When is ℒ(X) locally compact? II. The non-consonant case and the non-second-countable case

Let OX be the lattice of open subsets of a topological space X. Let us write O(X)σ, or just OσX, for OX with the Scott topology of the inclusion ordering ⊆. ℒ(X) is the lattice of lower semicontinuous maps from X to the extended non-negative real numbers, and ℒ(X)σ is ℒ(X) with the Scott topology of the pointwise ordering.

Last time, we have seen that, given any ⊙-consonant second-countable space X, the following are equivalent:

  1. X is core-compact;
  2. O(X)σ is core-compact (equivalently, locally compact, equivalently stably compact);
  3. ℒ(X)σ is core-compact (equivalently, locally compact, equivalently stably compact).

We had also seen that this equivalence also holds if X is a poset with its Scott topology.

On November 23rd, 2024, a few days after I posted these results, Matthew de Brecht sent me an email, giving me a proof that the result still holds if we only assume that X is second-countable: ⊙-consonance is not needed. His proof is pretty short, and relies on results from [1, 2]. I will give another proof below, which relies on results from the book and results that we have seen on this blog, most notably… M. de Brecht and T. Kawai’s results on the double hyperspace construction, and the Lyu-Jia theorem.

Then, still using double hyperspace techniques, I will show that the result of last time also holds if we assume X to be consonant (not ⊙-consonant, not second-countable).

Both are improvements over the result of last time. Hence, for a space X that is either consonant or second-countable, it is equivalent for OσX to be locally compact, for ℒ(X)σ to be locally compact, or for X to be core-compact.

Some of the additional tricks we will need today consist in characterizing the cases where the Hoare hyperspace HX of a space X is locally compact, resp. a c-space. We start with the former without any further ado.

When is the Hoare hyperspace of a space locally compact?

For every space X, HX is the space of all closed subsets of X, with the lower Vietoris topology: its subbasic open subsets are of the form ♢U ≝ {C ∈ HX | C intersects U}, where U ranges over the open subsets of X.  HX is a T0 space, whose specialization ordering is ordinary inclusion, called the Hoare hyperspace of X.

A space X is core-compact if and only if OX is a continuous dcpo (Definition 5.2.3 in the book). Equivalently, X is core-compact if and only if for every point x of X, for every open neighborhood U of x in X, there is an open set V such that x ∈ V ⋐ U, where ⋐ is the way-below relation on OX (Proposition 5.2.6 in the book).

Lemma A. For every core-compact space X, HX is locally compact, and even stably compact.

Proof. We have already seen this near the end of the March 2015 post, where we showed this by proving that HX is even the de Groot dual of the space of open subsets of X with the lower topology. You can also prove that HX is locally compact by hand: first realize that you can replace X by its sobrification, namely that HSX is homeomorphic to HX; then you can assume that X is locally compact, and therefore you can use Lemma C and Lemma D from last time (exercise!). ☐

Lemma B. For every topological space X, if HX is locally compact then X is core-compact.

Proof. The function η : X → HX that maps every point x to ↓x is continuous, since the inverse image of any subbasic open set ♢U under η is equal to U. We could say a lot more about η: it is the unit of the H monad, it is a topological embedding if X is T0, and so on, but we will not need any of that.

Let x be an arbitrary point of X, and U be an open neighborhood of x in X. Then the closed set η(x)=↓x is in ♢U. Since HX is locally compact, there is a compact saturated subset Q of HX such that ↓x is in the interior V of Q (in HX), and Q is included in ♢U. Let V ≝ η–1(V). Since η is continuous, V is open in X, and since η(x)=↓x is in V, x is in V.

We claim that V ⋐ U. Let (Ui)i ∈ I be a directed family of open subsets of X whose union contains U. The ♢ operation commutes with arbitrary unions, and is monotonic, so ♢U ⊆ ∪i ∈ I ♢Ui; also, the latter union is directed. Since Q is included in ♢U and is compact, Q is included in ♢Ui for some i ∈ I. Then V = η–1(V) ⊆ η–1(Q) ⊆ η–1(♢Ui) = Ui. ☐

The case where X is second-countable

Let us embark on showing that ℒ(X)σ is locally compact if and only if X is core-compact, for every second-countable space X (no ⊙-consonance, no consonance involved).

Let us write OpX for OX with the pointwise topology (instead of the Scott topology for OσX = O(X)σ), whose basic open subsets are [x ∈] ≝ {U ∈ OX | x ∈ U}, for each point x of X. The Scott topology is always finer than the pointwise topology.

We will need to use quite a few results that we have seen in the past. For the most part, I will recall them during the proof. We will also need the notion of a c-space: that is a space in which for every point x, every open neighborhood U of x contains int(↑y) for some point y of U. The c-spaces are exactly the spaces whose sobrification are continuous dcpos, with their Scott topology (Corollary 8.3.44 in the book); in particular, the sober c-spaces are exactly the continuous dcpos with their Scott topology.

Proposition C. For every second-countable space X, the following are equivalent:

  1. X is core-compact;
  2. O(X)σ is core-compact (equivalently, locally compact, equivalently stably compact);
  3. ℒ(X)σ is core-compact (equivalently, locally compact, equivalently stably compact).

Proof. The fact that O(X)σ (resp. ℒ(X)σ) is core-compact if and only if it is locally compact if and only if it is stably compact was proved near the beginning of the post of last time. Similarly for the implications 3 ⇒ 2 and 1 ⇒ 3. It remains to show 2 ⇒ 1.

Just as last time, we use the following result from the October 2022 post, which comes from [1], and applies to any topological space X at all. I will give all missing definitions shortly.

Proposition. φX and ψX form a pair of mutually inverse continuous maps between QHX and OpOσX. We have φX–1([U ∈]) = ☐♢U, and ψX–1(☐♢U) = [U ∈] for every open subset U of X.

For every space Y, QY is the space of all compact saturated subsets of Y, with the upper Vietoris topology: its basic open subsets are of the form ☐U ≝ {Q ∈ QY | Q ⊆ U}, where U ranges over the open subsets of Y.  QY is a T0 space, whose specialization ordering is reverse inclusion, called the Smyth hyperspace of Y. The function φX maps every Q ∈ Q(H(X)) to the collection of open subsets U of X that intersect every C ∈ Q; ψX maps every U ∈ OpOσX to the intersection of the sets ♢U, where U ranges over U. We will just use the fact that there is a homeomorphism between QHX and OpOσX, and we will not care about the precise definitions of φX and ψX.

Since X is second-countable, so is HX. Indeed, given a countable base B of the topology of X, every open subset U of X can be written as a union of elements Bi of B, and then ♢U is the corresponding unions of elements ♢Bi. In other words, the open sets ♢B with B in B form a (countable) subbase of the topology of HX.

By a result of Schalk that we have presented in the March 2015 post (look for the very first Proposition there), HX is sober. In particular, it is well-filtered (Proposition 8.3.5 in the book). In the December 2018 post (look for “Proposition (de Brecht, Kawai)”), we have seen that for every well-filtered, second-countable space Y, the upper Vietoris and Scott topologies coincide on QY. The result is due to de Brecht and Kawai [1]—you had guessed it. Hence the (upper Vietoris) topology QHX is in fact the Scott topology of its specialization ordering.

As a parenthesis, it follows that the homeomorphic space OpOσX has the same property: its (pointwise) topology is the Scott topology of its specialization ordering. The latter is inclusion, so OpOσX = OσOσX. We obtain that φX and ψX form a pair of mutually inverse continuous maps between QHX and OσOσX (not OpOσX), when X is second-countable, and that is Proposition 6.11 of [1].

Let us return to our task. We assume that OσX is core-compact. Then O(OσX) is a continuous dcpo: that is the definition of core-compactness (Definition 5.2.3 in the book). Therefore QHX is a continuous dcpo, too. More precisely, it is a continuous dcpo when equipped with its specialization ordering, which is reverse inclusion. In particular, QHX is a c-space—in its Scott topology, hence in its upper Vietoris topology, since they are the same.

This allows us to use the Lyu-Jia Theorem (in the January 2022 post), which states, among other things, that for every space Y, QY (with its upper Vietoris topology) is a c-space if and only if Y is locally compact. Hence, under the assumption that OσX is core-compact, we now know that HX is locally compact. By Lemma B, X is core-compact, and this finishes to prove 2 ⇒ 1. ☐

Baire space NN is second-countable and not core-compact, so we retrieve the result (seen as Corollary K last time) that O(NN)σ, ℒ(NN)σ are not core-compact. The space Q of rational numbers is second-countable, but not consonant, so our results of last time did not apply. But Proposition C applies, and Q is not core-compact (Example 5.2.14 in the book), so we obtain the following.

Corollary D.  O(Q)σ, ℒ(Q)σ are not core-compact.

We will now prove that the conclusion of Proposition C also holds if instead of assuming X second-countable, we assume it consonant. This is what M. de Brecht drew my attention to.

When is the Hoare powerdomain of a space a continuous dcpo?

We have defined HY as a topological space, with its lower Vietoris topology. Its specialization ordering is inclusion. In the following lemma, we look at HY as a poset, with the inclusion ordering.

Lemma E. For every topological space Y, HY is a continuous dcpo (with respect to the inclusion ordering) if and only if Y is a c-space.

Proof. If Y is a c-space, then HY is a continuous dcpo, and its topology is the Scott topology of inclusion. This is part of the final Proposition in the March 2015 post.

Conversely, we assume that (the poset underlying) HY is a continuous dcpo, and we claim that Y is a c-space. The way-below relation ≪ on HY is defined as usual by C ≪ C’ if and only if every directed family of closed sets whose supremum is above C’ contains an element that is above C. The way-way-below relation ⋘ is defined in the same way, without the “directed” part: C ⋘ C’ if and only if every (arbitrary) family of closed sets whose supremum is above C’ contains an element that is above C.

We note that: (i) for every y ∈ Y, for every C ∈ HY, ↓y ⋘ C if and only if ↓y ≪ C. It is clear that ↓y ⋘ C implies if ↓y ≪ C: every directed family of closed sets whose supremum is above C is in particular a family of closed sets whose supremum is above C. Conversely, let us assume ↓y ≪ C. Let (Ci)i ∈ I be a family of closed subsets of Y whose supremum C” is above C. We write C” as the supremum of the directed family of sets CJ ≝ ∪i ∈ J Ci, where J ranges over the finite subsets of I. Since ↓y ≪ C, ↓y is included in CJ for some finite subset J of I. Then y is in CJ, hence in Ci for some i ∈ J. Since Ci is downwards-closed, ↓y ⊆ Ci.

Then: (ii) for every C ∈ HY, C is both the supremum and the union of the family of closed sets ↓y, where y ranges over C. It is the union of those sets because C is downwards-closed. Since that union is equal to C, it is closed, hence is automatically the supremum of the same family.

A complete lattice is prime-continuous if and only if every element is the supremum of the elements way-way-below it. (The definition of the way-way-below relation makes sense in any complete lattice, not just HY.) By Raney’s Theorem (Exercise 8.3.16 in the book), a complete lattice L is completely distributive if and only if it is prime-continuous—but we will not use this.

We recall that we are now assuming that (the poset underlying) HY is a continuous dcpo. Given any C ∈ HY, C is the supremum of a directed family (Ci)i ∈ I of closed subsets of Y each way-below C, by definition of continuous dcpos. By (ii), each Ci is the supremum of the closed sets ↓y where y ranges over Ci. For every y ∈ Ci, we have ↓y ⊆ Ci ≪ C, so ↓y ≪ C, and therefore ↓y ⋘ C, by (i). Hence C is the supremum of the family of closed sets ↓y, where y ranges over ∪i ∈ I Ci, and each one is way-way-below C. Therefore HY is prime-continuous.

Now a complete lattice L is prime-continuous if and only if its opposite lattice Lop is prime-continuous (Exercise 8.3.18 in the book). Hence the opposite lattice of HY is also prime-continuous. But HY is order-isomorphic to the opposite of the lattice OY, through the function that sends every closed set to its complement. Hence OY is prime-continuous.

Lemma 8.3.42 of the book states that the spaces Y such that OY is prime-continuous are exactly the c-spaces (and that the sober spaces Y such that OY is prime-continuous are the continuous dcpos with their Scott topology), and this allows us to conclude. ☐

Consonant spaces

Let me write ■Q, for every compact saturated subset Q of a space X, for the set of open neighborhoods of Q. A space X is consonant if and only every Scott-open subset of OX is a union of sets of the form ■Q, where each Q is compact saturated in X.

We have already recalled that there is a pair of mutually inverse continuous maps φX and ψX between QHX and OpOσX. Another result from [1], cited in the same October 2022 post, is the following.

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

  1. X is consonant;
  2. σX is surjective;
  3. σX is a homeomorphism of HQX onto QHX, with inverse τX.

In that theorem, σX maps every C ∈ HQX to the collection of closed subsets C of X that intersect every Q ∈ C, and τX maps every Q ∈ QHX to the collection of compact saturated subsets Q of X that intersect every C ∈ Q.

Now, if X is any consonant space (not necessarily ⊙-consonant or second-countable), then φX o σX is a homeomorphism of HQX onto OpOσX, whose inverse is τX o ψX. In particular, φX o σX defines an order-isomorphism of HQX (ordered by inclusion) onto O(OσX) (also ordered by inclusion).

If additionally OσX is core-compact, then by definition O(OσX) is a continuous dcpo, so HQX is also a continuous dcpo with the inclusion ordering. We have seen in Lemma E that this entails that QX is a c-space. We now use the Lyu-Jia Theorem (already mentioned above, or see the January 2022 post): QX is a c-space if and only if X is locally compact.

Hence we obtain the following strengthening of Proposition J of last time, and a variant of Proposition C above.

Proposition E. For every consonant space X, the following are equivalent:

  1. X is core-compact;
  2. O(X)σ is core-compact (equivalently, locally compact, equivalently stably compact);
  3. ℒ(X)σ is core-compact (equivalently, locally compact, equivalently stably compact);
  4. X is locally compact.

Proof. We have just proved that 2 implies 4, and this was the difficult direction. 4 implies 1 (Theorem 5.2.9 in the book), 1 implies 3 (as recalled near the beginning of the post of last time), and 3 implies 2 (that was one of the first corollaries in the post of last time). ☐

The improvement over Proposition J of last time is three-fold. First, we do not require X to be ⊙-consonant, namely that all finite coproducts of X with itself be consonant; consonance is a strictly weaker property. Second, we do not require X to be second-countable. Third, we have a new equivalent item 4. In particular, we obtain the following corollary. The result is due to Yu Chen, Hui Kou and Zhenchao Lyu . (I thank Xiaodong Jia for pointing out this reference me. I have already cited their paper here.)

Corollary F [3, Proposition 5.4]. For a consonant space, it is equivalent to be core-compact or locally compact.

A similar property was known for sober spaces (Theorem 8.3.10 and Theorem 5.2.9 in the book). Corollary F has a short proof on its own, given by Chen, Kou and Lyu. I cannot resist reproducing it here. Let X be a consonant, core-compact space, and let x be a point of X and U an open neighborhood of x in X. Since X is core-compact, there is an open neighborhood V of x such that V ⋐ U. We use interpolation to find an open set V0 such that V ⋐ V0 ⋐ U, then again to find an open set V1 such that V ⋐ V1 ⋐ V0, etc. Hence V ⋐ … ⋐ Vn ⋐ … ⋐ V1 ⋐ V0. The family F ≝ {W open in X | Vn ⋐ W for some n ∈ N} is Scott-open (it is a filter of open sets, too, which follows from the fact that F is also the collection of open subsets of X that contain some Vn, but we will not need that). F contains U, too, so by consonance there is a compact saturated subset Q of X such that Q ⊆ U and such that every open neighborhood W of Q belongs to F; in particular, every open neighborhood of Q contains some Vn. Hence Q, which is saturated hence equal to the intersection of its open neighborhoods, contains the intersection of the open sets Vn, which itself contains V. Therefore x ∈ V ⊆ Q ⊆ U. ☐

Let me summarize what we know. The case of posets with the Scott topology is Theorem B of last time. The cases of second-countable and of consonant spaces are from Propositions C and E, respectively.

Theorem G. Let X be a topological space. If X is:

  1. a poset with its Scott topology,
  2. or a second-countable topological space,
  3. or a consonant space,

then the following are equivalent:

  1. X is core-compact;
  2. O(X)σ is core-compact (equivalently, locally compact, equivalently stably compact);
  3. ℒ(X)σ is core-compact (equivalently, locally compact, equivalently stably compact).

As applications of Theorem G, we now know that O(NN)σ, ℒ(NN)σ, O(Q)σ, ℒ(Q)σ are not core-compact. The case of the Sorgenfrey line Rℓ still resists! Indeed, Rℓ is not a poset with its Scott topology, and it is neither second-countable nor consonant.

  1. Matthew de Brecht and Tatsuji Kawai. On the commutativity of the powerspace constructions. Logical Methods in Computer Science, 15(3), 2019.
  2. Matthew de Brecht, Matthias Schröder, and Victor Selivanov. Base-Complexity Classifications of Qcb0-Spaces. Computability 1, 2012, pages 1–26.
  3. Yu Chen, Hui Kou, and Zhenchao Lyu. Two topologies on the lattice of Scott closed subsets. Topology and its Applications, 306, 107918, 2022.
jgl-2011

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