a note on the topology generated by scott open

A NOTE ON THE TOPOLOGY

GENERATED BY SCOTT OPEN FILTERS

Martin Maria Kovar

(University of Technology, Brno)

August 4, 2004

Abstract. In Problem 527 of Open Problems in Topology J. Lawson and M. Misloveask, for which DCPO’s the Scott topology has a basis of open filters and for whichDCPO’s the topology generated by the Scott open filters is T0. In this paper weintroduce a notion of a Hofmann-Mislove DCPO, for which we can partially answerthe question.

0. Introduction and Terminology

In this section we explain the most of used terminology, with an exception of someprimitives and notions that we touch only marginally. These notions are not essentialfor understanding the paper. However, for more complex and detailed explanationand introductory to the topics the reader is referred to the books and monographs[G+], [LM] and [Vi].

Let P be a set, ≤ a reflexive and transitive, but not necessarily antisymmet-ric binary relation on P . Then we say that ≤ is a preorder on P and (P ≤) isa preordered set. For any subset A of a preordered set (P,≤) we denote ↑A ={x| x > y for some y ∈ A} and ↓A = {x| x 6 y for some y ∈ A}. An important ex-ample of a preordered set is given by a preorder of specialization of a topologicalspace (X, τ), which is defined by x ≤ y if and only if x ∈ cl {y}. This preorder is apartial order in the usual sense if and only if the space (X, τ) is T0. For any x ∈ Xit is obvious that ↓{x} = cl {x}. A set is said to be saturated in (X, τ) if it is anintersection of open sets. One can easily verify that a set A ⊆ X is saturated in(X, τ) if and only if A = ↑A, that is, if and only if A is an upper set with respectto the preorder of specialization of (X, τ). Thus for every set B ⊆ X, the set ↑B wecall a saturation of B. The family of all compact saturated sets in (X, τ) is a closed

1991 Mathematics Subject Classification. 54F05, 54H99, 6B30.Key words and phrases. DCPO, Scott topology, Scott open filter, de Groot dual.The author acknowledges support from Grant no. 201/03/0933 of the Grant Agency of the

Czech Republic and from the Research Intention MSM 262200012 of the Ministry of Education ofthe Czech Republic

Typeset by AMS-TEX

1

2 MARTIN MARIA KOVAR

base for a topology τd, which is called de Groot dual of the original topology τ . Atopological space is said to be sober if it is T0 and every irreducible closed set is aclosure of a singleton.

Let (X,≤) be a partially ordered set, or briefly, a poset. If (X,≤) has, in addition,finite meets, then any element p ∈ X is said to be prime if x ∧ y ≤ p implies x ≤ por y ≤ p for every x, y ∈ X. The set P of all prime elements of (X,≤) is calledthe spectrum of (X,≤). If (X,≤) is a frame and P ⊆ X is its spectrum, by a localecorresponding to (X,≤) we mean the quadruple (X, P,≤, ²), where ² is a binaryrelation defined by p ² x if and only if x � p. Then the elements of X we call opensand the elements of P the abstract or localic points of the locale (X, P,≤, ²). A localeis called spatial if it can arise from the frame of open sets of some topological space.Then X can be represented as a topology on P , ≤ is the set inclusion and the meaningof the relation ² is the same as of ∈. In that case, the topological space (P, X) issober. It is not difficult to show that a locale is spatial if and only if it has enoughabstract points to distinguish the opens, i.e. u = v if and only if (p ² u) ⇔ (p ² v) forevery p ∈ P and u, v ∈ X. For more detail regarding the other possible definitions ofa locale, its properties and relationship to the topological spaces we refer the readerto [Vi].

We say that the poset (X,≤) is directed complete, or DCPO, if every directedsubset of X has a least upper bound – a supremum. A subset U ⊆ X is said tobe Scott open, if U =↑U and whenever D ⊆ X is a directed set with sup D ∈ U ,then U ∩D 6= ∅. One can easily check that the Scott open sets of a DCPO form atopology. This topology we call the Scott topology. Thus a set A ⊆ X is closed inthe Scott topology if and only if A =↓A and if D ⊆ A is directed, then sup D ∈ A. Itfollows from Zorn’s Lemma that in a DCPO, every element of a Scott closed subsetis comparable with some maximal element. It is easy to see that the closure of asingleton {x} in the Scott topology is ↓{x}, thus the original order ≤ of X can berecovered from the Scott topology as the preorder of specialization. In particular,the Scott topology on a DCPO is always T0.

Let us describe some other important topologies on DCPO’s. The upper topology[G+], which is also referred as the weak topology [LM] or the lower interval topology[La] has the collection of all principal lower sets ↓{x}, where x ∈ X, as the subbasefor closed sets. The preorder of specialization of the lower interval topology coincidewith the original order of (X ≤). Hence, the saturation of a subset A ⊆ X withrespect to this topology is ↑A. Similarly, the lower topology, also referred as theweakd topology [LM] or the upper interval topology [La], arises from its subbase forclosed sets which consists of all principal upper sets ↑{x}, where x ∈ X. Note thatthe weakd topology is not the de Groot dual of the weak topology in general; theweakd topology is the weak topology with respect to the inverse partial order. Thepreorder of specialization of the upper interval topology is a binary relation inverseto the original order of (X,≤). Consequently, the saturation of a subset A ⊆ Xwith respect to this topology is ↓A. The topology on the spectrum P of a directedcomplete ∧-semilattice (X,≤), induced by the upper interval topology, is called thehull-kernel topology [LM].

The set F ⊆ X is called a filter, if F =↑F and every finite subset of F has alower bound in (X,≤). An interesting topology on X can arise from the family of all

A NOTE ON THE TOPOLOGY . . . 3

filters in (X,≤), which are open in the Scott topology. We can take this family as asubbase for open sets of the new topology. This topology is strictly weaker than theScott topology in general. However, J. Lawson and M. Mislove stated the questionwhen these two topologies coincide as a part of the Problem 527 in Open Problemsin Topology [LM]. The complete question was stated as follows:

Problem 527. Characterize those DCPO’s(i) for which the Scott topology has a basis of open filters, and(ii) for which the topology generated by Scott open filters is T0.

Analogously, characterize those T0 topological spaces X for which the Scott topologyon the lattice O(X) of open sets satisfies (i) or (ii).

Note that both conditions are clearly true in the first case if the DCPO is contin-uous and, consequently, in the second case if X is core compact.

1. Preliminaries

We start with study of the relationship between compact sets and Scott openfilters in DCPO’s and we also introduce the notions of a Hofmann-Mislove DCPOand its generalized spectrum. A source of inspiration of this section, among others,was an interesting paper [KP] of K. Keimel and J. Paseka. In the following definitionwe slightly adjust the notion of a prime element to be more applicable also for thoseDCPO’s which do not have finite meets.

Definition 1.1. Let (X,≤) be a poset, L ⊆ X. We say that L is prime if for everya, b ∈ X

↓{a}∩ ↓{b} ⊆↓L ⇒ (a ∈↓L) ∨ (b ∈↓L).

It can be easily seen that if (X,≤) has finite meets, any element p ∈ X is prime ifand only if the singleton {p} is prime as a set. Hence, we can extend the notion of aprime element also to those posets which do not necessarily have finite meets. Thusin the following text we mean that an element p of a poset (X,≤) is prime if andonly if {p} is prime in the sense of the previous definition. As the following lemmashows, the notions of a prime set and of a filter are dual.

Lemma 1.2. Let (X,≤) be a poset. Then L ⊆ X is prime if and only if F = Xr ↓Lis a filter.

Proof. Let L ⊆ X be prime. Then F = Xr ↓L is a lower set. Let a, b ∈ F . Thena, b /∈↓L, which implies that there is some c ∈↓{a}∩ ↓{b} such that c /∈↓L, i.e. c ∈ F .Conversely, let F be a filter. Suppose that ↓{a}∩ ↓{b} ⊆↓L for some a, b ∈ X. Thena, b /∈↓L implies a, b ∈ F which means that there is some c ≤ a, c ≤ b, c ∈ F . Thenc ∈↓{a}∩ ↓{b}, but c /∈ L, which is a contradiction. ¤

Definition 1.3. Let X be a set, Φ ⊆ 2X . We say that K ⊆ X is up-filtered compactwith respect to the family Φ if K ∩ (

⋂ϕ) 6= ∅ for every filter base ϕ ⊆ Φ such that

every its element meets K. In a DCPO (X,≤) we say that K ⊆ X is up-filteredcompact, if it is up-filtered compact with respect to the family {↑{x} |x ∈ X} ofprincipal upper sets.


If the DCPO (X,≤) has finite meets, up-filtered compact means the same ascompact with respect to the upper interval topology. Note that this clone of com-pactness we refer as up-compact in [Ko]. The following lemma shows that thereis some ”de Groot-like” duality between the upper interval topology and the Scotttopology.

Lemma 1.4. Let (X,≤) be a DCPO. Then K ⊆ X is Scott closed if and only if Kis saturated in the upper interval topology and up-filtered compact.

Proof. Let K ⊆ X be Scott closed. Then F = X r K is an upper set and soF =

⋃a∈F ↑{a} which means that K = Xr

⋃a∈F ↑{a} =

⋂a∈F (Xr ↑{a}). Then K

is an intersection of open sets in the upper interval topology, so it is saturated. Letϕ = {↑{a} | a ∈ A} be a filter base such that K∩ ↑{a} 6= ∅ for every a ∈ A. Since ϕis a filter base then if a, b ∈ A there exists c ∈ A such that ↑{c} ⊆↑{a} ∩ {b}, thatis, c ≥ a, b. So A is directed. Further, if a ∈ A, then there is some x ∈ K∩ ↑{a}. Itfollows that a ≤ x and since K is a lower set, we have a ∈ K. Hence A ⊆ K. Thenu = sup A ∈ K since K is Scott closed. But then u ∈ K ∩ (

⋂a∈A ↑{a}). It means

that K is up-filtered compact.Conversely, let K ⊆ X be up-filtered compact and saturated in the upper interval

topology. Then there exists F ⊆ X such that K =⋂

a∈F (Xr ↑{a}) and, conse-quently, K is a lower set. Let A ⊆ K be directed. Then Φ = {↑{a} | a ∈ A} is aclosed filter base and every its element clearly meets K. Since K is up-filtered com-pact, there exists t ∈ K ∩ (

⋂a∈A ↑{a}). Then t ≥ a for every a ∈ A, so t ≥ supA.

Since K is a lower set, sup A ∈ K. Hence, K is Scott closed. ¤

Proposition 1.5. Let (X,≤) be a DCPO, ω the upper interval topology on X, ωP

the induced topology on P ⊆ X. The following conditions (i) and (ii) are equivalent:(i) There exist P ⊆ X such that:

(1) For every Scott open filter F ⊆ X, if we denote ψ(x) = Pr ↑{x} andL =

⋂a∈F ψ(a), the set L is up-filtered compact, saturated in (P, ωP )

and F = {x|x ∈ X, L ⊆ ψ(x)}.(2) For every up-filtered compact and saturated L ⊆ P in (P, ωP ), the set

F = {x|x ∈ X,L ⊆ ψ(x)} is a Scott open filter.

(ii) There exist P ⊆ X such that:(1) For every Scott closed prime set K ⊆ X, the set L = P ∩K is up-filtered

compact, saturated in (P, ωP ) and K =↓L.(2) For every up-filtered compact and saturated L ⊆ P in (P, ωP ), the set

↓L is Scott closed and prime.

Proof. It is obvious that F ⊆ X is a Scott open filter if and only if K = X r F is aScott closed prime set. Further,

⋂a∈F ψ(a) =

⋂a∈F (Pr ↑{x}) = P r

⋃a∈F ↑{a} =

P r F = P ∩ K and Xr ↓L = {x|x ∈ X, x /∈↓L} = {x|x ∈ X,L∩ ↑{x} = ∅} ={x|x ∈ X, L ⊆ ψ(x)}. Now it is clear that (ii) is only a reformulation of (i). ¤


Definition 1.6. Let (X,≤) be a DCPO. We say that (X,≤) is Hofmann-Mislove,if (X,≤) satisfies any of the conditions (i) or (ii) of Proposition 1.5. The set P ⊆ Xfrom (i) or (ii) we call a generalized spectrum of (X,≤); the topology ωP we call thegeneralized hull-kernel topology on P .

The next proposition shows that in a Hofmann-Mislove DCPO, the generalizedspectrum is determined uniquely.

Proposition 1.7. Let (X,≤) be a Hofmann-Mislove DCPO, S ⊆ X its any gener-alized spectrum. Then S = {p| p ∈ X, p is prime} = {m|m is a maximal element ofa Scott closed prime subset of X}.Proof. Let M = {m|m is a maximal element of a Scott closed prime set}, P ={p| p ∈ X, p is prime}. Let p ∈ P be a prime element. Then ↓{p} is a Scott closedprime set and p its maximal element. Then P ⊆ M . Let m ∈ M and let K ⊆ X be aScott closed prime set such that m ∈ K is its maximal element. Then K =↓(K ∩S),so m ∈↓(K ∩ S). Then there exists t ∈ K ∩ S with m ≤ t. But m is maximal in K,so m = t ∈ S. Hence, M ⊆ S. Let s ∈ S. Then {s} is clearly up-filtered compact, aswell as ↓{s}. The set L = S∩ ↓{s} is saturated in (S, ωS) and ↓L =↓{s}. Hence, L isup-filtered compact. Then ↓L is Scott closed and prime, which means, in particular,that s is prime. Therefore, S ⊆ P . Now we have P ⊆ M ⊆ S ⊆ P , which completesthe proof. ¤

Remark 1.8. If (X,≤) is a DCPO with finite meets, that is, a directed complete ∧-semilattice, then the natural requirement (2) of the conditions (i) or (ii) in Proposition1.5 is fulfilled automatically. Indeed, from Proposition 1.7 we know that P must bethe set of all prime elements. Let L ⊆ P be any set and let ↓{a}∩ ↓{b} ⊆↓L forsome a, b ∈ X. Then a∧ b ∈↓L, so there is some p ∈ L such that a∧ b ≤ p. But thena ≤ p or b ≤ p since p is prime. But then a ∈↓L or b ∈↓L. Hence, L and also ↓L areprime sets. If L is, in addition, up-filtered compact, then so ↓L and by Lemma 1.4,↓L is Scott closed. ¤

Hence, from the condition (i) of Proposition 1.5 we can see that the Hofmann-Mislove DCPO’s are – in other words – exactly those DCPO’s which satisfy thewell-known Hofmann-Mislove Theorem.

Corollary 1.9. Let (X,≤) be a DCPO with finite meets. Then (X,≤) is Hofmann-Mislove if and only if every maximal element of a Scott closed prime set is prime.

Proof. The ”only if” part follows from Proposition 1.7. Let K ⊆ X be a Scott closedprime set, P the set of all prime elements. Let L = P ∩K. It follows from Lemma 1.4that K is up-filtered compact and saturated in the upper interval topology, hence Lis saturated in the hull-kernel topology on P . We will show that K =↓L. Obviously,↓L ⊆ K since K is a lower set. But every x ∈ K is comparable with some maximalelement m ∈ K, which is prime by the assumption. Hence x ≤ m ∈ L, so m ∈↓L.It holds K =↓L, but then L is also up-filtered compact. Then the part (1) of thecondition (ii) in Proposition 1.5 is fulfilled and the part (2) is fulfilled automatically.This completes the proof. ¤


The following corollary is a reformulation of the previously mentioned Hofmann-Mislove Theorem. In addition to the assumptions of Corollary 1.9 we will supposethat the DCPO (X,≤) is a distributive lattice. But this is equivalent to the assump-tion that (X,≤) is a frame.

Corollary 1.10. Let (X,≤) be a frame. Then (X,≤) is Hofmann-Mislove.

Proof. Let K ⊆ X be a Scott closed prime set, m ∈ K its maximal element. Supposethat a ∧ b ≤ m for some a, b ∈ X. Then (a ∨m) ∧ (b ∨m) = (a∧ b) ∨ (a ∧m) ∨ (m ∧b) ∨ (m ∧m) = m. Then ↓{a ∨m}∩ ↓{b ∨m} ⊆ K, but K is lower and prime, soa ∨m ∈ K or b ∨m ∈ K. But then a ∨m = m or b ∨m = m, i.e. a ≤ m or b ≤ m,since m is maximal. Hence, m is a prime element. ¤

2. Scott open sets and filters

Now we give a partial answer to the Problem 527 in terms of Hofmann-MisloveDCPO’s, their generalized spectra and the de Groot-like duality between the upperinterval topology and the Scott open topology. If not otherwise specified, thoroughthis section all saturations of sets as well as up-filtered compactness are assumedwith respect to the upper interval topology on a DCPO.

Theorem 2.1. Let (X,≤) be a Hofmann-Mislove DCPO, P ⊆ X its generalizedspectrum. The following conditions are equivalent:

(i) The Scott topology on X has a base of Scott open filters.(ii) For every up-filtered compact saturated set K ⊆ X there is a family {Ki}i∈I

of up-filtered compact sets such that K =⋂

i∈I Ki and Ki =↓(P ∩Ki).(iii) Every up-filtered compact saturated set in X is an intersection of saturations

of up-filtered compact subsets of the generalized spectrum of X.

Proof. The conditions (ii) and (iii) are clearly equivalent since L ⊆ P is up-filteredcompact if and only if ↓L is up-filtered compact. Suppose (i). Let K ⊆ be a up-filtered compact set, saturated in the upper interval topology. Then, by Lemma 1.4,F = X r K is a Scott open set. By (i), F =

⋃i∈I Fi, where every Fi is a Scott

open filter. Denote Ki = X r Fi. Then K =⋂

i∈I Ki and by Lemma 1.2 everyKi is a Scott closed prime set. Since (X ≤) is Hofmann-Mislove, by the definitionLi = P ∩Ki is up-filtered compact subset of the generalized spectrum and Ki =↓Li,which gives (iii).

Suppose (ii). Let F ⊆ X be a Scott open set. We put K = X r F . It followsfrom Lemma 1.4 that K is up-filtered compact and saturated in the upper intervaltopology. By (ii), K =

⋂i∈I Ki, where every Ki is up-filtered compact as well as

Li = P ∩Ki and Ki =↓Li. Since (X ≤) is Hofmann-Mislove, by the definition Ki

is Scott closed and prime, so by Lemma 1.2 Fi = X rKi is a Scott open filter andclearly, F =

⋃i∈I Fi, which implies (i). ¤

Corollary 2.2. Let (X,≤) be a frame. Then the Scott topology on X is the de Grootdual ωd of the upper interval topology ω. Moreover, the Scott topology on X has abase of Scott open filters if and only if the saturations of compacts subsets of thespectrum of X form a closed base of ωd.


Now we characterize those DCPO’s, whose topology generated by Scott open filtersis T0.

Theorem 2.3. Let (X,≤) be a Hofmann-Mislove DCPO, P ⊆ X its generalizedspectrum. The following conditions are equivalent:

(i) The topology on X generated by the Scott open filters is T0.(ii) For every x, y ∈ X there exists up-filtered compact L ⊆ P such that one of

the points x, y is and the other is not contained in the saturation of L.(iii) For every x, y ∈ X, x 6= y there is a prime element p ∈ X such that x ≤ p,

y � p or x � p, y ≤ p .

Proof. Let x, y ∈ X. Then, by Lemma 1.2 there is a Scott open filter F ⊆ X suchthat x ∈ F , y /∈ F if and only if there is a Scott closed prime set K ⊆ X suchthat x /∈ K, y ∈ K. But in a Hofmann-Mislove DCPO, every Scott closed primeset is a saturation of a up-filtered compact subset of the generalized spectrum, and,conversely, a saturation of a up-filtered compact subset of the generalized spectrumis Scott closed and prime. Thus (i) and (ii) are equivalent. The implication (iii) ⇒(ii) is clear. It remains to show that (ii) ⇒ (iii).

Suppose(ii). Let x, y ∈ X and let L ⊆ P be up-filtered compact, such that x ∈↓Land y /∈↓L. Then ↓L is a Scott closed prime set, so ↓L has some maximal element,say p, such that x ≤ p. Clearly, y � p since otherwise y ∈↓L. But by Proposition 1.7p is prime. Hence, (iii) holds. ¤

For frames we can say even more. The T0 separation property of the topologygenerated by Scott open filters on a frame (X,≤) is equivalent to the spatiality ofthe corresponding locale, as it follows from the next two corollaries.

Corollary 2.4. Let (X, τ) be a topological space, (τ,⊆) the frame of open sets. Thenthe topology on τ generated by the Scott open filters is T0.

Proof. Let U, V ∈ τ and U 6= V . Then U * V or U + V . Suppose the firstpossibility (the other one is analogous). Then there is some p ∈ U such that p /∈ V .Then p ∈ X r V , which implies cl {p} ⊆ X r V and so V ∩ cl {p} = ∅. DenoteP = X r cl {p}. Then V ⊆ P and U * P . We will show that P ∈ τ is prime. LetR, S ∈ τ such that R ∩ S ⊆ P . Then p /∈ R ∩ S, so p /∈ R or p /∈ S. Since R, S areopen, it follows that R ∩ cl {p} = ∅ or S ∩ cl {p} = ∅. But then R ⊆ P or S ⊆ P .Hence, P is prime. The Theorem 2.3 now completes the proof. ¤

Corollary 2.5. Let (X,≤) be a frame and let the topology on X generated by Scottopen filters be T0. Then the corresponding locale is spatial.

Proof. Let P ⊆ X be the spectrum of (X,≤). Then the abstract points of thecorresponding locale (X,P,≤, ²) can be represented as the elements of P . For everyp ∈ P and u ∈ X we have

p ² u ⇔ u � p ⇔ p ∈ Xr ↑{u} .

It follows from Theorem 2.3 that any two different opens x, y ∈ X of the locale(X,P,≤, ²) have an abstract point p ∈ P such that either p ² x and p 2 y, or p 2 xand p ² y. Hence (X, P,≤,²) is spatial. ¤


Since the Scott topology on a DCPO is always T0, we have the following corollary.

Corollary 2.6. Let (X,≤) be a frame whose Scott topology has a base of open filters.Then the corresponding locale is spatial, or equivalently, (X,≤) can be represented asa topology of some sober space.

References

[G+] Gierz G., Hofmann K. H., Keimel K., Lawson J. D., Mislove M., Scott D. S., A Compendiumof Continuous Lattices, Springer-Verlag, Berlin, Heidelberg, New York, 1980, pp. 372.

[Ko] Kovar M. M., On iterated de Groot dualizations of topological spaces, Topology and Appl.,1-7 (to appear).

[KP] Keimel K., Paseka J., A direct proof of Hofmann-Mislove Theorem, Proc. Am. Math. Soc.120, 1 (1994), 301-303.

[La] Lawson J. D., The upper interval topology, property M and Compactness, Electronic Notesin Theoretical Computer Science, http://www.elsevier.nl/locate/entcs/volume13.html, 13(1998), 1-15.

[LM] Lawson J.D., Mislove M., Open Problems in Topology (van Mill J., Reed G. M., eds.), North-Holland, Amsterdam, 1990, pp. 349-372.

[Vi] Vickers S., Topology Via Logic, Cambridge University Press, Cambridge, 1989, pp. 200.

Department of Mathematics, Faculty of electrical engineering and communication,University of Technology, Technicka 8, Brno, 616 69, Czech Republic

E-mail address: [email protected]

THE DE GROOT DUAL FOR

TOPOLOGICAL SYSTEMS AND LOCALES

Martin Maria Kovar

(University of Technology, Brno)

September 2, 2004

Abstract. A topology is the de Groot dual of another topology, if it has a closed baseconsisting of all compact saturated sets. In this paper we extend the de Groot dual alsoto topological systems and locales. We compare the behavior of this extension with theproperties of the de Groot dual of a topological space, and study the similarities as wellas the differences. We state a natural open question: For which locales is the extendedde Groot dual operator of finite order?

0. Introduction

J. D. Lawson and M. Mislove stated a question whether a sequence of iteratedduals of a topology terminates by two topologies, which are dual to each other (1990,Problem 540 of Open Problems in Topology,[LM]). It should be noted that for T1

spaces, the problem was solved a long time before it was formulated by Lawson andMislove, by G. E. Strecker, J. de Groot and E. Wattel (1966, [GSW]). In T1 spaces,the dual studied by Lawson and Mislove coincides with another dual, studied by deGroot, Strecker and Wattel more than 30 years ago. That is why the dual studied byLawson and Mislove is now often called the de Groot dual, although de Groot originallyintroduced and studied the dualization for T1 spaces only. This natural unification ofthe terminology is due to R. Kopperman [Kop] who probably came with it first. In2000, it was B. Burdick who proved that for some topologies on hyperspaces (whichare not T1 in general) there can arise at most four distinct topologies from iteratingthe dualization process: the original topology τ , then τd, τdd and τddd, since for thesetopologies held τdd = τdddd. Finally, this result was generalized for all topologicalspaces by the author (see [Kov]) in the spring of 2001.

Although the original Problem 540 of Lawson and Mislove arose from certain ques-tions in domain theory, the connections between this part of general topology and oneof its main sources of motivation, the theoretical computer science, need not be obvi-ous to everyone. Perhaps that is (among other reasons) also because the language oftheoretical computer science is more adjusted to use the terms of order, lattices, frames

1991 Mathematics Subject Classification. saturated set, dual topology, compactness operator.Key words and phrases. 6A12, 6D22, 54B99, 54D30, 54E55.The author acknowledges support from Grant no. 201/03/0933 of the Grant Agency of the Czech

Republic and from the Research Intention MSM 262200012 of the Ministry of Education of the CzechRepublic

Typeset by AMS-TEX

1


or domains, and if it uses some topological principles, rather than classical topologicalnotions it is related to topological systems and locales, in which the thinking need notbe spatial. In this paper we try to bring the de Groot dual from topological spaces toits implicit origins in theoretical computer science and study what happens with thedual if we leave the realm of spatial thinking.

1. Dualizations of Topological Systems

Before we start, let us recall some notions which will be essential for our furtherconstructions. Let (X, τ) be a topological space. By τd we denote a topology dualto τ , that is, the topology generated by the compact saturated sets in (X, τ), as itsclosed base. This topology is also often called co-compact and the operator d we callde Groot dual operator, as it has been explained in the introduction. Let A be a frame,X a set, |=⊆ X × A. We write x |= a for (x, a) ∈|= and say ”x satisfies a”. Let thefollowing conditions are satisfied:

(i) If B ⊆ A, then (x |= ∨B) ⇔ (x |= b for some b ∈ B).

(ii) If C ⊆ A is finite, then (x |= ∧C) ⇔ (x |= c for every c ∈ C).

Then we say that the triple (X,A, |=) forms a topological system. The elements ofA we call opens [Vi]. Obviously, if (X, τ) is a topological space, then (X, τ,∈) is atopological system. Conversely, let (X, A, |=) be a topological system. Let intX(a) ={x|x ∈ X, x |= a} for every a ∈ A and let τX(A) = {intX(a)| a ∈ A}. One can easilycheck that τX(A) is a topology on X and so (X, τX(A)) is a topological space. Wecall τX(A) the underlying topology on X induced by A. The corresponding topologicalsystem (X, τX(A),∈) is said to be the spatialization of (X, A, |=). The notions ofcompactness and saturation can be naturally and without any difficulties extendedfrom topological spaces to topological systems. A set K ⊂ X is said to be compact(saturated, respectively) in (X, A, |=) if K is compact (saturated, respectively) in thetopological space (X, τX(A)). Let us denote ↑K the intersection of all open sets of(X, τX(A)), containing K. We call ↑K the saturation of K in (X,A, |=). Obviously,↑K is saturated in (X, τX(A)). A set U ⊂ X is said to be co-compact (co-saturated,respectively) in (X,A, |=) if X r U is compact (saturated, respectively) in (X, A, |=).Let (X, A,`) and (Y, B, |=) be topological systems. Let f : X → Y a mapping,g : B → A be a frame morphism such that x ` g(b) if and only if f(x) |= b for everyx ∈ X, b ∈ B. Then the couple (f, g) is called a morphism or a continuous map from(X,A,`) to (Y,B, |=). We denote by 2 = {⊥,>} the Serpinski frame, consisting ofthe two elements > – the top and ⊥ – the bottom.

However, the way how the dualization of a topological system can be defined consis-tently with the dualization of a topological space, need not be unique. Independentlyon its definition, it seems to be reasonable if the dualization again is a topological sys-tem and if it commutes with the spatialization if possible. To cover the most generalcase, we say that the topological system (X ′, P, |=) is a dualization of (X, A,`) rep-resented by the frame P if there exists a morphism (f, e) of topological systems from(X, (τX(A))d,∈) to (X ′, P, |=) such that e : P → (τX(A))d is a frame epimorphism.The dualization (X ′, P, |=) of (X,A,`) is strict if f is a bijection. The basic propertiesof the defined concepts are described by the following theorem.

Theorem 1.1. Let (X, A,`) be a topological system, P be a frame. The followingconditions are equivalent:

(i) P represents a dualization of (X,A,`).(ii) P represents a strict dualization of (X,A,`).

THE DE GROOT DUAL FOR TOPOLOGICAL SYSTEMS AND LOCALES 3

(iii) There exists a frame epimorphism e : P → (τX(A))d.

Proof. The implications (ii) ⇒ (i) ⇒ (iii) are trivial. Suppose (iii). For every x ∈ X,p ∈ P we define x |= p if and only if x ∈ e(p). Suppose that x ∈ X and S ⊆ P . Thenx |= ∨

S ⇔ x ∈ e(∨

S) =⋃

s∈S e(s) ⇔ (∃s ∈ S) : (x ∈ e(s)) ⇔ (∃s ∈ S) : (x |= s).Let F ⊆ P be finite. Then x |= ∧

F ⇔ x ∈ e(∧

F ) =⋂

s∈F e(s) ⇔ (∀s ∈ F ) : (x ∈e(s)) ⇔ (∀s ∈ F ) : (x |= s). Hence, (X,P, |=) is a topological system which is a strictdualization of (X, A,`). ¤

Now we show how the dualization commutes with the spatialization. Let (X ′, P, |=)be a dualization of (X, A,`). Then there exist f : X → X ′ and a frame epimor-phism e : P → (τX(A))d such that (f, e) is a morphism from (X, ((τX(A))d,∈) to(X ′, P, |=). The spatialization of (X,A,`) is (X, τX(A),∈), whose dualization (in topo-logical spaces) is (X, (τX(A))d,∈). The dualization of (X, A,`) is (X ′, P, |=), whosespatialization is (X ′, τX′(P ),∈). Let p ∈ P . Then f−1(intX′(p)) = e(p) ∈ (τX(A))d.Hence, f is a continuous mapping of topological spaces (X, (τX(A))d) and (X ′, τX′(P )).Moreover, if U ∈ (τX(A))d, then there exists p ∈ P such that e(p) = U . Thenf(U) = intX′(p)∩f(X), so f : X → f(X) is an open mapping. Hence, the dualization(X ′, P, |=) is strict if and only if f is a homeomorphism. This can be illustrated bythe following diagram, in which d denotes the de Groot dual operator, ”Spat” is thespatialization and ”Dual” stands for the studied dualization:

(X,A,`) Dual //

Spat

²²

(X ′, P, |=)

Spat

²²(X, τX(A),∈)

d// (X, (τX(A))d,∈)

f//

(f,e)66lllllllllllll

(X ′, τX′(P ),∈)

Corollary 1.2. Suppose that P is a frame that represents a (strict) dualization of thetopological system (X,A,`). If there is a frame S and a frame epimorphism f : S → P ,then also S represents a (strict) dualization of (X, A,`).

By the definition, for every dualization (X ′, P, |=) of (X,A,`) there exists a mor-phism (f, e) from (X, (τX(A))d,∈) to (X ′, P, |=) which, however, need not be unique.In this sense, (X, (τX(A))d,∈) can be interpreted as the least or (weakly) initial du-alization of (X, A,`), which we will consider as the extension of the de Groot dual totopological systems. In other words, the de Groot dual for topological systems is thecomposition

d ◦ Spat .

Let Loc(X ′, P, |=) be the localification of (X ′, P, |=). Then Loc(X ′, P, |=) is again adualization of (X, A,`). Let (X ′, P, |=) be a locale. Then (f, e) can be extendedover Loc(X, (τX(A))d,∈), so Loc(X, (τX(A))d,∈) can be interpreted as the least or(weakly) initial localic dualization of (X, A,`). This dualization we will take as theanalogue of the de Groot dual for locales. In other words, the de Groot dual for localeswe will define as the composition

l = Loc ◦ d ◦ Spat,

where ”Loc” means the localification. On the other hand, it can be easily seen thatthe ”greatest” or ”terminal” counterpart in dualizations of topological systems doesnot exist. The result of the previous considerations we can formulate as a corollary.


Corollary 1.3. Every topological system has its least dualization and its least localicdualization.

The next following corollaries summarize some other facts about dualizations oftopological systems.

Corollary 1.4. Let (Xi, Ai, |=i) be a topological system and Pi a frame representingits (strict) dualization for every i ∈ I. Then the product frame P =

∏i∈I Pi represents

a (strict) dualization of (Xi, Ai, |=i) for every i ∈ I.

We say that the topological system (Xn, An, |=n) is the n-th iterated (strict) dual-ization of (X, A,`) if there exist topological systems (Xi, Ai, |=i) for i = 1, 2, . . . , n−1such that (X1, A1, |=1) is a (strict) dualization of (X,A,`) and (Xi+1, Ai+1, |=i+1) isa (strict) dualization of (Xi, Ai, |=i) for i = 1, 2, . . . , n− 1.

Corollary 1.5. Let (X, A, |=) be a topological system. Then there exists a frame Prepresenting its n-th iterated (strict) dualization for every n ∈ N.

The last corollary follows from the fact that the strict dualization commutes withthe spatialization and the fact that τdd = τdddd holds for every topological space (X, τ)[Kov].

Corollary 1.6. A frame P represents the n-th iterated strict dualization of a topologi-cal system (X,A,`) if and only if P represents its (n+2k)-th iterated strict dualization,for every n = 2, 3, . . . and k = 1, 2, . . . .

It is a natural question whether every dualization of a locale is again a localictopological system. However, it is easy to answer this question in the negative. To givea proper counterexample, we need to recall some notions. A subset A of a topologicalspace is irreducible if it is not a union of two proper closed subsets. A topological spaceis said to be sober if it is T0 and every irreducible closed set is a closure of a singleton.It is well-known that for spatial topological systems (i.e. topological spaces), soberand localic mean the same.

Example 1.7. (i) Let X = N, A = {0, 1, . . . , ω} with its natural linear order (ω isthe first infinite ordinal) and we put x ` a ⇔ x ≤ a for every x ∈ X and a ∈ A.Then (X,A,`) is a spatial topological system. It follows that τ = τX(A) = {∅, X} ∪{{1, 2, . . . , n} |n ∈ N} is T0 and the specialization order is inverse to the natural orderof X = N. Every non-empty closed subset of (X, τ) is of the form Gk = {k, k + 1, . . . },where k ∈ N, it is irreducible and Gk = cl {k}. So the topological space (X, τ) is soberand (X, A,`) is a locale.

(ii) The compact saturated subsets of (X, τ) are exactly the finite lower closed sets.Then τd = {∅} ∪ {{n, n + 1, . . . , } |n ∈ N} an hence, every non-empty closed set iseither finite or equal to X. Hence, X is irreducible, but there is no point x ∈ X withX = cld {x}. Hence, (X, τd) is not sober and so (X, τd,∈) is not a locale.

(iii) The soberification of (X, τd) can be interpreted as the topological space (Y, σ),where Y = {1, 2, . . . , ω} and σ = {∅}∪{{n, n + 1, . . . , ω} |n ∈ N}. Indeed, f : τd → 2is a frame morphism if and only if it is monotone. All the morphisms are representedby the points of X with the exception of the morphism fω for which fω(U) = > if andonly if U 6= ∅. This morphism is represented in Y by the new point ω which satisfiesall non-empty opens. Every non-empty closed set in (Y, σ) is irreducible and has theform Fk = {1, 2, . . . , k} = clσ {k} or equals to Y = clσ {ω}.

(iv) The specialization order of (Y, σ) coincides with its natural linear order andthe compact saturated sets in (Y, σ) are exactly all upper-closed subsets of Y . Hence,


σd = {∅,N, Y } ∪ {{1, 2, . . . , n} |n ∈ N}. The non-empty closed sets in (Y, σd) areirreducible and have the form {ω} = clσd {ω} and Hk = {k, k + 1, . . . , ω} = clσd {k}for every k ∈ N. Since σd is T0, (Y, σd) is sober.

(v) The specialization order of (Y, σd) is inverse to its natural linear order. Thenevery compact saturated subset of (Y, σd) equals to Y if it contains ω and is finitelower closed otherwise. Then σdd = {∅} ∪ {{n, n + 1, . . . , ω} |n ∈ N} = σ. Hence, interms of the de Groot dual for locales, τ l = σ = σll = τ lll. ¤

According to Example 1.7 (v), the following problem, analogous to Problem 540 ofLawson and Mislove that we mentioned in the introduction, has its natural place here.

Problem 1.8. Characterize those locales (topological spaces, respectively) for whichthe process of iterating the dual operator l can generate at most finitely many non-isomorphic frames (up to frame isomorphism distinct topologies, respectively).

2. Dualizations represented by frames of functions

The way through the de Groot dual composed with the spatialization and the lo-calification which we described in the previous section is not very straightforward. Analternative way how we can dualize topological systems and locales will be studied inthis section. But firstly, let us mention a one of possible motivations with a backgroundin the computer science.

Let X be a set of some computer programs and A be a poset of their possible initial outputsa ∈ A. We write x ` a if the program x has the output a. The set K ⊆ X of programs is compactin some sense if the following condition is satisfied: If the programs in K has their initial outputs insome directed B ⊆ A, then there is some b ∈ B such that every program from K has the output b.The set K ⊆ X is saturated if it is the largest set with the same initial outputs. The program x maybe considered independent on a compact set of programs K if the event that some program in K hasan output a does not imply the event that also x has the output a. From the observations whethera program x is independent on a compact saturated set K may arise a new structure which may beinterpreted as the dual (in some sense) of the original triple (X, A,`).

Now we formalize the construction that we have sketched above. Let 2A be the setof all mappings of A to 2, where we put y ≤ z if and only if y(a) ≤ z(a) for everya ∈ A. Obviously, 2A is a frame isomorphic to the power set 2A ordered by inclusion.Let P ⊆ 2A. By False and True we denote the constant functions on A identicallyequal to ⊥ and >. Let us denote [P ] =

⋂ {S|P ⊆ S ⊆ 2A, S is closed under all joins

and finite meets in2A}.

Lemma 2.1. Let A be a frame, P ⊆ 2A. The following statements are fulfilled:(i) [P ] is a subframe of 2A.(ii) If P is closed under finite meets, then every element of [P ] is a join of some

Y ⊆ P .(iii) If P is closed under finite meets and every its element distinct from True

preserves (directed, non-empty, non-empty and directed, respectively) joins,then every element of [P ] distinct from True preserves (directed, non-empty,non-empty and directed, respectively) joins.

(iv) If P is closed under finite meets and finite joins, then every element of [P ] isa join of some directed Y ⊆ P .

Proof. To show (i), we will prove that [P ] is closed under all joins and finite meets.Let Φ =

{F |P ⊆ F ⊆ 2A, F is closed under all joins and finite meets in 2A

}. Let


Y ⊆ [P ] and let Z ⊆ [P ] be finite. Let F ∈ Φ. Then Y, Z ⊆ F and hence∨

Y ∈ Fand

∧Z ∈ F , which implies that

∨Y,

∧Z ∈ ⋂

F∈Φ F = [P ].Let us prove (ii). If P is closed under finite meets, then True =

∧∅ ∈ P . Let

F ={y| y ∈ 2A, y =

∨Y for some Y ⊆ P}. Then False =

∨∅ ∈ F and True =∨ {True} ∈ F . Let Y ⊆ F be non-empty. For every y ∈ Y there exists Yy ⊆ P such

that y =∨

Yy. Denote w =∨

Y and W =⋃

y∈Y Yy and let v ∈ W . There is somey ∈ Y such that v ∈ Yy. Then v ≤ y ≤ w, so w is an upper bound of W . Let u ∈ 2A

be an upper bound of W . Then u is an upper bound of Yy for every y ∈ Y . It followsthat u ≥ y for every y ∈ Y , so u ≥ w. It follows that w =

∨W . Since W ⊆ P ,

we have w ∈ P . Let s, z ∈ F . There are some Ys, Yz ⊆ P such that s =∨

Ys andz =

∨Yz. Denote V = {p ∧ q| p ∈ Ys, q ∈ Yz}. Since P has binary meets, it follows

that V ⊆ P . We will show that s ∧ z =∨

V . Obviously, s, z are upper bounds ofV , so s ∧ z is an upper bound of V . Let u ∈ 2A be an upper bound of V . Thens∧ z = (

∨Ys)∧ z =

∨p∈Ys

(p∧ z) =∨

p∈Ys

∨q∈(p∧ q) ≤ u. It follows that s∧ z =

∨V

and hence s ∧ z ∈ F . It follows that F is closed under all joins and binary meets, soP ⊆ [P ] ⊆ F . Hence, every element of [P ] is a join of some Y ⊆ P .

Now, let us show (iii). Let y ∈ [P ], y 6= True. Since P has finite meets, by (ii)there exists some Y ⊆ P such that y =

∨Y . Then z 6= True for every z ∈ Y .

Let B be a (directed, non-empty, non-empty and directed, respectively) subset of A.If B = ∅ and the elements of P distinct from True preserve the empty join, theny(

∨∅) = y(⊥) = (

∨Y )(⊥) =

∨z∈Y z(⊥) =

∨z∈Y ⊥ = ⊥. If B 6= ∅, it follows

that y(∨

B) = (∨

Y )(∨

B) =∨

z∈Y z(∨

B) =∨

z∈Y

∨b∈B z(b) =

∨b∈B

∨z∈Y z(b) =∨

b∈B(∨

z∈Y z)(b) =∨

b∈B(∨

Y )(b) =∨

b∈B y(b).Finally, let us prove (iv). If P has finite meets and joins, by (ii) it follows that

there is some Y ⊆ P such that y =∨

Y . We put V = {∨ F |∅ 6= F ⊆ Y, F is finite}.It follows that V is directed, V ⊆ P and y =

∨V . ¤

Now, let (X, A, |=) be a topological system, a ∈ A and M ⊆ X. We denote

wM (a) ={ >, if x |= a for every x ∈ M ,⊥, otherwise.

Lemma 2.2. Let (X, A, |=) be a topological system and K be compact in (X,A, |=).If K 6= ∅ then wK preserves directed joins and finite meets; w∅ = True.

Proof. It is obvious that w∅ = True. Let K 6= ∅. Then wK(>) = > and wK(⊥) = ⊥.Let B ⊆ A be non-empty and directed. Suppose that wK(

∨B) = >. Then x |= ∨

Bfor every x ∈ K. Since (X,A, |=) is a topological system, it follows that for everyx ∈ K there is some bx ∈ B such that x |= bx. It follows that {intX(bx)|x ∈ K} is anopen cover of K. Since K is compact, there exist x1, x2, . . . , xk ∈ K such that K ⊆⋃k

i=1 intX(bxi). Let bK ∈ B be such element that bK ≥ bxi for every i = 1, 2, . . . , k.Then K ⊆ intX(bK). It follows that wK(bK) = > and hence

∨b∈B wK(b) = >.

Conversely, suppose that∨

b∈B wK(b) = >. Then, there is some bK ∈ B such thatwK(bK) = >. If follows that K ⊆ intX(bK). Since bK ≤ ∨

B it follows that K ⊆intX(

∨B), which gives wK(

∨B) = >.

Let C ⊆ A be non-empty and finite. Then wK(∧

C) = > if and only if x |= ∧C

for every x ∈ K. This is true if and only if x |= c for every x ∈ K and c ∈ C.But this is equivalent to wK(c) = > for every c ∈ C. This holds if and only if∧

c∈C wK(c) = >. ¤


Let y : A → 2 be a mapping and (X,A, |=) a topological system. We say that yis a set (compact, finite, point, respectively) function in (X, A, |=) if there exists a set(a compact set, a finite set, a singleton, respectively) K ⊆ X such that y = wK . IfK = {p}, we write wK = w{p} = wp. The set containing all compact functions in(X,A, |=) we denote by C(X, A). For every mapping y : A → 2 we put

c(y) ={ {x|x ∈ X, (∀a ∈ A) : (y(a) = >) ⇒ (x |= a)} , if y 6= True,∅, if y = True.

Lemma 2.3. Let (X, A, |=) be a topological system and y ∈ 2A. Then the set c(y) issaturated in (X, A, |=).

Proof. Obviously, ∅ is saturated. Let c(y) 6= ∅. Suppose that z ∈ X r c(y). Thenthere is some a0 ∈ A such that y(a0) = > and z 6|= a0. Then for every x ∈ c(y) itfollows that x |= a0, which implies that c(y) ⊆ intX(a0) and z /∈ intX(a0). Therefore,c(y) =

⋂ {intX(a)| a ∈ A, c(y) ⊆ intX(a)}. ¤

Lemma 2.4. Let (X,A, |=) be a topological system and K ⊆ X be compact in (X, A, |=). Then wK = w↑K and c(wK) =↑K.

Proof. If a ∈ A, then wK(a) = > ⇔ K ⊆ intX(a) ⇔↑K ⊆ intX(a) ⇔ w↑K(a) = >.Hence wK = w↑K . Let x ∈ c(wK). It follows that if K ⊆ intX(a) for some a ∈ A,then x ∈ intX(a). Then x ∈ ⋂ {intX(a)|K ⊆ intX(a)} =↑K. Conversely, let x ∈↑K.If wK(a) = > for some a ∈ A, then K ⊆ intX(a), which implies that x ∈ intX(a) andso x |= a. Hence x ∈ c(wK). ¤

Lemma 2.5. Let (X, A, |=) be a topological system. The set C(X,A) is closed underfinite meets in 2A.

Proof. From Lemma 2.2 it follows that∧∅ = True = w∅ ∈ C(X, A). Let y, z ∈

C(X, A). Denote K = c(y) ∪ c(z) and w = wK . It follows that w(a) = > ⇔ K =c(y) ∪ c(z) ⊆ intX(a) ⇔ (y(a) = >) ∧ (z(a) = >) ⇔ (y ∧ z)(a) = >. It follows thatw = y ∧ z, which implies that y ∧ z ∈ C(X, A). ¤

We denote by Φd(X, A) the set of all compact saturated sets in a topological system(X,A,`) and by Σd(X,A) the set of all co-compact co-saturated sets in (X, A,`).Obviously, Φd(X,A) is a union semilattice and Σd(X, A) is an intersection semilattice,isomorphic to Φd(X, A).

Corollary 2.6. Let (X, A, |=) be a topological system. Then c : C(X, A) → Φd(X, A)is a semilattice isomorphism.

Proof. In the proof of Lemma 2.5 we proved that c(y ∧ z) = c(y) ∪ c(z) for y, z ∈C(X, A). From Lemma 2.4 it follows that c is a one-to-one. ¤

Let (X,A,`) be a topological system. We will say that x ∈ X is independent ony ∈ [C(X, A)] and write x |= y if there is some a ∈ A such that y(a) = > and x 0 a.

Theorem 2.7. Let (X, A,`) be a topological system. Then (X, [C(X,A)], |=) is atopological system which is a strict dualization of (X, A,`).

Proof. Firstly, will prove that (X, [C(X, A)], |=) forms a topological system. SinceFalse(a) = ⊥ for every a ∈ A, it follows x |= False =

∨∅ for no x ∈ X. Let x ∈ X.


Since x 0 ⊥ and True(⊥) = > it follows x |= True =∧∅. Let Y ⊆ [C(X, A)] be

non-empty. It follows that x |= ∨Y if and only if there exists a ∈ A such that x 0 a

and (∨

Y )(a) =∨

y∈Y y(a) = >. But this is true if and only if there is some a ∈ A andy ∈ Y , such that x 0 a and y(a) = >. This is equivalent to existence of some y ∈ Ysuch that x |= y. Let Z ⊆ [C(X,A)] be non-empty and finite. It follows that x |= ∧

Zif and only if there exists a ∈ A such that x 0 a and (

∧Z)(a) =

∧z∈Z z(a) = >. But

this holds if and only if there is some a ∈ A such that x 0 a and z(a) = > for everyz ∈ Z. This implies that x |= z for every z ∈ Z. Conversely, let x |= z for everyz ∈ Z. Then, for every z ∈ Z, there exists az ∈ A such that z(az) = > and x 0 az.We put a =

∨z∈Z az. Then z(a) = > and x 0 a for every z ∈ Z, which is equivalent

to x |= ∧Z. Therefore, (X, [C(X, A)], |=) is a topological system.

Now, we will show that [C(X, A)] induces the co-compact topology on X. We puts(U) = wXrU for every U ∈ Σd(X,A). Let U ∈ Σd(X,A) and x ∈ intX(s(U)).Then x |= w(XrU), which implies that there exists a ∈ A such that w(XrU)(a) = >and x 0 a. It follows that X r U ⊆ intX(a) and hence x /∈ X r U . Hence x ∈ U .Conversely, let x ∈ U . Then x /∈ X rU = K, where X rU is compact and saturated.Hence, there exists some a ∈ A such that x /∈ intX(a) and X r U ⊆ intX(a). Thens(U)(a) = w(XrU)(a) = > and x 0 a, which implies that x |= s(U) and so x ∈intX(s(U)). It follows that U = intX(s(U)) for every U ∈ Σd(X, A). But Σd(X, A)forms a base of the dual topology (τX(A))d, and hence (τX(A))d ⊆ τX([C(X, A)]). Lety ∈ [C(X, A)] and x ∈ intX(y). Then x |= y. It follows from Lemma 2.1 (ii) thatthere is some Y ⊆ C(X, A) such that y =

∨Y . Then x |= yx for some yx ∈ Y and

so x ∈ intX(yx). It follows that yx is a compact function, so there exist K compactand saturated in (X, A,`) such that yx = wK . Let U = X rK. Then U ∈ (τX(A))d

and s(U) = yx. If follows that x ∈ U = intX(s(U)) = intX(yx) ⊆ intX(y). Hence,intX(y) is a union of elements of (τX(A))d, and so (τX(A))d = τX([C(X,A)]). Now,the pair (f, e) = (idX , Spat) is the requested morphism of topological systems fromthe definition of the strict dualization. ¤

Let A be a poset. We denote by 〈A → 2〉 ⊆ 2A the set of all functions A → 2that preserve all directed joins and finite meets, whenever they exist. We also denote〈A → 2〉T = 〈A → 2〉 ∪ {True}. We write [A → 2] instead of [〈A → 2〉].Proposition 2.8. Let (X,A,`) be a topological system. The following statements areequivalent:

(i) C(X, A) = 〈A → 2〉T .(ii) Every element of 〈A → 2〉 is a compact function.(iii) Every element of 〈A → 2〉 is a set function.(iv) Every element of 〈A → 2〉 is a meet of point functions.

Proof. Obviously, (i) ⇒ (ii) ⇒ (iii). Suppose (iii). Let y ∈ 〈A → 2〉. There is someM ⊆ X such that y = wM . We will show that y =

∧p∈M wp. Suppose that y(a) = >

for some a ∈ A. Then M ⊆ intX(a), which implies that wp(a) = > for every p ∈ M .Hence, (

∧p∈M wp)(a) =

∧p∈M wp(a) = >. Conversely, let (

∧p∈M wp)(a) = >. Then

wp(a) = > and hence p ∈ intX(a) for every p ∈ M . It follows that M ⊆ intX(a),which implies that y(a) = wM (a) = >. It follows that y =

∧p∈M wp, so (iv) is true.

Suppose (iv). Let y ∈ 〈A → 2〉. There is some M ⊆ X such that y =∧

p∈M wp =wM . We will show that M is compact. Suppose that B ⊂ A is directed and M ⊆⋃

b∈B intX(b) = intX(∨

B). Then wM (∨

B) =∨

b∈B wM (b) = >. It follows that there


is some b1 ∈ B such that wM (b1) = >. Hence, M ⊆ intX(b1), which implies that Mis compact, so y = wM ∈ C(X,A). It follows (i). ¤

Let A be a frame. Recall that a map y : A → 2 is a frame morphism if y preservesall joins and all finite meets.

Lemma 2.9. Let A be a frame. Then every element y ∈ 〈A → 2〉 is a meet of framemorphisms.

Proof. We denote F = {u|u : A → 2 is a frame morphism, u ≥ y} and z =∧

F . Itfollows that z ≥ y. To show that z = y we will prove that if y(a) = ⊥ for some a ∈ A,then also z(a) = ⊥. Let us denote M = {b| b ∈ A, a ≤ b, y(b) = ⊥}. Let L ⊆ M bea linearly ordered chain. Then y(

∨L) =

∨b∈L y(b) = ⊥, so

∨L ∈ M and L has an

upper bound. Hence, by Zorn’s Lemma, M has some maximal element m ∈ M . Theny(m) = ⊥, which implies that m 6= > ∈ A.Let

u(b) ={ ⊥, if b ≤ m,>, otherwise.

Obviously, u(⊥) = ⊥ and u(>) = >. Let ∅ 6= B ⊆ A. Then u(∨

B) = ⊥ ⇔ ∨B ≤

m ⇔ (∀b ∈ B) : (b ≤ m) ⇔ (∀b ∈ B) : (u(b) = ⊥) ⇔ ∨b∈B u(b) = ⊥. Let C ⊆ A be

non-empty and finite. Then u(∧

C) ≤ ∧c∈C u(c), since u is monotone. Suppose that

u(∧

C) = ⊥ and u(c) = > for every c ∈ C. Then, for any c ∈ C, it follows that c � mand hence m < m ∨ c. Then > =

∧c∈C y(m ∨ c) = y(

∧c∈C(m ∨ c)) = y(m ∨ (

∧C)),

which implies that∧

C � m. This is a contradiction to u(∧

C) = ⊥. ¤

Theorem 2.10. Let (X, A,`) be a locale. Then (X, [A → 2], |=) is a strict (notnecessarily localic) dualization of (X, A,`).

Proof. The points of X are exactly the frame morphisms of A → 2, so they coincidewith the point functions. The theorem now follows from Theorem 2.7, Proposition 2.8and Lemma 2.9. ¤

Note that the points of the locale (X, A,`) are fully determined by the correspond-ing frame A, so we can identify (X,A,`) with A and the localic dualization of (X,A,`)we can similarly identify with the frame [A → 2].

Corollary 2.11. Let (X, A,`) be a locale. Then the iterated localic dualizations of(X,A,`) are represented by the sequence of frames [A → 2], [[A → 2] → 2], [[[A →2] → 2] → 2], . . . .

Acknowledgment. The author is thankful to all colleges who contributed byquestions, advice, references, ideas or inspiration to the informal discussion relatedto the topic in the two conferences – The Second Workshop on Formal Topology inVenice and the Workshop on Topology in Computer Science in New York, both in thespring of 2002. It was, in the alphabetic order, B. S. Burdick, A. Jung, R. Kopperman,J. D. Lawson, M. B. Smyth and others.

References

[Bu] Burdick, B. S., A note on iterated duals of certain topological spaces, preprint (2000), 1-8.

[G] de Groot J., An Isomorphism Principle in General Topology, Bull. Amer. Math. Soc. 73(1967), 465-467.


[GHSW] de Groot J., Herrlich H., Strecker G.E., Wattel E., Compactness as an Operator, Compo-sitio Mathematica 21, Fasc. 4 (1969), 349-375.

[GSW] de Groot J., Strecker G.E., Wattel E., Proceedings of the Second Prague Topological Sym-phosium, Prague, 1966, pp. 161-163.

[Kop] Kopperman, R., Assymetry and duality in topology, Topology and Appl. 66(1) (1995),1-39.

[Kov] Kovar, M. M., At most 4 topologies can arise from iterating the de Groot dual, Topologyand Appl. 130 (2003), 175-182.

[LM] Lawson J.D., Mislove M., Open problems in topology (van Mill J., Reed G. M., eds.),North-Holland, Amsterdam, 1990, pp. 349-372.

[Vi] Vickers S., Topology Via Logic, Cambridge University Press, Cambridge, 1989, pp. 200.

Department of Mathematics, Faculty of electrical engineering and communication,University of Technology, Technicka 8, Brno, 616 69, Czech Republic

E-mail address: [email protected]

a note on the topology generated by scott open

Documents