Extensions of closure spaces

@ Applied General Topology c Universidad Polit´ecnica de Valencia Volume 4, No. 2, 2003 pp. 223–241 Extensions of closure spaces D. Deses∗, A. De Groot-Van der Voorde, E. Lowen-Colebunders Dedicated to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. A closure space Xis a set endowed with a closure operator P(X)→ P(X), satisfying the usual topological axioms, except finite additivity. A T1closure extension Yof a closure space Xinduces a structure γon Xsatisfying the smallness axioms introduced by H. Herrlich [?], except the one on finite unions of collections. We’ll use the word seminearness for a smallness structure of this type, i.e. satisfying the conditions (S1),(S2),(S3) and (S5) from [?]. In this paper we show that every T1seminearness structure γon Xcan in fact be induced by aT1closure extension. This result is quite different from its topological counterpart which was treated by S.A. Naimpally and J.H.M. Whitfield in [?]. Also in the topological setting the existence of (strict) extensions satisfying higher separation conditions such as T2and T3has been completely characterized by means of concreteness, separatedness and regularity [?]. In the closure setting these conditions will appear to be too weak to ensure the existence of suitable (strict) extensions. In this paper we introduce stronger alternatives in order to present internal characterizations of the existence of (strict) T2or strict regular closure extensions. 2000 AMS Classification: 54A05, 54D35, 54E15, 54E17, 54D10. Keywords: closure space, seminearness, separation, regularity, (strict) extension, minimal small stack. 1. Introduction. The structures we will be dealing with, can be defined in various equivalent ways, from which we shall use frequently two particular descriptions, namely by small collections and by uniform covers. ∗The first author is ‘aspirant’ of the F.W.O.-Vlaanderen. 224 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders 1.1. Let Xbe a set. A nonempty collection A⊂P(X), not containing ∅ is said to be a stack if A∈ A whenever there exists B∈ A with B⊂A. If A⊂P(X) is an arbitrary collection of nonempty subsets then we put stack A={A⊂X|∃B∈ A :B⊂A} and sec A={A⊂X|∀B∈ A :A∩B6=∅} We write ˙xfor stack {{x}}. In [?] H. Herrlich considered the following smallness axioms, which can be expressed in terms of stacks in the following way. For γ⊂ P2(X), a collection of stacks, consider the conditions (S1) If A⊂Band A ∈ γthen B ∈ γ (S2) ∀x∈X: ˙x∈γ (S3) γ6=P2(X) (S4) If (A ∪ B)∈γthen A ∈ γor B ∈ γ (S5) If sec {cl A|A∈ A} ∈ γthen sec A ∈ γwhere cl A ={x∈X|sec {A, {x}} ∈ γ} A structure γsatisfying (S1),(S2),(S3) is called a prenearness structure, if (S4) is added γis a merotopic structure and if γsatisfies all five of the conditions then it is a nearness structure. We will not be dealing with axiom (S4) but we will assume that γsatisfies (S1), (S2), (S3) and (S5). Then γis called a seminearness structure and (X, γ) is a seminearness space. As in [?] a function f: (X, γ)→(X0, γ0) between seminearness spaces is said to be uniformly continuous if it preserves smallness in the sense that A ∈ γ⇒stack {f(A)|A∈ A} ∈ γ0 Seminearness spaces and uniformly continuous maps form a topological construct in the sense of [?]. We refer to the original papers [?], [?] for a systematic study of prenearness and nearness spaces. On the latter a selfcontained textbook ”Uniforme Ra¨ume” appeared [?]. Another textbook by G. Preuss [?] also contains an introduction to nearness spaces and to some of the more general structures such as prenearness and merotopic structures. In [?] however the latter are called seminearness spaces, so that the terminology used in [?] differs from the one we use here. 1.2. In [?] an equivalent way of describing the structure was presented in terms of uniform covers. If Xis a set then the following conditions on µ⊂ P2(X) are considered (U1) If U ≺ V and U ∈ µthen V ∈ µ(where ≺denotes the classical refinement relation) (U2) If U ∈ µthen Uis a cover of X (U3) ∅6=µ6=P2(X) (U4) If U ∈ µand V ∈ µthen {U∩V|U∈ U, V ∈ V} ∈ µ (U5) If U ∈ µthen {intµU|U∈ U} ∈ µwhere intµU={x∈X|{U, X − {x}} ∈ µ} The covers in µare called uniform covers. In our setting we will not be dealing with (U4), so our covering structures µsatisfy (U1), (U2), (U3) and Extensions of closure spaces 225 (U5). (X, µ) then forms an equivalent way for the description of a seminearness space and the translation between (X, γ) and (X, µ) is as usual: U ∈ µ⇐⇒ ∀A ∈ γ:U ∩ A 6=∅ A ∈ γ⇐⇒ ∀U ∈ µ:U ∩ A 6=∅ 1.3. Some of the examples we will construct in the last section of the paper, satisfy even stronger conditions than the seminearness axioms. A uniform space, described in terms of covers satisfies (U1), (U2), (U3), (U4) and the condition (U5’), saying that every uniform cover has a uniform star refinement, which is in fact stronger than (U5). If we leave out (U4), as we did before, and retain (U1), (U2), (U3) and (U5’) then we still have a (covering) seminearness space. In this case we will say that the seminearness space is a uniform seminearness space . A special case of this situation is the following. Let Xbe a set and let {Ui|i∈I} be any collection of partitions of Xthen U ∈ µ⇐⇒ ∃i∈I:Ui≺ U defines a (covering) uniform seminearness structure on X. It is said to be zero dimensional since it is generated by a collection of partitions. These structures are investigated in more detail in [?]. 1.4. A closure space (X, C) is a pair, where Xis a set and Cis a subset of the power set P(X) satisfying the conditions that Xbelongs to Cand that C is closed for arbitrary unions. The sets in Care called open sets. A function f: (X, C)→(Y, D) between closure spaces (X, C) and (Y, D) is said to be continuous if f−1(D)∈ C whenever D∈ D.Cl is the construct with closure spaces as objects and continuous maps as morphisms. Some isomorphic descriptions of Cl are often used f.i. by giving the collection of all closed sets (the so called Moore family [?]) where, as usual, the closed sets are the complements of the open ones and continuity is defined accordingly. Another isomorphic description is obtained by means of a closure operator [?]. The closure operation cl :P(X)→ P(X) associated with a closure space (X, C) is defined in the usual way by x∈cl A ⇐⇒ (∀C∈ C :x∈C⇒C∩A6=∅) where A⊂X and x∈X. This closure need not be finitely additive, but it does satisfy the conditions cl ∅=∅,(A⊂B⇒cl A ⊂cl B), A ⊂cl A and cl(cl A) = cl A whenever Aand Bare subsets of X. Continuity is then characterized in the usual way. Finally closure spaces can also be equivalently described by means of neighborhood collections of the points. These neighborhood collections satisfy the usual axioms, except for the fact that the collections need not to be filters. So in a closure space the neighborhood collection V(x) of a point xis a stack, where every V∈ V(x) contains xand V(x) satisfies the open kernel condition. In the sequel we will just write Xfor a closure space and we’ll choose the most convenient form for its explicit structure. 226 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders Motivations for considering closure spaces can be found in several applications. We refer to [?] and [?] for applications in geometry, to [?] for applications in lattice theory, to [?], [?] and [?] for the use of closures in the development of representations of physical systems, to [?] for the use in social sciences and to [?] for applications in the context of knowledge representation. The introduction of [?] contains some more details on motivation. A closure space satisfies the R0symmetry axiom if x∈cl {y} ⇐⇒ y∈cl {x}, ∀x, y ∈Xand it satisfies T1if {x}is closed for every x∈X. If (X, γ) is a seminearness space then the closure defined in paragraph 1.1 by cl A ={x∈X|sec {A, {x}} ∈ γ} is an R0closure in our sense. This closure is the underlying closure of (X, γ) and we also say that it is compatible with (X, γ). Whenever we consider neighborhood collections Vγ(x), convergence or open sets for a seminearness space (X, γ), we are in fact referring to the underlying closure. As for nearness spaces we have in this more general context that the neighborhood collections Vγ(x) are minimal small stacks (where minimality refers to the inclusion order). Next we further illustrate the relation between R0closure spaces and seminearness spaces. 1.5. Let Ybe an R0closure space and define a seminearness structure by A ∈ γ⇐⇒ ∃y∈Y:V(y)⊂ A Remark that the underlying closure of the seminearness γcoincides with the given closure on Y. The construct of R0closure spaces is bicoreflectively embedded in the construct of seminearness spaces, cfr. [?], [?]. 1.6. Let Ybe an R0closure space and let Xbe a subset of Y. The closure structure of Yinduces a seminearness structure on Xas follows A ∈ γ⇐⇒ ∃y∈Y:V(y)⊂stackYA The underlying closure of γon Xcoincides with the closure structure induced by Yon X. If Xis dense in Y(clYX=Y) then Yis said to be a closure extension of Xand we say that (X, γ) is induced by the extension Yof X.Xis said to be strictly dense in Yif {clYB|B⊂X}is a base for the closed subsets of Y, in the sense that every closed set of Ycan be obtained by intersecting sets from the base. In that case Yis said to be a strict extension of Xand (X, γ) is said to be induced by a strict extension. The meaning of 1.6 is that, given an R0closure extension Yof a closure space X, a seminearness structure γis induced on Xwhich is compatible with the given closure on X. The first question we will be dealing with in this paper is, whether every seminearness γcompatible with Xas a closure space, can be induced by some R0closure extension Yof X. Extensions of closure spaces 227 The parallel question in the setting of topological spaces is whether every compatible nearness space can be induced by some R0topological extension. This question was answered negatively by S.A. Naimpally and J.H.M. Whitfield in [?]. A thorough study on extensions of topological spaces was later carried out by H.L. Bentley and H. Herrlich in [?], in particular giving internal characterizations for nearness spaces to be induced by T1,T2or T3(strict) extensions. In this paper we’ll deal with the closure counterparts of such questions. 2. Extensions. In this section, starting from a seminearness space (X, γ) we construct two types of enlargements, one type are the so called ”loose” enlargements and the other type is a strict one. 2.1. Construction of a loose enlargement. Let (X, γ) be a seminearness space and let {yA|A ∈ α}be a collection of points, not belonging to Xand in one to one correspondence to a collection αof nonconvergent small stacks with an open base. Let X0=X∪ {yA|A ∈ α}. On X0we define a closure structure cl0by determining the neighborhood collections of the points as follows: V0(x) = stackX0Vγ(x) for x∈X V0(yA) = stackX0A ∩ ˙yAfor A ∈ α Clearly (X0, cl0) is an R0closure space of which Xis a dense subset. Moreover if Dis a stack on Xand stackX0Dconverges in X0, then Dis small in (X, γ). It is clear that in order to obtain an extension of (X, γ) in the sense of 1.6 the condition D ∈ γ⇒ ∃A ∈ α:A⊂Dhas to be fulfilled. The following proposition is relevant in this respect since it shows that in fact openbased stacks determine the structure. Proposition 2.1. If (X, γ)is a seminearness space then for every A ∈ γthere exists a B ∈ γsuch that Bhas an open base and such that B ⊂ A. Proof. For A ∈ γlet B=stack {B⊂X|Bopen, B ∈ A}. If U ∈ µthen so is {int U|U∈ U}. So finally B ∩ U 6=∅. Proposition 2.2. Let (X, γ)be a seminearness space and let α={A ∈ P2(X)|A is a small openbased nonconvergent stack} Then (X0, γ0)is an R0closure extension of (X, γ). We conclude from this fact that every seminearness space can be induced by an R0closure extension. Remark that by exactly the same construction one has that every T1seminearness space is induced by some T1extension. Remark also that these results deviate from their well known topological counterparts, cfr. [?], [?], [?]. In general, given a T1space (X, γ) there can be many different T1extensions inducing γ. On the other hand, as we will see, strict T1extensions need not exist. However, if there exist T1strict extensions, then they are essentially 228 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders unique. The reason for this is explained in the next results on minimal small stacks, where again minimality refers to the inclusion order on stacks. The following result is quite parallel to its topological counterpart developed in [?]. Proposition 2.3. (1) If (X, γ)is a seminearness space induced by an R0extension Yof X, then every minimal small stack is a trace V(y)|Xfor some y∈Y. (2) If (X, γ)is a seminearness space induced by a strict R0extension Yof Xthen {V(y)|X|y∈Y} is the collection of all minimal small stacks. Proof. (1) If Mis minimal small and stackYMconverges to y∈Ythen V(y)|X⊂ Mand hence V(y)|X=M. (2) Let y∈Ybe some point of a strict extension Y, and suppose that A⊂V(y)|Xand that Ais small. Suppose that stackYAconverges to z, i.e. V(z)|X⊂ A. It follows that z∈clY{y}. Indeed otherwise there would exist a subset B⊂Xsuch that y∈clYBand z6∈ clYB. And this is impossible. Hence we can conclude that V(y) = V(z) and so A=V(y)|X.  Corollary 2.4. If (X, γ)is a seminearness space and Yis a strict T1extension then the points of Yare in one to one correspondence to the minimal small stacks in γ. Also the following construction is quite similar to its topological counterpart [?],[?]. 2.2. Construction of a strict enlargement. Let (X, γ) be a seminearness space and let ˆ X=X∪ {yM|M nonconvergent minimal small} where again different points yMare chosen to be outside of Xand in one to one correspondence with the minimal small nonconvergent stacks. For A⊂Xput O(A) = int A ∪ {yM|A∈ M} and let ˆ cl on ˆ Xbe the closure having as an open base {O(A)|A⊂X}. Remark that {ˆ cl K|K⊂X}is a base for the closed sets where ˆ cl K = cl K ∪ {yM|K∈sec M}. Clearly ( ˆ X, ˆ cl) is an R0closure space and it contains Xas a strictly dense subset. Moreover if Dis a stack on Xand stack ˆ XDconverges in ˆ Xthen: Either Vˆ X(yM)⊂stack Dfor Mminimal small and not convergent, then we Extensions of closure spaces 229 have M ⊂ D. Or Vˆ X(x)⊂stack Dand then VX(x)⊂ D. So in any case D ∈ γ. In order to obtain an extension of (X, γ) we need to impose the following condition (cfr. [?]). Definition 2.5. A seminearness space is concrete if the minimal small stacks determine the structure in the following sense ∀A ∈ γ:∃M minimal small M ⊂ A Proposition 2.6. (X, γ)is induced by a strict R0extension if and only if it is a concrete seminearness space. Proof. If (X, γ) is a concrete seminearness space we make the construction developed in paragraph 2.2 and we prove that ( ˆ X, ˆ cl) is an extension. So it remains to show that if Dis a small stack on Xit converges in ( ˆ X, ˆ cl). Choose M ⊂ D minimal small. Either M=V(x) for some x∈Xand then for A⊂Xwith x∈int A we have int A ∈ D and hence O(A)∈ D. So we have Vˆ X(x)⊂ D. Or Mdoes not converge. Then we prove that stack ˆ XD ⊃ V ˆ X(yM) Let A⊂Xsuch that yM∈O(A), i.e. A∈ M. In view of proposition 2.1 the stack Mhas an open base. Then also int A ∈ M and finally int A ∈ D. So again we can conclude that O(A)∈stack ˆ XD. Conversely, suppose that (X, γ) has a strict R0extension (Y, clY). Let D be small in (X, γ) then stackYDconverges to some y∈Y. Then clearly D ⊃ VY(y)|Xand in view of proposition 2.3 we have that VY(y)|Xis a minimal small stack. Hence (X, γ) is concrete.  Remark that using exactly the same construction one has that (X, γ) is induced by a strict T1closure extension if and only if it is T1and concrete. Remark that if Yis a strict T1extension of (X, γ) then Yis unique up to an isomorphism leaving Xpointwise fixed. It can easily be seen that the function φ:Y→ˆ Xmapping y∈Yto yMwith M=VY(y)|Xif y6∈ Xand mapping x∈Xto x, is bijective and satisfies φ(clYB) = ˆ cl B, for every B⊂X. Therefore we also have φ(clYZ) = ˆ cl(φ(Z)) for every Z⊂Y. The previous results on R0and T1strict extensions are completely analogous to their topological counterparts. In the next section, where higher separation is considered, the parallelism with the topological situation does not go through. 3. Separation and extensions. In this section we introduce higher separation conditions for seminearness spaces. The notion ”separatedness” was introduced in [?] in the setting of prenearness spaces and it proved to be very useful in the study of topological extensions. However, in our setting, in order to produce Hausdorff closure 230 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders extensions, ”separatedness” will no longer be strong enough. We briefly recall some definitions and results. If (X, γ) is a seminearness space then a stack Ais said to be near if sec Ais small. For instance, if TA∈A clγA6=∅then Ais near. A stack Ais said to be concentrated if it is small and near. For example, the neighborhood collections in (X, γ) are concentrated. Small filters are also always concentrated. Definition 3.1. [?] A seminearness space (X, γ) is separated if for every concentrated stack Aalso B={B⊂X|stackX{B}∪Anear} is near. The proof of the following proposition is similar to the one of proposition 10.5 in [?] and can be found in [?]. Proposition 3.2. For a seminearness (X, γ)the following are equivalent (1) (X, γ)is separated (2) Every concentrated stack contains a unique minimal small stack Proposition 3.3. If (X, γ)is separated, Mis a minimal small concentrated stack and M 6=V(x)then ∃A∈ V(x) : ∃M∈ M :M∩A=∅ Proof. If on the other hand every A∈ V(x) intersects every M∈ M then V(x)∪ M would be concentrated and then V(x) = Min view of the previous proposition.  Corollary 3.4. If (X, γ)is separated and T1then in the underlying closure, distinct points have disjoint neighborhoods. In particular a closure space (considered as a seminearness space) is separated and T1if and only if distinct points have disjoint neighborhoods. We use the label ”Hausdorff” or T2for this property. The following conditions (i) and (ii) clearly are strengthening those formulated in proposition 3.3. Proposition 3.5. For a seminearness space (X, γ)the following are equivalent (i) for Mand Nminimal small concentrated stacks, M 6=Nthen ∃M∈ M :∃N∈ N :N∩M=∅ (ii) for Mand Nminimal small stacks, M 6=Nthen ∃M∈ M :∃N∈ N :N∩M=∅ Proof. That (i) implies (ii) follows from the observation that when Nis small but not concentrated, then sec Nis not small. So if Mis small we have M 6⊂ sec N. Therefore Mand Ncontain disjoint sets.  Definition 3.6. A seminearness space (X, γ) satisfies (S) if it fulfills one (and hence both) of the conditions formulated in proposition 3.5 Extensions of closure spaces 231 Remark that if (X, γ) satisfies (S) and Mand Nare different minimal small stacks, M∈ M and N∈ N satisfying (ii) can be taken to be disjoint and open. Hence in that case we have ∀x∈X:M6∈ V(x) or N6∈ V(x) Next we generalize these ideas in order to introduce an even stronger separation condition. Let (X, γ) be a seminearness space and let Σ = {˙x|x∈X} ∪ {M| minimal small nonconvergent} Definition 3.7. Subsets Dand Bare said to be γ-disjoint if ∀P ∈ Σ : D6∈ P or B6∈ P Clearly Dand Bare γ-disjoint if and only if (i) D∩B=∅ (ii) for every Mminimal small nonconvergent stack, D6∈ M or B6∈ M. Definition 3.8. A seminearness space (X, γ) is said to satisfy (T) if minimal small stacks M 6=Ncontain sets M∈ M and N∈ N that are γ-disjoint. Conditions (S) and (T) will play an important role in the investigation of Hausdorff closure extensions. First we discuss the relation between the various separation conditions. Proposition 3.9. (1) In a seminearness space we have (T) ⇒(S) (2) In a concrete seminearness space we have (T) ⇒(S) ⇒separated Proof. (1) Let Mand Nbe (concentrated) minimal small, choose γ-disjoint sets M∈ M and N∈ N . Then we have M∩N=∅. (2) (X, γ) is concrete and satisfies (S). Let Abe concentrated and let M be a minimal small stack, M ⊂ A. Consider sec Awhich is small and a minimal small stack N ⊂ sec A. It follows that M=N. Finally by proposition 3.2 the space (X, γ) is separated.  From the proof of (2) we immediately have the following. Corollary 3.10. If (X, γ)is concrete and satisfies (S) then every minimal small concentrated stack Msatisfies M ⊂ sec M i.e. Mis a linked system in the sense of [?]. 238 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders R Regular Separated T S Uniform 5.7 5.7 5.8 5.6 5.6 5.8 5.8 5.7 Figure 2. Implications for concrete seminearness spaces 5.4. For concrete seminearness spaces we proved the implications in figure ??. Again no other implications hold except for those obtained by transitivity. The numbers refer to the examples presented below and these are counterexamples for the reversed arrows. 5.5. For general seminearness spaces the diagram is as in figure ??. R Separated T Regular S Uniform Figure 3. Implications for seminearness spaces As a counterexample showing that (S) does not imply separated in the nonconcrete case, we again refer to example 3 in [?]. 5.6. A concrete and separated T1seminearness space which does not satisfy (S). Therefore it is nonregular, nonuniform and satisfies neither (R) nor (T). Let X=R2and define M=stack {pr−1 1(c)|c∈R} N=stack {pr−1 2(c)|c∈R} Extensions of closure spaces 239 For a stack Awe define A ∈ γ⇐⇒ M ⊂ A or N ⊂ A or ˙x⊂ A for some x∈X The underlying closure is the discrete one. Clearly every small stack contains a unique minimal small stack, but Mand Ndo not contain disjoint sets and so (X, γ) does not satisfy (S). 5.7. A concrete T1seminearness space which is uniform (and even zerodimensional) and so it is regular and satisfies (S) and is separated. However it satisfies neither (T) nor (R). Let A, B, C be three pairwise disjoint sets with more than one point and X=A∪B∪C. In order to define γon Xconsider the following stacks M=stack {A, B} N=stack {B, C} P=stack {A, C} Define a stack Ato be small if and only if M ⊂ A or N ⊂ A or P ⊂ A or ˙x⊂ A for some x∈X Then the pointfilters ˙xfor x∈Xare the only concentrated minimal stacks, and the other minimal stacks M,N,Pare not concentrated. (X, γ) does not satisfy (T) since for instance for Mand Nneither of the disjoint sets Aand B,Aand Cor Band Care γ-disjoint. It follows that (X, γ) does not satisfy (R). However (X, γ) is uniform since its collection µof uniform covers is generated by the following collection µ0={U1,U2,U3}of partitions U1={A, B} ∪ {{x}|x∈C} U2={B, C} ∪ {{x}|x∈A} U3={A, C} ∪ {{x}|x∈B} It follows that (X, γ) is regular, also separated and satisfies (S). So the strict extension ( ˆ X, ˆ cl) is a T1extension that is not Hausdorff and not regular. Remark however that the loose extension constructed by adding different points for the minimal small stacks that do not converge, is regular and T1. 5.8. A concrete T1seminearness space which is uniform (and even zerodimensional) and therefore is regular. It satisfies (T) and hence also (S) and it is separated. However it does not satisfy (R). Let A, A0, P, P 0, Q and Q0be pairwise disjoint sets with more than one point and let X=A∪A0∪P∪P0∪Q∪Q0 240 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders In order to define γon Xconsider the following stacks A=stack {A, A0} P=stack {P, P 0, A} Q=stack {Q, Q0, A} B=stack {P, Q} Define a stack Sto be small if and only if A⊂S,P ⊂ S,Q ⊂ S,B ⊂ S or ˙x⊂ S for some x∈X Clearly (X, γ) is concrete and T1. Use the fact that A0, P 0, Q0are sets belonging to just one nonconvergent minimal small stack to see that (X, γ) satisfies (T). Again (X, γ) is uniform and in fact (X, µ) is generated by a collection of partitions. So (X, γ) is regular. However (R) is not satisfied. Let’s concentrate on Aand consider A∈ A. The sets Pand Aare not γ-disjoint since they both belong to P. Also Qand Aare not γ-disjoint. It follows that for U={A}∪{D|Dand A γ-disjoint} we have U ∩B =∅. So U 6∈ µand therefore A <6<µA. Clearly this implies that A<<µ6∈ γ. It follows that the strict extension ( ˆ X, ˆ cl) of (X, γ) is a Hausdorff closure space that is not regular. Remark however that the loose extension constructed by adding different points for the minimal small stacks that do not converge, is a regular T1extension. References [1] J. Ad´amek, H. Herrlich and G. Strecker, Abstract and concrete categories, Wiley and sons, New York (1990). [2] D. Aerts, Foundations of quantum physics: a general realistic and operational approach, Internat. J. Theoret. Phys., 38(1) (1999), 289-358. [3] G. Aumann, Kontaktrelationen, Bayer. Akad. Wiss. Math. - Nat. Kl. Sitzungber. (1970), 67-77. [4] M.K. Bennett, Affine and projective geometry, Wiley and Sons, New York (1995). [5] H.L. Bentley, Nearness spaces and extensions of topological spaces, Studies in Topology, Academic Press (1975), 47-66. [6] H.L. Bentley and H. Herrlich, Extensions of topological spaces, in Topology, Proc. Memphis State Univ. Conf., Marcel Decker (1976), 129-184. [7] H.L. Bentley, H. Herrlich, and E. Lowen-Colebunders, Convergence, J. of Pure and Appl. Algebra, 68 (1990), 27-45. [8] H.L. Bentley and E. Lowen-Colebunders, Completely regular spaces, Commentat. Math. Univ. Carol., 32(1) (1991), 129-153. [9] G. Birkhoff, Lattice Theory, American Mathematical Society, Providence, Rhode Island (1940). [10] V. Claes, E. Lowen-Colebunders and G. Sonck, Cartesian closed topological hull of the construct of closure spaces, Theory Appl. Categ., 8(2001), 481-489. [11] D. Deses and E. Lowen-Colebunders, On completeness in a non-Archimedean setting, via firm reflections, accepted for publication, Bulletin Belgian Math. Soc. Extensions of closure spaces 241 [12] M. Ern´e, Lattice representations for categories of closure spaces, Categorical topology, Heldermann Verlag, Berlin (1984), 197-222. [13] C.A. Faure and A. Fr¨olicher, Modern projective geometry, Kluwer Academic Publishers, Dordrecht (2000). [14] B. Ganter and R. Wille, Formal Concept Analysis, Springer Verlag, Berlin (1998). [15] J. de Groot, G.A. Jensen and A. Verbeek, Superextensions, Math. Centrum, Amsterdam (1968). [16] A. De Groot - Van der Voorde, Separation axioms in extension theory for closure spaces and their relevance to state property systems, PhD thesis, Vrije Universiteit Brussel, July 2001. [17] H. Herrlich, A concept of nearness, Gen. Topol. Appl., 4(1974), 191-212. [18] H. Herrlich, Topological structures, Math. Centre Tracts, 52 (1974), 59-122. [19] H. Herrlich, Topologie II: Uniforme R¨aume, Heldermann Verlag, Berlin (1987). [20] S.A. Naimpally and J.H.M. Whitfield, Not every near family is contained in a clan, Proc. of the Amer. Math. Soc., 47 (1975), 237-238. [21] D.J. Moore, Categories of representations of physical systems, Helv. Phys. Acta, 68 (1995), 658-678. [22] C. Piron, Recent developments in quantum mechanics, Helv. Phys. Acta, 62 (1989), 82-90. [23] G. Preuss, Theory of Topological Structures, D. Reidel Publishing Company, Dordrecht (1989). Received November 2001 Revised October 2002 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders Department of Mathematics, Vrije Universiteit Brussel, 1050 Brussels, Belgium E-mail address:[email protected], [email protected], [email protected]