Meet-continuous spaces

Meet-continuous dcpos were defined and studied by Hui Kou, Ying-Ming Liu, and Mao-Kang Luo [1] about 14 years ago, and their importance only starts to be appreciated now.  One of the leading results in the theory of meet-continuous dcpos is that a dcpo is continuous if and only if it is quasi-continuous and meet-continuous.  I have already talked briefly about meet-continuous dcpos here, where I notably mentioned Weng Kin Ho, Achim Jung and Dongsheng Zhao’s new proof of that theorem through Stone duality.  Today, I would like to talk about yet another proof, mentioned by Xiaodong Jia [3, Theorem 3.1.12] in his remarkable PhD thesis, and first appearing in [7, Proposition III-3.10].  This is a much more elementary proof.  Then, I will modify it to suit my personal taste, and my exposition will progressively diverge.

Meet-continuity

A lattice is meet-continuous if and only if the meet, or infimum operation ⋀ is Scott-continuous.  Meet-continuous lattices seem to have been first studied by John Isbell [2].

Since a function of two arguments is Scott-continuous if and only if it is Scott-continuous in each argument separately, a lattice L is meet-continuous if and only if, for every y in L, the function x ↦ x ⋀ y is Scott-continuous.  In other words, a meet-continuous complete lattice is nothing else than a frame… but that is not the direction I want to take.

Since the function x ↦ x ⋀ y is always monotonic, Scott-continuity reduces to require that for every y in L, for every directed family (xi)i ∈ I, (supi ∈ I xi) ⋀ y ≤ supi ∈ I (xi ⋀ y).  (The converse inequality always holds.)

Kou, Liu and Luo observed that one can give an equivalent definition that does not use the  ⋀ operator at all.  Let us reconstruct one way of doing so.  Let D be a directed family (xi)i ∈ I of elements of L.  If L is a meet-continuous lattice, and y ≤ sup D, then y = (sup D) ⋀ y ≤ supi ∈ I (xi ⋀ y) ≤ sup (↓D ∩ ↓y).  That implies that y is in cl (↓D ∩ ↓y), hence that ↓y = (↓D ∩ ↓y).

This is Kou, Liu and Luo’s definition of a meet-continuous dcpo: one in which ↓y = (↓D ∩ ↓y) for every y ≤ sup D, or equivalently:

(meet-continuity in Kou, Liu and Luo’s sense) for every directed family D and every y ≤ sup D, y ∈ cl (↓D ∩ ↓y) .

We have just checked that every meet-continuous complete lattice is meet-continuous in Kou, Liu, and Luo’s sense.  Let us check the converse.  We consider a complete lattice in which y ∈ cl (↓D ∩ ↓y), for every directed family D and every y ≤ sup D.  We take a directed family D = (xi)i ∈ I, and a point y.  Then z = (supi ∈ I xi) ⋀ y, which is ≤ sup D, must be in cl (↓D ∩ ↓z) by assumption.  We wish to show that z ≤ supi ∈ I (xi ⋀ y), and for that it is enough to show that every open neighborhood U of z contains supi ∈ I (xi ⋀ y).  Since z is in cl (↓D ∩ ↓z), U intersects cl (↓D ∩ ↓z).  Therefore U also intersects ↓D ∩ ↓z, say at z’.  Then z’ ≤ xi for some i, and z’ ≤ z by definition, so z’ ≤ xi ⋀ y.  Since z’ is in U, we conclude.

Some of Xiaodong Jia’s characterizations of meet-continuity

Xiaodong’s PhD thesis sets meet-continuity at the very center of domain theory, as the title shows.  One of the things he does is give various characterizations of the notion, and we shall see some of them—and also a few others.  I will not run through all of Xiaodong’s characterizations, and I will leave those that are based on so-called forbidden structures for a later post.

Before I start, let me restate a useful lemma from [4], where it appears as Lemma 3.8.  For a poset Z, let me write Zσ for Z with its Scott topology.  Generalizing that, given a topological space Z, let me write Zσ for Z with the Scott topology of its specialization preordering.  Let us agree to call Z a quasi-monotone convergence space if the topology of Zσ is finer than that of Z.

As a vacuous example, every poset is a quasi-monotone convergence space in its Scott topology.  A much more interesting observation—which I won’t make use of in this post however—is that every monotone convergence space is a quasi-monotone convergence space, and every sober space is a monotone convergence space.

Lemma. For every quasi-monotone convergence space Z, and every topological space Z’, every continuous map f from Z to Z’ is also continuous from Zσ to Z’σ (i.e., Scott-continuous).

Proof. Since f is continuous, it is monotonic, hence the only thing that we have to show is that f(sup D) ≤ sup f(D) for every directed family D in Z.  In order to do so, it suffices to show that every open neighborhood U of f(sup D) contains sup f(D).  Any such U has the property that sup D is in f-1(U), which is open in Z, hence in Zσ, since Z is a quasi-monotone convergence space.  Therefore some element x of D is in f-1(U), hence f(x) is in U.  Then f(sup D), which is even larger, is also in U.   ☐

Given a topological space X, we consider the space H(X) of closed subsets of X. I have already mentioned that this is the Hoare hyperspace of X.   Its topology—the lower Vietoris topology—has a subbase consisting of sets ◊U, U open in X, where by definition ◊U is the set of closed sets F of X that intersect U.  The specialization ordering of H(X) is inclusion, and there is a continuous map η : x ∈ X ↦ ↓x ∈ H(X).

Under inclusion H(X) is a complete lattice, and it is traditional in domain theory to give it the Scott topology.  In general the Scott topology is finer than the lower Vietoris topology, and it coincides with it when X is a c-space (see last proposition in the above mentioned post). Since we do not assume X to be a c-space, one should really be careful and distinguish H(X) from H(X)σ.

Proposition. Given a dcpo X, the following are equivalent:

  1. X is meet-continuous;
  2. for every y in X, the map ηy :  x ∈ X ↦ ↓x ∩ ↓y is continuous from Xσ to H(X);
  3. for every y in X, the map ηy :  x ∈ X ↦ ↓x ∩ ↓y is (Scott-)continuous from Xσ to H(X)σ;
  4. for every y in X, for every (Scott-)open subset U of X, ↑(U ∩ ↓y) is (Scott-)open;
  5. H(X) is a meet-continuous complete lattice;
  6. for all downwards closed subsets A and B, cl (A) ∩ cl (B) = cl (A ∩ B);
  7. for all upwards closed subset A and B, int (A) ∪ int (B) = int (A ∪ B).

The equivalence between 1 and 3 is Proposition 3.1.5 of [3].  The equivalence between 1 and 5 is Proposition 3.2.9.  Item 4 is mentioned in Theorem 3.2.1.

Proof.  1⇒2.   Take an arbitrary point y in X, an arbitrary open subset U of X.  We wish to show that ηy-1(◊U) is Scott-open.  It is easy to see that it is upwards-closed.  We take an arbitrary directed family D, and we assume that sup D is in ηy-1(◊U).  Hence ↓sup D ∩ ↓y intersects U, say at x.  Since x ≤ sup D and 1 holds, x is in cl (↓D ∩ ↓x).  It follows that the latter intersects U, hence also ↓D ∩ ↓x intersects U.  Pick a point in that intersection: it is in U, below some point z of D, and below x.  Hence ↓z ∩ ↓y intersects U, showing that z (a point of D) is in ηy-1(◊U).

2⇒3.  Xσ is a quasi-monotone convergence space: then apply the previous Lemma.

3⇒4.  Assuming 3, item 2 holds because the Scott topology on H(X) is finer than the Vietoris topology.  Hence for every y in X, ηy-1(◊U) is Scott-open.  However, ηy-1(◊U) is the set of points x such that ↓x ∩ ↓y intersects U, equivalently the set of points above some point in ↓y ∩ U, and that is ↑(U ∩ ↓y).

4⇒1.  Assume y ≤ sup D, where D is some directed family of elements of X, and let U be the complement of cl (↓D ∩ ↓y).  We wish to show that y is in cl (↓D ∩ ↓y).  If that were not the case, then y would be in U, hence also in ↑(U ∩ ↓y).  The latter is Scott-open by 4, and since y ≤ sup D, some element z of D must be in ↑(U ∩ ↓y).  Hence there is a point x in U such that x≤y and x≤z.  But now x is in ↓D ∩ ↓y, hence in cl (↓D ∩ ↓y), and also in U: contradiction.

It remains to show that any of the equivalent items 1—4 is equivalent with 5, 6, and 7.  We show 4⇒5⇒6⇒7⇒4.

4⇒5.  We only have to show that intersection is Scott-continuous on H(X).  Let C be a closed subset of X, and (Ci)i ∈ I be a directed family of closed subsets of X, with supremum C’.  We must show that C ∩ C’ ⊆ supi (C ∩ Ci), where sup denotes closure of union.  (The converse inclusion is immediate since intersection is monotonic.)  In order to do so, we take an arbitrary open set U that intersects C ∩ C’, and we show that it intersects supi (C ∩ Ci).  Since U intersects C ∩ C’, there is a point y in U that is in C and in C’.  By item 4, ↑(U ∩ ↓y) is open, and intersects C’ at y.  Since it intersects C’ = cl (∪i Ci) and it is open, it intersects ∪i Ci, hence some Ci, say at z.  This z is in ↑(U ∩ ↓y), hence is above some point x of U such that x≤y.  Since Ci is downwards closed, x is in Ci, and since C is downwards closed and contains y, x is also in C.  This shows that U intersects C ∩ Ci (at x), hence also the larger set supi (C ∩ Ci).

5⇒6.  The inclusion cl (A ∩ B) ⊆ cl (A) ∩ cl (B) is always true.  In the converse direction, cl (A) is the supremum in H(X) of the directed family of elements of the form ↓F, where F ranges over the finite subsets of A, and cl (B) is the supremum of the directed family of elements of the form ↓G, where G ranges over the finite subsets of B.  By item 5, cl (A) ∩ cl (B) is then equal to the supremum of the family of all subsets of the form ↓F ∩↓G, where F and G are as above, and those subsets are all included in A ∩ B, hence in cl (A ∩ B).

6⇒7.  Follows directly by complementation, and the realization that the complements of downwards closed sets are exactly the upwards closed sets.

7⇒4.  For every open subset U, U is included in ↑(U ∩ ↓y) ∪ V, where V is the complement of ↓y, as one checks by realizing that every element of U is either below y or not.  Hence U is included in the interior of ↑(U ∩ ↓y) ∪ V, which is equal to int (↑(U ∩ ↓y)) ∪ int (V) by item 7.  Now V is open, so int (V) = V.  From U ⊆ int (↑(U ∩ ↓y)) ∪ V, we obtain U ∩ ↓y ⊆ int (↑(U ∩ ↓y)), hence ↑(U ∩ ↓y) ⊆ int (↑(U ∩ ↓y)), and that shows that ↑(U ∩ ↓y) is open.    ☐

That proposition suggests several ways of generalizing the notion of meet-continuous dcpo to all topological spaces.  Maybe you have noticed that the proofs I gave of the implications 4⇒5⇒6⇒7⇒4 work in any topological space.  Hence I would like to suggest the following.

Definition (meet-continuous topological space).  A topological space is meet-continuous if and only if any of the following equivalent conditions hold:

  • for every y in X, for every open subset U of X, ↑(U ∩ ↓y) is open;
  • H(X) is a meet-continuous complete lattice;
  • for all downwards closed subsets A and B, cl (A) ∩ cl (B) = cl (A ∩ B);
  • for all upwards closed subset A and B, int (A) ∪ int (B) = int (A ∪ B).

The definition should be compared to Erné’s notion of c-space.  A c-space is the same thing as a topological space in which the interior of an arbitrary union of upwards-closed subsets is the union of the interiors (Exercise 5.1.38 in the book).  A meet-continuous topological space is a special case of that, where only finite unions are considered.

Locally finitary compact + meet-continuous = c-space

Since arbitrary unions can be organized as directed unions of finite unions, one may wonder what a space might be in which interior distributes over directed unions of upwards-closed subsets.

Proposition. Let X be a topological space.  The following are equivalent:

  1. X is locally finitary compact;
  2. for every directed family (Ai)i ∈ I of upwards-closed subsets, int (∪i Ai) = ∪i int (Ai).

Proof.  1⇒2.  Only the inclusion int (∪i Ai) ⊆ ∪i int (Ai) needs proof.  For every point x in int (∪i Ai), there is a finite set F such that x is in int (↑F) and F is included in int (∪i Ai), since X is locally finitary compact.  In particular, F is included in ∪i Ai, and since F is finite, it is included in some Ai.  It follows that x is in int (Ai), hence in ∪i int (Ai).

2⇒1.  Let x be a point of X, and U be an open neighborhood of x.  Since U is upwards closed, it is equal to the union of the directed family of sets of the form ↑F, where F ranges over the finite subsets of U.  Since U is its own interior, U = int (∪F ↑F), where F ranges over the finite subsets of U.  By 2, U is also equal to ∪F int (↑F), so x is in int (↑F) for some finite subset F of U.    ☐

Since arbitrary unions are directed unions of finite unions, we can now conclude.

Theorem.  A topological space is a c-space if and only if it is locally finitary compact and meet-continuous.

Among all topological spaces, look at the sober spaces.  A sober c-space is the same thing as a continuous dcpo (Proposition 8.3.36 in the book), a sober locally finitary compact space is the same thing as a quasi-continuous dcpo (Exercise 8.3.39 in the book).  In both cases, the topology is the Scott topology.  Hence:

Corollary.  A sober space is a continuous dcpo if and only if it is a meet-continuous quasi-continuous dcpo.

Since quasi-continuous dcpos are sober (see Exercise 8.2.15 in the book), another formulation of that corollary is the usual one [1, 3]: a quasi-continuous dcpo is continuous if and only if it is meet-continuous.

We retrieve the main result of [5], too.  This is because:

  • the Stone duals of continuous posets, and more generally of c-spaces, are the completely distributive complete lattices (equivalently, the prime continuous complete lattices),
  • the Stone duals of meet-continuous posets are the join-continuous lattices (i.e., the complete lattices whose opposite poset is a frame) [6],
  • and the Stone duals of quasi-continuous dcpos, and more generally of locally finitary compact spaces, are the hypercontinuous lattices.

That also goes the other way around: the above theorem can be deduced from [5] by running Stone duality in the converse direction.

  1. Hui Kou, Ying-Ming Liu, and Mao-Kang Luo. On Meet-Continuous Dcpos. In G. Zhang, J. Lawson, Y. Liu, and M. Luo, editors, Domain Theory, Logic and Computation, volume 3 of Semantic Structures in Computation, pages 117–135. Springer Netherlands, 2003.
  2. John R. Isbell. Meet-continuous lattices, Symposia Mathematica 16 (1975), pp. 41–54, convegno sulla Topologica Insiemsistica e Generale, INDAM, Roma, Marzo 1973.
  3. Xiaodong Jia.  Meet-Continuity and Locally Compact Sober Spaces.  PhD thesis, University of Birmingham, 2018.
  4. Jean Goubault-Larrecq.  QRB-Domains and the Probabilistic Powerdomain.  Logical Methods in Computer Science 8(1:14), 2012.
  5. Weng Kin Ho, Achim Jung, and Dongsheng Zhao.  Join-Continuity + Hypercontinuity = Prime Continuity. arXiv report 1607.01886, v1, July 2016.
  6. X. Mao and L. Xu. Meet continuity properties of posets. Theoretical Computer Science, 410:4234–4240, 2009.
  7. Gerhard Gierz, Karl-Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael Mislove, and Dana S. Scott. Continuous Lattices and Domains. Encyclopedia of Mathematics and Its Applications, vol. 93, 2003. Cambridge University Press.

Thanks to Xiaodong for pointing out to me that the proof of Proposition 3.1.12 in [3] is from [7]!

Added, June 19th, 2023: what I have called meet-continuous spaces here had been invented by Marcel Erné under the name of web spaces five years prior, see: M. Erné, Infinite distributive laws versus local connectedness and compactness properties, Topology and its Applications 156 (2009), pages 2054–2069.

— Jean Goubault-Larrecq (February 26th, 2018)jgl-2011