Stone duality for preordered topological spaces I. Bonsangue-Jacobs-Kok duality for preorders

Stone duality is about an adjunction O ⊣ pt between the category Top of topological spaces and continuous maps on the one hand, and the opposite (namely, with arrows reversed) category Loc of the category Frm of frames and frame homomorphisms. The functor O maps every topological space to its frame of open sets. See Section 8.1 in the book for an introduction to the subject.

Nesta van der Schaaf has spent the year in my lab, although in a different team, concerned with quantum programming languages, quantum information, and quantum circuit description languages. We have met once or twice, and discussed the problem of obtaining a general notion of locales with a notion that would mirror a notion of partial order put on top of a topological space. In other words, is there a Stone duality for (pre)ordered topological spaces, namely just topological spaces with a preordering? He has a proposal in his PhD thesis [1], but it has a few problems. Nesta explained to me what they are, but I have forgotten about them (sorry).

We are not requiring the preordering and the topology to be related in any way. For example, we are not considering pospaces, which would be topological spaces X with a partial ordering ≤ that is compatible with the topology in the sense that the graph (≤) of the partial ordering is closed in X × X. We are not even considering semiclosed partial orderings, as defined in the March 2025 post, a weaker notion, although I will come back to those at some point (but not today).

There is a simple way of finding such a theory of Stone duality, which I will start describing in today’s post. Here it is. We know that a preordering is the same thing as an Alexandroff-discrete topology, namely a topology in which arbitrary intersections of open sets are open (see below). Then a preordered topological space will simply be a bitopological space, where one of the topologies is Alexandroff-discrete. Finally, in order to find the desired duals, we rely on Jung-Moshier duality for bitopological spaces. That’s it.

Well, I am somehow lying: the resulting constructions are pretty complex. Jung-Moshier duality, even in the form given by T. Jakl in his PhD thesis, is complex, and this will be even more complex. I don’t know yet whether I will describe these constructions in full: we will see next month. For this month, I will concentrate on describing a suitable duality for Alexandroff-discrete spaces. This is due to Bonsangue, Jacobs and Kok [2, Section 6.4].

Alexandroff-discrete spaces, preorders, and why the usual form of Stone duality is not exactly what we need here

Given any set X with a preordering ≤, we can topologize X by defining the open sets as being the upwards-closed subsets of X with respect to ≤; this is the Alexandroff topology. The topology we obtain is Alexandroff-discrete: every intersection of open sets (upwards-closed sets) is upwards-closed, hence open. Conversely, given any Alexandroff-discrete space X, we obtain a preordering ≤ by looking at the specialization preordering of X, and the two constructions are inverse of each other.

Now we can reproduce the O ⊣ pt duality and restrict it to Alexandroff-discrete spaces, and this will automatically define a duality between preordered sets and (the opposite category of) certain frames. For every Alexandroff-discrete space X, OX is not just a frame, but a completely distributive complete lattice.

This can be seen as follows. We use Raney’s theorem (Exercise 8.3.16 in the book), which says that the completely distributive complete lattices are exactly the prime-continuous complete lattices. A complete lattice L is prime-continuous if and only if every element of L is the supremum (not the directed supremum) of the family of elements way-way-below it in L. An element u of L is way-way-below an element v of L, in notation u ⋘ v, if and only if every (not necessarily directed) family D whose supremum lies above v contains an element above u. Now, given a preordered set X, seen as an Alexandroff-discrete topological space, and letting L ≝ OX:

• U ⋘ V, where U, V ∈ OX, if and only if there is a point x of V such that U ⊆ ↑x. The if direction is easy: if U ⊆ ↑x and x ∈ V, there for every family of upwards-closed subsets O_i of X whose supremum (=union) contains V, this union will contain x, hence x will be in some O_i, and therefore U ⊆ O_i; so U ⋘ V. Conversely, if U ⋘ V, then we consider the family D of all (open) sets of the form ↑x with x ∈ V; the union of all elements of D contains V, so by definition of ⋘, U is included in one of them.
• Every element V ∈ OX can be written as the union of the sets ↑x with x ∈ V. We have just seen that ↑x ⋘ V for every such x. Hence OX is prime-continuous.

So far so good. However, the pt part of the adjunction does not work so well. Indeed, for every completely distributive complete lattice L, pt L is not in general Alexandroff-discrete. The best we can say is that is a continuous dcpo with its Scott topology (Theorem 8.3.43 in the book).

What went wrong? Well, in fact OX is a completely distributive complete lattice not just when X is Alexandroff-discrete, but even when X is a continuous dcpo with its Scott topology, and even more generally, when X is a c-space (Lemma 8.3.41 in the book). So maybe we should find some further properties of OX when X is Alexandroff-discrete? We can find some. For example, we will see that OX is algebraic (Lemma G, near the end of this post), but algebraicity of L is still not enough to ensure that pt L is Alexandroff-discrete.

The solution was worked out by Bonsangue, Jacobs and Kok [2, Section 6.4], and is pretty simple. The issue is not with the properties of OX, but with what we allow as morphisms on the localic side.

Instead of considering the category Frm of frames and frame homomorphisms, or its full subcategory of completely distributive complete lattice (and frame homomorphisms), they consider the category CDCL of completely distributive complete lattices and complete lattice homomorphisms; namely, functions that preserve all suprema (as with frame homomorphisms) and all infima (not just finite infima, as frame homomorphisms do).

Bonsangue-Jacobs-Kok duality: O^a and pt^a

We already have one half of the Bonsangue-Jacobs-Kok duality: there is a functor O^a from Pre, the category of preordered sets and monotonic maps (equivalently, of Alexandroff-discrete spaces and continuous maps), to CDCL^op. For every preordered set X, O^aX is the completely distributive complete lattice of upwards-closed subsets of X (open subsets in the Alexandroff topology). For every monotonic map f : X → Y between preordered sets, O^af is the sup-and-inf-preserving function that maps every upwards-closed subset V of Y to f^–1(V). We write O^a instead of O to stress the fact that we are taking the Alexandroff-open subsets.

How about the reverse direction? We imitate the pt functor of usual Stone duality: for every frame L, pt L is the set of points of L, defined as the completely prime filters of L, with some topology. The points of a frame L are exactly the characteristic functions of frame homomorphisms from L to the two-element lattice {0, 1} (with 0<1) (see the June 2025 post, end of the ‘Stone duality in just a few seconds’ section): a completely prime filter of L has a characteristic function, seen as a map to {0, 1}, that is a frame homomorphism, and conversely every frame homomorphism φ from L to {0, 1} is the characteristic funtion of φ^–1({1}), which is a completely prime filter (exercise!). Also, {0, 1} is simply O1, where 1 is any one-element space, equating 0 with the empty set and 1 with 1.

Here, O^a1 can also be equated with {0, 1}. Given a completely distributive complete lattice L, we define our modified notion of points of L as the sets φ^–1({1}), where φ ranges over the morphisms from L to {0, 1} in the category CDCL. Then φ preserves arbitrary suprema and arbitrary, not just finite, infima.

• Since φ is monotonic, φ^–1({1}) is upwards-closed.
• Since φ preserves arbitrary infima, φ^–1({1}) is closed under arbitrary infima. Hence it is a complete filter. A filter, by comparison, is only closed under finite infima (we usually say is is closed under binary infima and contains the top element ⊤ of L, and this is equivalent).
• Since φ preserves arbitrary infima, φ^–1({1}) is closed under arbitrary suprema. Hence given any family (u_i)_i ∈ I of elements of L whose supremum u is in φ^–1({1}), some u_i must already be in φ^–1({1}): otherwise φ(u_i)=0 for every i ∈ I, and therefore φ(u)=0, by preservation of suprema. In other words, φ^–1({1}) is a completely prime complete filter.

Conversely, every completely prime complete filter x in L is of the form φ^–1({1}), where φ is the characteristic function of x. I will let you prove it as an exercise.

For every completely distributive complete lattice L, we therefore define pt^a L as the set of completely prime complete filters x of L (not just completely prime filters as with pt L).

We need to put a preordering on pt^a L. In order to do so, we will imitate the definition of the topology on pt L, and we will put a similarly defined topology on pt^a L; then we will observe that this topology is Alexandroff-discrete.

On pt L, there is one open subset for each element u of L exactly. Similarly, given a completely distributive complete lattice L, for every u ∈ L, we define O_u as {x ∈ pt^a L | u ∈ x}, and we declare that the sets O_u are the open subsets of pt^a L.

Lemma A. For every completely distributive complete lattice L, the sets O_u, u ∈ L, are the open sets of an Alexandroff-discrete topology. This topology is the Alexandroff topology of inclusion ⊆. The map u ∈ L ↦ O_u preserves arbitrary suprema and arbitrary infima.

Proof. We start by the last of those statements.

• For every family (u_i)_i ∈ I of elements of L, letting u ≝ ∨_i ∈ I u_i, we have O_u = ∪_i ∈ I O_ui. Indeed, for every completely prime complete filter x on L, x ∈ O_u if and only if u ∈ x, if and only if some u_i is in x (since x is completely prime), if and only if x is in some O_ui.
  In particular, any union of open sets O_u is open, namely, also of this form.
• For every family (u_i)_i ∈ I of elements of L, letting u ≝ ⋀_i ∈ I u_i, we have O_u = ∩_i ∈ I O_ui. Indeed, for every completely prime complete filter x on L, x ∈ O_u if and only if u ∈ x, if and only if every u_i is in x (since x is a complete filter), if and only if x is in every O_ui.
  In particular, any intersection of open sets O_u is open, namely, also of this form.

It immediately follows that the sets O_u form an Alexandroff-discrete topology.

What is the specialization preordering of this topology on pt^a L? Writing it as ≤, we have x ≤ y if and only if every open neighborhood O_u of x contains y, by definition. In other words, x ≤ y if and only if for every u ∈ L, u ∈ x implies u ∈ y. The specialization preordering is simply inclusion ⊆, and therefore our fancy topology on pt^a L is simply the Alexandroff topology of inclusion. ☐

Then pt^a defines a functor from CDCL^op to Pre, whose action on morphisms φ : L → L’ (namely, complete lattice homomorphisms φ from L’ to L) is the function pt^a φ : pt^a L → pt^a L’ that maps every completely prime complete filter x on L to φ^–1(x). I will let you check that φ^–1(x) is also a completely prime complete filter. The function pt^a φ is monotonic because x ⊆ y implies φ^–1(x) ⊆ φ^–1(y).

The adjunction O^a ⊣ pt^a

Proving that there is an adjunction O^a ⊣ pt^a then works exactly as with O ⊣ pt.

The unit η^a_X : X → pt^a O^a X maps every x ∈ X to the completely prime complete filter of upwards-closed subsets of X that contain x. This is not just a (completely prime) filter, but a complete filter: every intersection of upwards-closed subsets of X that contain x is upwards-closed and contains x. It is also clear that η^a_X is monotonic: if x ≤ y, then every upwards-closed subset of X that contains x also contains y.

We now use the characterization of adjunctions F ⊣ U of Diagram (5.2), Section 5.5.1 of the book. Given a functor F : C → D, and an assignment of an object UL for every object L of D, plus a morphism η_X : X → UFX for every object X of C, F is left adjoint to U with unit η if and only if for every morphism f : X → UL in C, there is a unique morphism f^! : FX → L in D such that Uf^! o η_X = f. We consider the case where C=Pre, D=CDCL^op, F=O^a, U=pt^a, and η_X=η^a_X. Please pay attention that morphisms f^! : FX → L in D are morphisms f^!: O^aX → L in CDCL^op, namely complete lattice homomorphisms f^! : L → O^aX: be careful about arrow reversals.

Proposition B. For every monotonic map f : X → pt^a L, where X is a preordered set and L is a completely distributive complete lattice, there is a unique complete lattice homomorphism f^! : L → O^aX such that pt^a f^! o η^a_X = f.

Proof. The condition pt^a f^! o η^a_X = f means that for every x ∈ X, f^!–1(η^a_X(x))=f(x), namely that for every u ∈ L, u ∈ f(x) if and only if f^!(u) ∈ η^a_X(x), and the latter is equivalent to x ∈ f^!(u). In particular, if f^! exists, then for every u ∈ L, f^!(u) must be {x ∈ X | u ∈ f(x)}. This shows that f^! is unique if it exists.

Additionally, u ∈ f(x) is equivalent to f(x) ∈ O_u, so f^!(u) must be equal to f^–1(O_u). This leads us to define f^!(u) ≝ f^–1(O_u) for every u ∈ L. Now by Lemma A, u ∈ L ↦ O_u is a complete lattice homomorphism, and so is f^–1, so f^! is a complete lattice homomorphism, too. ☐

So, here we are:

Theorem C. There is an adjunction O^a ⊣ pt^a between the category Pre of preordered sets and monotonic maps and the category CDCL of completely distributive complete lattices and complete lattice homomorphisms. Its unit η^a_X : X → pt^a O^a X maps every x ∈ X to the completely prime complete filter of upwards-closed subsets of X that contain x.

This is Bonsangue-Jacobs-Kok duality [2].

Sobriety for preordered sets?

The O ⊣ pt adjunction is idempotent, and so is the adjunction between bitopological spaces and d-frames that we have seen in the June 2025 post. An adjunction F ⊣ U such that the unit at every object of the form U(-) is an isomorphism is called an idempotent adjunction. In the case of O ⊣ pt, this is very much related to the fact that the sobrification SX of a space X, which up to natural homeomorphism is pt OX, is sober, namely, that the unit at pt OX is a homeomorphism.

Hence we may call a-sober any preordered set X such that η^a_X is an isomorphism of preordered spaces. We will see that a-sober sets are a familiar notion. But first, let us make a simplification of the notion of completely prime complete filter.

Lemma D. For every preordered set X, every completely prime complete filter of upwards-closed subsets of X is the collection of upwards-closed subsets of X that contain x, for some x ∈ X. In other words, η^a_X is surjective.

Proof. Let F be a completely prime complete filter of upwards-closed subsets of X. Since it is a complete filter, the intersection U ≝ ∩F of all the elements of F is in F, and then U is the least element of F. In particular, F is the collection ↑U of all the upwards-closed subsets of X that contain U.

Now U is upwards-closed, hence is equal to the union of the sets ↑x where x ranges over U. Since F is completely prime, one of those is in F: ↑x is in F for some x ∈ U. Since ↑x ∈ F, U ⊆ ↑x, and since x ∈ U and U is upwards-closed, ↑x ⊆ U; so U = ↑x. Hence the upwards-closed sets in F are exactly those that contain x. ☐

This is rather different from the situation with the O ⊣ pt adjunction, whose unit is surjective at a topological space X if and only X is quasi-sober, namely if and only if every irreducible closed subset of X is the closure of some (not necessarily unique) point. Here, the unit is always surjective.

Lemma E. For every preordered set X, η^a_X is an isomorphism of preordered spaces if and only if it is injective, if and only if X is a partial order.

Proof. Since η^a_X is a morphism in Pre, it is monotonic. It reflects preorderings, in the sense that η^a_X(x) ⊆ η^a_X(y) implies x≤y: if η^a_X(x) ⊆ η^a_X(y), then every upwards-closed subset of X that contains x contains y, in particular ↑x, so y ∈ ↑x. Then, η^a_X(x) ⊆ η^a_X(y) is equivalent to x≤y, for all x, y ∈ X.

This entails that η^a_X(x) = η^a_X(y) is equivalent to x≤y and y≤x, so η^a_X is injective if and only if X is a partial order. It is surjective by Lemma D, hence an isomorphism if and only if X is a partial order. ☐

Hence there is no need to invent a new term such as “a-sober”: partial order is what we need.

We have seen that pt^a L is not just preordered, but ordered by inclusion (Lemma A), for every completely distributive complete lattice L. Hence pt^a L is a-sober… oops, a partial order, and therefore:

Proposition F. The O^a ⊣ pt^a adjunction is idempotent.

Sobrification for preordered sets?

The sobrification SX of a topological space is a simplifed description of pt OX. Similarly, we can simplify the description of pt^a O^a X for every preordered set X, up to natural order-isomorphism. It should come as no surprise that we get the order quotient of X this way, namely, the quotient of X under the equivalence relation ≡ defined by x ≡ y if and only if x ≤ y and y ≤ x.

Indeed, given two completely prime complete filters in pt^a O^a X, which we can write as η^a_X(x) and η^a_X(y) thanks to Lemma D, η^a_X(x) = η^a_X(y) precisely when x ≡ y. Let us check this: η^a_X(x) = η^a_X(y) if and only if x and y belong to the same upwards-closed subsets of X. This is clearly the case if x ≡ y, and conversely, if x and y belong to the same upwards-closed subsets of X then x ∈ ↑y and y ∈ ↑x, so x ≡ y.

There is no need to invent a concept such as “a-sobrification”, then. The “a-sobrification” of a preordered set X is its order quotient.

Spatiality for completely distributive complete lattices?

Symmetrically, with the O ⊣ pt adjunction comes the notion of spatial frames: a frame L is spatial if and only if the counit at L is a frame isomorphism.

In our case, the counit ε_L^a of the O^a ⊣ pt^a adjunction is obtained as follows. There is an identity morphism id_{pt^a L} from pt^a L to itself in CDCL. By Proposition B, there is a unique complete lattice homomorphism id_{pt^a L}^! : L → O^a pt^a L such that pt^a id_{pt^a L}^! o η^a_{pt^a L} = id_{pt^a L}. This must be ε_L^a, since among the two equations that the counit and unit must obey, we find the equation pt^a ε_L^a o η^a_{pt^a L} = id_{pt^a L} (see Section 5.5.2 in the book).

The way f^! is built, for every monotonic map f : X → pt^a L, where X is a preordered set and L is a completely distributive complete lattice, is by defining f^!(u) as f^–1(O_u) for every u ∈ L. This is what we did in the proof of Proposition B. We take X ≝ pt^a L and f ≝ id_{pt^a L}, and we obtain that ε_L^a(u) must be equal to O_u for every u ∈ L.

If ε_L^a is an isomorphism, then L must be an algebraic completely distributive complete lattice, because of the following.

Lemma G. For every preordered set X, O^aX is an algebraic dcpo, whose finite (in the sense of domain theory, namely those that are way-below themselves) elements are the sets ↑E, where is a finite (in the ordinary sense) subset of X.

Proof. Given a finite subset E of X, ↑E is finite, namely way-below itself in O^aX: given any directed family (U_i)_i ∈ I of upwards-closed subsets of X whose union contains E, and writing E as {x₁, …, x_n}, some U_i1 must contain x₁, …, some U_in must contain x_n, and by directedness there must be some i ∈ I such that U_i1, …, U_in ⊆ U_i; whence ↑E ⊆ U_i.

Conversely, let U be an upwards-closed subset of X that is finite in the sense of being way-below itself. The family of all the sets of the form ↑E with E finite (in the ordinary sense) and included in U is directed and its supremum is equal to U, so one of those sets ↑E contains U, by definition of the way-below relation. Since ↑E ⊆ U by assumption, ↑E = U.

Finally, and repeating that every U ∈ O^aX is the union (supremum) of the sets of the form ↑E with E finite and included in U, we see that U is the supremum of a family of finite elements below U. Therefore O^aX is algebraic. ☐

Hence, assuming that we are tempted to call a-spatial those completely distributive complete lattices L such that ε_L^a is an isomorphism, the a-spatial completely distributive complete lattices are simply the ones that are algebraic. This is what Proposition I below states.

We need a final auxiliary result. By Lemma G, the finite elements of O^aX are the sets ↑E with E finite. What are the sets ↑x, with just a single point x? There are the finite join-primes. A join-prime element of L is an element p of L, distinct from the least element ⊥, and such that for all u, v ∈ L such that p ≤ u ⋁ v, p ≤ u or p ≤ v already; equivalently, if p is less than or equal to a supremum of finitely many elements (possibly zero), then p is less than or equal to one of those finitely many elements.

Lemma H. In an algebraic completely distributive complete lattice L, every element u of L is the supremum of the finite join-primes below u.

Proof. Let us cheat. The official way of proving this resembles the techniques used in Exercise 8.3.47 of the book and in the surrounding sections, but why imitate proofs when you can use the theorems? Since L is a completely distributive complete lattice, it is the lattice of Scott-open subsets OX of a continuous dcpo X (Theorem 8.3.43 in the book; this is up to isomorphism, and I will simply omit any explicit mention of the isomorphism). By Exercise 8.3.22 in the book, and since L is an algebraic distributive complete lattice, X must be strongly locally compact in its Scott topology, meaning that the Scott topology must have a base of compact-open subsets, namely of subsets that are both compact and open.

Now let us consider a compact-open subset KU of X with the Scott topology.

• Since X is a continuous dcpo and KU is Scott-open, KU is equal to the union of the sets ↟x where x ranges over the points of KU, and each set ↟x is Scott-open (Theorem 5.1.17 in the book).
• Since KU is also compact, there is a finite subset E of such points x in KU such that KU = ∪_x ∈ E ↟x. Let us take E of smallest possible cardinality. In particular, we cannot have x ≤ y for two distinct elements x and y of E: otherwise ↟y would be included in ↟x and we could therefore remove y from E. Hence E is a finite antichain—a finite set whose elements are pairwise incomparable.
• For every x ∈ E, x is in KU hence in ↟y for some y ∈ E. But then y ≤ x, and we have just argued that this was impossible if x and y are distinct; so y=x, and therefore x ∈ ↟x.
• We have just proved that every element of E is finite (x ≪ x). Hence ↟x = ↑x for every x ∈ E, and therefore KU = ↑E.

Every Scott open subset is a union of such compact-open sets KU = ↑E, where E consists of finite elements of X, as we have just shown. Therefore every Scott-open subset U is a union (a supremum) of open sets of the form ↑x with x finite. Every such set ↑x is a join-prime, and it remains to see that it is a finite element of L = OX: for every directed family (U_i)_i ∈ I of Scott-open subsets whose union contains ↑x, one of them contains x hence the set ↑x, so ↑x is finite indeed. (This proof also shows that X is an algebraic dcpo; I will let you finish the argument!) ☐

Proposition I. For every completely distributive complete lattice L, ε_L^a is a surjective, complete lattice homomorphism in CDCL. It is a bijection, equivalently an isomorphism in CDCL, if and only if L is algebraic.

Proof. We know that ε_L^a is a morphism in CDCL. Every element of O^a pt^a L is an upwards-closed subset of pt^a L, namely an open set in the Alexandroff topology of pt^a L, and all of them are of the form O_u for some u ∈ L, by Lemma A. Hence ε_L^a is surjective.

By Lemma G, if ε_L^a is an isomorphism, then L must be algebraic. Conversely, let us assume that L is algebraic. We show that ε_L^a is injective: this is all that remains to be proved.

For every finite join-prime p of L, we observe that ↑p is a completely prime complete filter of elements of L. (The converse holds, too, but we will not need it.) That it is a complete filter is clear: if (u_i)_i ∈ I is any family of elements of ↑p, then p ≤ u_i for each i ∈ I, so p ≤ ∧_i ∈ I u_i and therefore ∧_i ∈ I u_i ∈ ↑p. Let us show that it is completely prime. Given any family (u_i)_i ∈ I of elements whose supremum u is in ↑p, we can write this supremum as the supremum of the directed family (u_J)_{J finite ⊆ I}, where u_J ≝ ⋁_j ∈ J u_j for every finite subset J of I. Since p ≤ u and p is finite, there must be a finite subset J of I such that p ≤ u_J. Since p is join-prime, there must be a j ∈ J such that p ≤ u_j. Therefore u_j is in ↑p, showing that ↑p is completely prime.

For all u, v ∈ L, if O_u=O_v then the completely prime complete filters that contain u are exactly the completely prime complete filters that contain v. Hence, for every finite join-prime p, u ∈ ↑p if and only if v ∈ ↑p; equivalently, p ≤ u if and only if p ≤ v. By Lemma H, every element of L is a supremum of finite join-primes p, so u=v. ☐

Every idempotent adjunction F ⊣ U restricts to an (adjoint) equivalence of categories between the full subcategories of objects that are isomorphic to objects of the form FY and of objects that are isomorphic to objects of the form UX. For example, the idempotent adjunction O ⊣ pt restricts to an equivalence categories between the category of sober spaces and the opposite of the category of spatial frames. In the present case, we have the following.

Theorem J. The idempotent adjunction O^a ⊣ pt^a restricts to an (adjoint) equivalence of categories between the category Pos of posets and monotone maps and the opposite of the category aCDCL of algebraic completely distributive complete lattices and complete lattice homomorphisms.

Next time, we will combine this construction to produce a duality for preordered topological spaces, with a variant of the d-frames mentioned in the June 2025 post. I have told you that this would be more complicated. To let you in on what the issue is, without too many details: doing the “obvious” construction will give you an analogue d-frames where the second frame is a frame of upwards-closed subsets. But you won’t be able to talk about downwards-closed subsets. In other words, and since frames and d-frames are really some forms of logics, the “obvious” construction would allow to talk about the future of points, but not their pasts. There is a simple way out: we build three frames instead of two, one for the topology, one for ≤ (“future”), one for ≥ (“past”)… and we constrain those three frames through appropriate axioms of compatibility.

1. Nesta van der Schaaf. Towards Point-Free Spacetimes. PhD thesis, University of Edinburgh, 2024. Available as arXiv report 2406.15406v2.
2. Marcello M. Bonsangue, Bart Jacobs, and Joost N. Kok. Duality beyond sober spaces: Topological spaces and observation frames. Theoretical Computer Science, 151:79–124, 1995.

— Jean Goubault-Larrecq (December 20th, 2025)