8.3.26 in the book as saying that every well-filtered, locally compact space is conformal. A theorem due to Xiaoquan Xu and Zhongqiang Yang [9, Corollary 5.9] states that every well-filtered, first-countable space such that every compact saturated subset has countably many minimal elements is conformal. Here *L*_{fan} is neither core-compact nor first-countable, but is conformal nonetheless. (Note that its compact saturated subsets not only have countably many, but even finitely many minimal elements.)

I will conclude by the following summary: *L*_{fan} is an example of a conformal space in its Scott topology, which is not first-countable and not core-compact, but is a consonant, maximal limit space and a complete lattice.

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— Jean Goubault-Larrecq (June 20th, 2024)