Contact Algebras and Region-based Theory of ... - Semantic Scholar

Fundamenta Informaticae 74 (2006) 251–282

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IOS Press

Contact Algebras and Region-based Theory of Space: Proximity Approach – II Georgi Dimov∗ and Dimiter Vakarelov∗

†

Department of Mathematics and Computer Science University of Sofia Blvd. James Bourchier 5, 1126 Sofia, Bulgaria [email protected] [email protected]

Abstract. This paper is the second part of the paper [2]. Both of them are in the field of region-based (or Whitehedian) theory of space, which is an important subfield of Qualitative Spatial Reasoning (QSR). The paper can be considered also as an application of abstract algebra and topology to some problems arising and motivated in Theoretical Computer Science and QSR. In [2], different axiomatizations for region-based theory of space were given. The most general one was introduced under the name “Contact Algebra”. In this paper some categories defined in the language of contact algebras are introduced. It is shown that they are equivalent to the category of all semiregular T0 -spaces and their continuous maps and to its full subcategories having as objects all regular (respectively, completely regular; compact; locally compact) Hausdorff spaces. An algorithm for a direct construction of all, up to homeomorphism, finite semiregular T0 -spaces of rank n is found. An example of an RCC model which has no regular Hausdorff representation space is presented. The main method of investigation in both parts is a lattice-theoretic generalization of methods and constructions from the theory of proximity spaces. Proximity models for various kinds of contact algebras are given here. In this way, the paper can be regarded as a full realization of the proximity approach to the region-based theory of space. Keywords: Qualitative Spatial Reasoning, mereological relations, contact relations, contact algebras, region-based theories of space, RCC models, equivalent categories, semiregular Ti -spaces (i = 0, 1, 2), weakly regular spaces, N-regular spaces, regular spaces, OCE-regular spaces, finite semiregular T0 -spaces, compact (locally compact, completely regular) Hausdorff spaces, proximity spaces, proximity models. ∗

This paper was supported by the project NIP-1510 “Applied Logics and Topological Structures” of the Bulgarian Ministry of Education and Science. † Address for correspondence: Department of Mathematics and Computer Science, University of Sofia, Blvd. James Bourchier 5, 1126 Sofia, Bulgaria

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1. Introduction This paper is the second part of the paper [2]. The description of the contents of the both parts was given in [2]. In this part we present proximity models for all kinds of contact algebras introduced in [2] (see Section 1) and we give region-based formulations of different categories of topological spaces (including the category of all semiregular T0 -spaces and their continuous maps and its full subcategories having as objects all regular Hausdorff spaces or some generalizations of them (like weakly regular spaces (introduced in [4]) or N-regular spaces (introduced here)), all compact Hausdorff spaces or all locally compact Hausdorff spaces, all Tychonoff spaces or the connected versions of all classes of spaces mentioned here) by proving some equivalence theorems (see Theorems 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7 and 3.8). The paper is organized as follows. In Section 1 we introduce different classes of proximity spaces in order to obtain proximity models of all kinds of contact algebras which we investigate in the first part of this paper. In Section 2, using the different kinds of points from Section 2 of [2] and the language of contact algebras, we introduce the notions of a C-system, T-system, Ti -system, where i = 0, 1, 2, WR-system, WR1-system, WR2-system, NR-system, NR2-system, R-system, TI-system, Tych-system and the notion of an s-map between such systems. Then we prove that the category SK of all semiregular T0 -spaces and their continuous maps is equivalent to the category KS of all complete T0 -systems and all s-maps between them. Equivalence theorems are obtained also for the pairs of the full subcategories of the categories SK and KS having as objects respectively all semiregular T1 -spaces and all complete T1 -systems, all semiregular T2 -spaces and all complete T2 -systems, all weakly regular Ti -spaces (i = 0, 1, 2) and all complete WR-systems (WR1-systems, WR2-systems), all N-regular Ti -spaces (i = 1, 2) and all complete NR-systems (NR2-systems), all regular Hausdorff spaces and all complete R-systems, all compact Hausdorff spaces and all complete normal contact algebras, all locally compact Hausdorff spaces and all complete local contact algebras, all Tychonoff spaces and all Tych-systems, all C-semiregular spaces and all complete contact algebras, all CM-semiregular spaces and all complete contact algebras satisfying T1 Axiom, all C-weakly regular spaces and all complete extensional contact algebras, all CN-regular T1 -spaces and all complete extensional contact algebras satisfying N-regularity Axiom, all OCE-regular spaces and all complete extensional contact algebras satisfying Regularity Axiom. The connected versions of these theorems are proved as well. As a corollary, an algorithm for a direct construction of all different finite semiregular T0 -spaces of rank n, where n ∈ IN+ , is obtained. Finally, in Section 3 we present an example of an RCC model B (i.e., of a connected extensional contact algebra) for which there is no regular Hausdorff space (X, τ ) such that B can be densely embedded in the standard contact algebra of all regular closed subsets of X. With this we answer a question posed by D¨untsch and Winter in [4]. In this paper we use the notations introduced in the first part (see [2]), as well as the notions and the result from it. For all notions and notations which are not defined here, see [2, 1, 5, 10, 12].

2. Proximity Models Recall that a binary relation α defined on the power set of a set X is called a Leader or LE-proximity on X if it satisfies the following conditions: (LE1) A α (B ∪ C) iff A α B or A α C, and (A ∪ B) α C iff A α C or B α C;

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(LE2) A α B implies A 6= ∅ and B 6= ∅; (LE3) A α B and {b} α C for each b ∈ B together imply A α C; (LE4) A ∩ B 6= ∅ implies A α B. When A and B are not in the relation α, we will write A(−α)B. For notational convenience, we will write A ≺ B for A(−α)(X \ B), x α A for {x} α A and x ≺ A for {x} ≺ A, when x ∈ X. If in addition α satisfies (SYM) A α B iff B α A, then α is called a Lodato or LO-proximity. A LE-proximity α on X is said to be separated iff it satisfies the additional condition (SEP) For all x, y ∈ X, x α y implies x = y. The binary relation α is called an Efremoviˇc proximity if it satisfies axioms (LE1), (LE2), (LE4), (SYM) and the following one: (EFR) If A, B ⊆ X and A ≺ B then there exists a subset C of X such that A ≺ C ≺ B. An Efremoviˇc proximity that satisfies the axiom (SEP) is called an EF-proximity. It is well known that every Efremoviˇc proximity is a LO-proximity. Let α be a LE-proximity on a set X. For each A ⊆ X, define Aα = {x ∈ X : x α A}. This defines a Kuratowski closure operator on the set X. Thus α yields a topology τ = τ (α) on the set X. If T is a topology on the set X, we say that T and α are compatible if T = τ (α); in this case we will also say that α is a proximity on the topological space (X, T). As it follows from (LE3), A α B iff A α B α

(1)

(see, e.g., [10]). The following fact is well known (see, e.g., [10]): Fact 2.1. Every topological space (X, τ ) has a compatible LE-proximity α given by the formula A αB iff A ∩ cl(X,τ ) (B) 6= ∅.

(2)

Definition 2.1. A LE-proximity α on a set X which satisfies the following condition (CSYM) If U and V are two open subsets of the topological space (X, τ (α)) and cl(U ) α cl(V ) then cl(V ) α cl(U ), is called a CA-proximity. The pair (X, α), where X is a set and α is a CA-proximity on the set X, is referred to as a CA-proximity space. Obviously, every Lodato proximity is a CA-proximity. Proposition 2.1. Every topological space (X, τ ) has a compatible CA-proximity α given by the formula (2).

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Proof: This follows easily from Fact 2.1.

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Proposition 2.2. Let (X, α) be a CA-proximity space. Then (RC(X, τ (α)), α|RC(X) ) is a CA. Proof: The proof is straightforward.

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Proposition 2.3. For every CA B = (B, C) there exists a CA-proximity space (X, α) such that B is CA-isomorphic to a dense subalgebra of the CA (RC(X), α|RC(X) ) and (X, τ (α)) is a C-semiregular space. Proof: According to [2, Theorem 5.5], every contact algebra B = (B, C) can be densely embedded in a contact algebra (RC(X, τ ), CX ), where (X, τ ) is a C-semiregular space. By Proposition 2.1, there exists a CA-proximity α on the set X compatible with τ , where α is given by the formula (2). Since, for F, G ∈ RC(X, τ ), F α G iff F ∩ G 6= ∅ iff F CX G, the CA B is densely embedded in the CA (RC(X), α|RC(X) ). u t Remark 2.1. It seems naturally to express Proposition 2.3 in the following form: “Every CA has a CAproximity model”. Many other results similar to Proposition 2.3 will be proved bellow. We meant exactly these results when we spoke about “proximity models” in the Abstract and in the Introduction. Definition 2.2. Let (X, α) be a CA-proximity space. An open filter F on the space (X, τ (α)) will be called an e-filter on (X, α) if the conditions cl(X,τ (α)) (U )(−α)cl(X,τ (α)) (V ) and F meets U imply that X \ cl(X,τ (α)) (V ) ∈ F, where U and V are open subsets of (X, τ (α)). An e-filter is said to be minimal if it is minimal with respect to the inclusion. A CA-proximity α on a set X is called a TCA-proximity if it satisfies the following condition: (TCAP) If U is an open non-empty subset of the space (X, τ (α)) then there exists a minimal e-filter in (X, α) containing cl(X,τ (α)) (U ). If α is a TCA-proximity on a set X then the pair (X, α) will be called a TCA-proximity space. Proposition 2.4. Let (X, α) be a CA-proximity space. If F is an e-filter in (X, α) then F0 = F ∩ RC(X, τ (α)) is an E-filter in the contact algebra (RC(X, τ (α)), α|RC(X) ). Conversely, if F0 is an Efilter in the contact algebra (RC(X, τ (α)), α|RC(X) ) then the family {Int(X,τ (α)) (F ) : F ∈ F0 } has the finite intersection property and generates an e-filter F in (X, α) such that F0 = F ∩ RC(X, τ (α)). Proof: The proof is straightforward.

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Corollary 2.1. Let (X, α) be a CA-proximity space. If F is a minimal e-filter in (X, α) then F0 = F ∩ RC(X, τ (α)) is a minimal E-filter in (RC(X, τ (α)), α|RC(X) ). Conversely, if F0 is a minimal E-filter in the contact algebra (RC(X, τ (α)), α|RC(X) ) then the filter F in X generated by the family {Int(X,τ (α)) (F ) : F ∈ F0 } is a minimal e-filter in (X, α) and F0 = F ∩ RC(X, τ (α)).

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Proposition 2.5. If (X, α) is a TCA-proximity space then the contact algebra (RC(X, τ (α)), α|RC(X) ) is a TCA. Proof: This follows from Proposition 2.2, Corollary 2.1 and [2, Fact 3.3(ii)].

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Proposition 2.6. If (X, τ ) is a TE-semiregular space then (X, τ ) has a compatible TCA-proximity α given by the formula (2). Proof: This follows easily from Proposition 2.1, [2, Proposition 4.14], [2, Fact 3.3(ii)] and Corollary 2.1.

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Proposition 2.7. For every complete TCA B there exists a TCA-proximity space (X, α) such that B is CA-isomorphic to the CA (RC(X), α|RC(X) ) and (X, τ (α)) is a CM-semiregular space. For every TCA B there exists a CA-proximity space (X, α) such that B is CA-isomorphic to a dense subalgebra of the CA (RC(X), α|RC(X) ) and (X, τ (α)) is C-semiregular. Proof: By [2, Theorem 5.1], every complete TCA B is isomorphic to the contact algebra (RC(X, τ ), CX ), where (X, τ ) is a CM-semiregular space. Then, by [2, Corollary 4.2], (X, τ ) is a TE-semiregular space. Hence, Proposition 2.6 implies that there exists a compatible TCA-proximity α on (X, τ ) given by the formula (2). Since α and CX coincide on RC(X, τ ), we obtain that B is isomorphic to the contact algebra (RC(X, τ (α)), α|RC(X) ). The second assertion in our proposition follows from [2,Theorem 5.5] and Proposition 2.1. In this case the contact algebra (RC(X), α|RC(X) ) is not obliged to be a TCA — it is a CA and some subalgebra of it is a TCA. u t The next proposition is very similar to Theorem 19.10 from [10] and follows easily from the proof of this theorem. Proposition 2.8. A topological space (X, τ ) is a T1 -space iff it has a compatible separated Lodato proximity αw given by the formula A αw B iff cl(X,τ ) (A) ∩ cl(X,τ ) (B) 6= ∅.

(3)

Definition 2.3. A separated Lodato proximity α on a set X will be called an ECA-proximity if it satisfies the following condition: (ECAP) For every non-empty open subset U of the space (X, τ (α)) there exists an open non-empty subset V of (X, τ (α)) such that V ≺ U . If α is an ECA-proximity on a set X, the pair (X, α) will be called an ECA-proximity space. Remark 2.2. It is easy to see that the axiom (ECAP) in Definition 2.3 is equivalent to the following one: (ECAP0 ) If x ∈ X, A ⊆ X and x ≺ A then there exist y ∈ X and B ⊆ X such that y ≺ B ≺ A. Proposition 2.9. (RC(X, τ (α)), α|RC(X) ) is an ECA when (X, α) is an ECA-proximity space.

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Proof: This follows easily from Proposition 2.2, [2, Lemma 2.2(iv)], and (1).

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Proposition 2.10. Every weakly regular T1 -space (X, τ ) has a compatible ECA-proximity αw given by the formula (3). Proof: By Proposition 2.8, αw is a compatible separated Lodato proximity on (X, τ ). Let U ∈ τ and U 6= ∅. Since X is weakly regular, there exists an open non-empty subset V of X such that cl(V ) ⊆ U . Hence cl(V ) ∩ (X \ U ) = ∅, i.e. V (−αw )(X \ U ). Thus V ≺ U . So, αw is an ECA-proximity. u t Proposition 2.11. For every ECA B there exists an ECA-proximity space (X, α) such that B is CAisomorphic to a dense subalgebra of the ECA (RC(X), α|RC(X) ) and (X, τ (α)) is a C-weakly regular space. Proof: This follows easily from [2, Theorem 5.5] and Proposition 2.10, following the line of the proof of Proposition 2.3. u t Definition 2.4. A separated Lodato proximity α on a set X will be called an N-proximity if it satisfies the following condition: (NP) If x ∈ X, A ⊆ X and x ≺ A then there exists an open subset V of (X, τ (α)) such that x α V and V ≺ A. If α is an N-proximity on a set X, the pair (X, α) will be called an N-proximity space. Obviously, every N-proximity is an ECA-proximity. Definition 2.5. Let (X, α) be an N-proximity space. An open filter F in (X, τ (α)) is called a regular co-cluster in (X, α) if: (i) For every F ∈ F there exists an U ∈ τ (α) such that U ≺ F and X \ U 6∈ F, (ii) For every U, V ∈ τ (α), the conditions F meets U and U (−α)V imply that X \cl(X,τ (α)) (V ) ∈ F. Proposition 2.12. Let (X, α) be an N-proximity space. Let us put B = (RC(X, τ (α)), α|RC(X) ). Then B is a CA. If F is a regular co-cluster in (X, α) then F0 = F ∩ RC(X, τ (α)) is a co-cluster in B. Conversely, if F0 is a co-cluster in B then the family {Int(F ) : F ∈ F0 } has the finite intersection property and generates a regular co-cluster F in (X, α) such that F0 = F ∩ RC(X, τ (α)). Proof: (⇒) Let F be a regular co-cluster in (X, α). By Proposition 2.2, B is a CA. We will show that F0 = F ∩ RC(X) is a co-cluster in B. Let F, G ∈ F0 . Then F ∩ G ∈ F and since F is open filter, there is a V ∈ F ∩ τ (α) such that V ⊆ F ∩ G. Hence Int(F ∩ G) ∈ F and thus F.G ∈ F. So, F.G ∈ F0 . Then, obviously, F0 is a filter in the CA (RC(X), α|RC(X) ). Let F, G ∈ RC(X) and F (−α)G. Suppose that F ∗ (= cl(X \ F )) 6∈ F0 . Let U = Int(F ) and V = Int(G). We will show that F meets U . Indeed, suppose that there is an H ∈ F such that U ∩ H = ∅.

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Since F is an open filter, we obtain that W = Int(H) ∈ F. Since U ∩ W = ∅, we have that F ∩ W = ∅. Thus W ⊆ X \ F ⊆ F ∗ and hence F ∗ ∈ F0 — a contradiction. Since U (−α)V , Definition 2.5(ii) implies that X \ cl(V ) ∈ F, i.e. X \ G ∈ F. Thus cl(X \ G) = G∗ ∈ F0 . Therefore, F0 is an E-filter in (RC(X), α|RC(X) ). Finally, we have to show that F0 satisfies condition (Co-clust) (see [2, Definition 3.3]). Let F ∈ F0 . Then F ∈ F and, by Definition 2.5(i), there exists an U ∈ τ (α) such that U ≺ F and X \ U 6∈ F. Obviously, G = cl(U ) ∈ RC(X) and G∗ = cl(X \ G) ⊆ X \ U . Hence G 6∈ F, i.e. G 6∈ F0 . Since U ≺ F , (1) implies that G ≺ F , i.e. G(−α)X \ F . Then, using (1) again, we conclude that G(−α)cl(X \ F ), i.e. G  F . So, F0 is a co-cluster in (RC(X), α|RC(X) ). (⇐) Let F0 be a co-cluster in the CA (RC(X), α|RC(X) ). Then F0 is a filter and hence, for every F, G ∈ F0 , F.G 6= ∅; thus Int(F ) ∩ Int(G) 6= ∅. Therefore, the family ϕ = {Int(F ) : F ∈ F0 } has the finite intersection property. So, it generates a filter F in the set X. Obviously, F is an open filter in (X, τ (α)) and F ∩ RC(X) = F0 . So, we need only to show that F satisfies conditions (i) and (ii) from Definition 2.5. For checking condition (i), let A ∈ F. Then there exists a V ∈ ϕ such that V ⊆ A. Clearly, F = cl(V ) ∈ F0 . Since F0 is a co-cluster in the CA (RC(X), α|RC(X) ), there exists a G ∈ RC(X) such that G  F and G∗ 6∈ F0 . Hence G 6= ∅ and G(−α)(X \ V ) (because F ∗ = X \ V and G(−α)F ∗ ). Let U = Int(G). Then U ≺ V and thus U ≺ A. Since X \ U = G∗ , we obtain that X \ U 6∈ F. So, (i) is satisfied. For checking (ii), let U, V ∈ τ (α), U (−α)V and F meets U . Set F = cl(U ) and G = cl(V ). Then F, G ∈ RC(X) and F (−α)G (by (1)). Since F0 is an E-filter, we get that either F ∗ ∈ F0 or G∗ ∈ F0 . Suppose that F ∗ ∈ F0 . Then Int(F ∗ ) ∈ ϕ ⊆ F. Since Int(F ∗ ) = X \ F , we get that X \ F ∈ F, which is a contradiction because U ∩ (X \ F ) = ∅ and U meets F. So, G∗ ∈ F0 . Then Int(G∗ ) ∈ F and hence X \ cl(V ) ∈ F. Therefore, (ii) is satisfied. u t Definition 2.6. An N-proximity on a set X is called an NCA-proximity if for every U, V ∈ τ (α) such that U α V there is a regular co-cluster in (X, α) which meets both U and V . If α is an NCA-proximity on a set X, the pair (X, α) will be called an NCA-proximity space. Proposition 2.13. Every N-regular T1 -space (X, τ ) has a compatible NCA-proximity αw given by the formula (3). Proof: By Proposition 2.8, αw is a compatible separated Lodato proximity on (X, τ ). We will show that αw satisfies condition (NP) from Definition 2.4. Let x ≺ A. Then x ∈ Int(A). Hence, by N-regularity of (X, τ ), there is an F ∈ RC(X) such that x ∈ F and F ⊆ Int(A). Let V = Int(F ). Then F = cl(V ) and hence x αw V . Further, cl(V ) ⊆ Int(A) implies that V (−αw )A. So, condition (NP) is satisfied. Thus αw is an N-proximity on (X, τ ). Let us check that it is an NCA-proximity. Let U, V ∈ τ (= τ (αw )) and U αw V . Then there exists a point x ∈ cl(U ) ∩ cl(V ). By [2, Lemma 4.1] and [2, Proposition 4.1(i)], we have that νx is a co-cluster in (RC(X), CX ) = (RC(X), αw |RC(X) ). Let F be the regular co-cluster in (X, αw ) generated by νx (see Proposition 2.12). Then x ∈ Int(A), for every A ∈ F. This implies that F meets U and V . u t Proposition 2.14. (RC(X, τ (α)), α|RC(X) ) is an NECA when (X, α) is an NCA-proximity space.

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Proof: By Definition 2.4 and Proposition 2.9, (RC(X), α|RC(X) ) is an ECA. For showing that N-regularity Axiom (see [2, Definition 3.6]) is fulfilled, let F, G ∈ RC(X) and F α G. Put U = Int(F ) and V = Int(G). Then, by (1), U α V . Since U, V ∈ τ (α), there exists a regular co-cluster F in (X, α) which meets both U and V . Then, by Proposition 2.12, F0 = F ∩ RC(X) is a co-cluster in the CA (RC(X), α|RC(X) ). Hence σ = c(F0 ) is a cluster in (RC(X), α|RC(X) ) (see [2, Lemma 3.2(i)]). We will show that F, G ∈ σ. Suppose that F 6∈ σ. Then F ∗ ∈ F0 . Thus F ∗ ∈ F. Since F is an open filter, we get that Int(F ∗ ) ∈ F, i.e. X \ F ∈ F. This is a contradiction because F meets U . So, F ∈ σ. Analogously, we get that G ∈ σ. Therefore, (RC(X), α|RC(X) ) is an NECA. u t Proposition 2.15. For every NECA B there exists an NCA-proximity space (X, α) such that B is CAisomorphic to a dense subalgebra of the NECA (RC(X), α|RC(X) ) and (X, τ (α)) is a CN-regular T1 space. Proof: By [2, Theorem 5.5], there exists a CN-regular T1 -space (X, τ ) such that B is isomorphic to a dense subalgebra of the NECA (RC(X), CX ) (see also [2, Proposition 4.7]). Now Proposition 2.13 implies that αw , given by (3), is a compatible NCA-proximity on (X, τ ). Obviously CX coincides with αw |RC(X) . u t Hence B is densely embedded in the NECA (RC(X), αw |RC(X) ). Recall (see [15]) that a binary relation α defined on the power set of a set X is called a basic proximity on X iff it satisfies conditions (LE1), (LE2), (LE4) and (SYM). A basic proximity α is called an Rproximity ([7]) if it satisfies condition (SEP) and the following one: (RP) If x ∈ X, A ⊆ X and x ≺ A then there exists a B ⊆ X such that x ≺ B ≺ A. ˇ Let α be a basic proximity on a set X. Then Aα defines a Cech closure operator on the set X. Thus α α yields a topology τ = τ (α) = {X \ A : A = A } on the set X ([15]). When α is an R-proximity, Aα defines a Kuratowski closure operator on the set X ([7]). An R-proximity α which satisfies condition (LE3) is called an LR-proximity ([7]); in other words, a separated Lodato proximity satisfying condition (RP) is an LR-proximity. Let α be a basic proximity on a set X. A filter F in X is round if for any A ∈ F there exists a B ∈ F such that B ≺ A. A round filter F is a round end ([11]) if, for U, V ∈ τ (α), F meets U and cl(U )(−α)cl(V ) imply X \ cl(V ) ∈ F. A c-proximity α on a set X is an R-proximity satisfying the property that if U, V ∈ τ (α) and cl(U ) α cl(V ) then some round end in (X, α) meets both U and V ([11]). When α is a basic proximity (resp., R-proximity; LR-proximity; c-proximity), the pair (X, α) is called a basic proximity space (resp., R-proximity space; LR-proximity space; c-proximity space). Proposition 2.16. Let (X, α) be an LR-proximity space. Let us put B = (RC(X, τ (α)), α|RC(X) ). Then B is a CA. If F is a round end in (X, α) then F0 = F ∩ RC(X, τ (α)) is an end in B. Conversely, if F0 is an end in B then it generates a round end F in (X, α) such that F0 = F ∩ RC(X, τ (α)). Proof: (⇒) Let F be a round end in (X, α). Put F0 = F ∩ RC(X). Let F, G ∈ F0 . Then F ∩ G ∈ F. Hence, there exists some H ∈ F such that H ≺ F ∩G. Then H ⊆ Int(F ∩G). Thus F.G = cl(Int(F ∩G)) ∈ F0 .

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Now it is obvious that F0 is a filter in (RC(X), α|RC(X) ). It is a round filter because if F ∈ F0 then there exist G, H ∈ F such that H ≺ G ≺ F and hence H ⊆ Int(G); thus, by (1), G0 = cl(Int(G)) ∈ F0 and G0 ≺ F ; using again (1), we get that G0  F (i.e. G0 (−α)F ∗ ). Finally, we have to check that condition (E-fil) (see [2, Definition 3.2]) is fulfilled. Let F, G ∈ RC(X) and F (−α)G. Suppose that F ∗ 6∈ F0 . Let U = Int(F ). We will show that F meets U . Indeed, if there exists an H ∈ F such that U ∩ H = ∅ then we can find an H 0 ∈ F with H 0 ≺ H; this will imply that V = Int(H) ∈ F and hence U ∩ V = ∅; thus F ∩ H 0 = ∅, i.e. H 0 ⊆ F ∗ ; hence F ∗ ∈ F0 , which is a contradiction. So, F meets U . Since G = cl(Int(G)) and F (−α)G, we conclude that X \ G ∈ F. Therefore, G∗ ∈ F0 . So, F0 is an end in (RC(X), α|RC(X) ). (⇐) Let F0 be an end in (RC(X), α|RC(X) ). Then F0 has the finite intersection property and hence it generates a filter F in X. Obviously, F is a round filter in (X, α) and F∩RC(X) = F0 . Let U, V ∈ τ (α), F meets U and cl(U )(−α)cl(V ). Put F = cl(U ) and G = cl(V ). Then F, G ∈ RC(X) and F (−α)G. Hence either F ∗ ∈ F0 or G∗ ∈ F0 . Since U ∩ F ∗ = ∅, we get that F ∗ 6∈ F0 . Therefore, G∗ ∈ F0 . There exists an H ∈ F0 such that H  G∗ . Then H(−α)G and hence H ⊆ X \ G. Thus X \ cl(V ) ∈ F. So, F is a round end in (X, α). u t Definition 2.7. A Lodato c-proximity on a set X is called an RCA-proximity. When α is an RCAproximity on a set X, the pair (X, α) is called an RCA-proximity space. Proposition 2.17. For every regular T1 -space (X, τ ), the proximity αw , given by the formula (3), is an RCA-proximity on the space (X, τ ). Proof: By Proposition 2.8, αw is a separated Lodato proximity on the space (X, τ ). As it is shown in [7], αw is an R-proximity on X. Let us check that it is an RCA-proximity. Let U, V ∈ τ (= τ (αw )) and U αw V (equivalently, cl(U ) αw cl(V )). Then there exists a point x ∈ cl(U ) ∩ cl(V ). By [2, Lemma 4.2, Proposition 4.1(i), Lemma Lemma 3.2(iv)], we have that νx is an end in (RC(X), CX ) = (RC(X), αw |RC(X) ). Let F be the round end in (X, αw ) generated by νx (see Proposition 2.16). Then x ∈ Int(A), for every A ∈ F. This implies that F meets U and V . u t Proposition 2.18. (RC(X, τ (α)), α|RC(X) ) is an RECA when (X, α) is an RCA-proximity space. Proof: Since, obviously, every RCA-proximity is an ECA-proximity, Proposition 2.9 implies that the contact algebra (RC(X), α|RC(X) ) is an ECA. For showing that Regularity Axiom (see [2, Definition 3.6]) is fulfilled, let F, G ∈ RC(X) and F α G. Put U = Int(F ) and V = Int(G). Then F = cl(U ) and G = cl(V ). Since U, V ∈ τ (α), there exists a round end F in (X, α) which meets both U and V . Then, by Proposition 2.16, F0 = F ∩ RC(X) is an end in the CA (RC(X), α|RC(X) ). Hence σ = c(F0 ) is a co-end in (RC(X), α|RC(X) ) (see [2, Lemma 3.2(iv)]). We will show that F, G ∈ σ. Suppose that F 6∈ σ. Then F ∗ ∈ F0 . Thus F ∗ ∈ F. Since F is a round filter, it is an open filter and we get that Int(F ∗ ) ∈ F, i.e. X \ F ∈ F. This is a contradiction because F meets U . So, F ∈ σ. Analogously, we get that G ∈ σ. Therefore, (RC(X), α|RC(X) ) is an RECA. u t Proposition 2.19. For every RECA B there exists an RCA-proximity space (X, α) such that B is CAisomorphic to a dense subalgebra of the RECA (RC(X), α|RC(X) ) and (X, τ (α)) is OCE-regular.

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Proof: By [2, Theorem 5.5], there exists an OCE-regular space (X, τ ) such that B is isomorphic to a dense subalgebra of the RECA (RC(X), CX ) (see also [2, Proposition 4.10]). Now Proposition 2.17 implies that αw , given by (3), is a compatible RCA-proximity on (X, τ ). Obviously CX coincides with αw |RC(X) . Hence B is densely embedded in the RECA (RC(X), αw |RC(X) ). u t Theorem 2.1. Let (X, τ ) be a T1 -space, αw be the separated Lodato proximity on (X, τ ), given by the formula (3) (see Proposition 2.8), and Y be a dense subspace of X. Let δ be the restriction of αw on Exp(Y ). Then δ is a separated LO-proximity on Y compatible with the topology τ |Y and if (X, τ ) is a weakly regular (resp., N-regular; regular) space then (Y, δ) is an ECA-proximity space (resp., NCA-proximity space; RCA-proximity space), the contact algebra (RC(Y ), δ|RC(Y ) ) is an ECA (resp., NECA; RECA) and it is isomorphic to (RC(X), αw |RC(X) )(= (RC(X), CX )). Proof: Obviously, δ is a separated LO-proximity on the set Y . Since Aδ = {y ∈ Y : y δ A} = {y ∈ Y : y αw A} = Y ∩ Aαw , δ is compatible with the topology τ |Y (see Proposition 2.8). Let A, B ⊆ Y . Then we have that A δ B iff A αw B iff clX (A) ∩ clX (B) 6= ∅. (a) Let X be a weakly regular space and U be a non-empty open subset of Y . Then there exists an open subset U 0 of X such that U = U 0 ∩ Y . Since U 0 is non-empty and X is weakly regular, there exists a non-empty open subset V 0 of X with clX (V 0 ) ⊆ U 0 . Let V = V 0 ∩ Y . Then V is a non-empty open subset of Y and V ≺δ U . Indeed, we have that clX (V ) = clX (V 0 ) ⊆ U 0 and U 0 ∩clX (Y \U ) = ∅. Hence clX (V ) ∩ clX (Y \ U ) = ∅ and hence V (−δ)Y \ U , i.e. V ≺δ U . So, δ is an ECA-proximity on (Y, τ |Y ). The rest of the assertion follows from Proposition 2.9 and the fact that the map f : RC(Y ) −→ RC(X), defined by f (A) = clX (A), is a Boolean isomorphism (see [2, Lemma 4.3]). (b) Let X be an N-regular space. We will first show that δ is an N-proximity. Let A ⊆ Y and x ≺δ A. Then x ∈ IntY (A). There exists an open subset U 0 of X such that IntY (A) = Y ∩ U 0 . Then x ∈ U 0 and since X is N-regular, there exists an F ∈ RC(X) such that x ∈ F ⊆ U 0 . Let V 0 = IntX (F ) and V = Y ∩ V 0 . Then, obviously, x δ V . Since U 0 ∩ clX (Y \ A) = ∅ and F ⊆ U 0 , we obtain that clX (V ) ∩ clX (Y \ A) = ∅, i.e. V ≺δ A. So, δ is an N-proximity on (Y, τ |Y ). We will now show that δ is an NCA-proximity. Let U and V be open subsets of Y and U δ V . Then there exist open in X subsets U 0 and V 0 such that U = Y ∩ U 0 and V = Y ∩ V 0 . Obviously, U 0 αw V 0 . Since X is N-regular, Proposition 2.13 implies that there is a regular co-cluster F0 in (X, αw ) which meets both U 0 and V 0 . Let F = F0 ∩ Y (= {F ∩ Y : F ∈ F0 }). Then F meets both U and V . We need only to show that F is a regular co-cluster in (Y, δ). Let F ∈ F. Then there exists an F 0 ∈ F0 such that F = Y ∩ F 0 . Since F0 is a regular co-cluster in (X, αw ), there exists a W 0 ∈ τ such that W 0 ≺αw F and X \ W 0 6∈ F0 . Let W = Y ∩ W 0 . Then W is open in Y and W ≺δ F since clX (W ) ∩ clX (Y \ F ) ⊆ clX (W 0 ) ∩ clX (X \ F 0 ) = ∅. Suppose that Y \ W ∈ F. Then there exists a G ∈ F0 such that G ∩ Y = Y \ W . Obviously, G ∩ Y ⊆ X \ W 0 . Since F0 is an open filter, IntX (G) ∈ F0 . We have that clX (IntX (G)) = clX (Y ∩ IntX (G)) ⊆ X \ W 0 . Thus X \ W 0 ∈ F0 , which is a contradiction. So, Y \ W 6∈ F. Let U1 and V1 be open in Y , F meets U1 and U1 (−δ)V1 . Then there exist U10 , V10 ∈ τ such that 0 U1 ∩ Y = U1 and V10 ∩ Y = V1 . Obviously, U10 (−αw )V10 and F0 meets U10 . Since F0 is a regular co-cluster in (X, αw ), we obtain that X \ clX (V10 ) ∈ F0 . Hence Y \ clY (V1 ) ∈ F. So, F is a regular co-cluster in (Y, δ). Therefore, δ is an NCA-proximity in Y .

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The last part of the assertion follows from Proposition 2.14 (see also the end of the proof of (a)). (c) Let X be a regular space. Then, as it follows from Lemma 4.15(b) of [11], δ is a c-proximity on Y . Since it is LO-proximity, we obtain that δ is an RCA-proximity on Y . The last part of the assertion follows from Proposition 2.18 (see also the end of the proof of (a)). u t Finally, let us recall the following three assertions from [14]: Proposition 2.20. ([14]) If (X, α) is an EF-proximity space then the pair (RC(X, τ (α)), α|RC(X) ) is an IECA. Proposition 2.21. ([14]) For every IECA B there exists a compact EF-proximity space (X, α) such that B is CA-isomorphic to a dense subalgebra of the IECA (RC(X), α|RC(X) ). Proposition 2.22. ([14]) Let (X, τ ) be a compact T2 -space, αw be the EF-proximity on (X, τ ), given by the formula (3) (see Proposition 2.8), and Y be a dense subspace of X. Let δ be the restriction of αw on Exp(Y ). Then δ is a separated EF-proximity on Y compatible with the topology τ |Y , the contact algebra (RC(Y ), δ|RC(Y ) ) is an IECA and is isomorphic to (RC(X), αw |RC(X) ) = (RC(X), CX ). Definition 2.8. ([9]) A local proximity space (X, β, B) consists of a set X, a basic proximity β on X, and a boundedness B in X subject to the following axioms: (a) If A ∈ B, C ⊆ X and A  C (where  is defined with respect to β) then there exists a B ∈ B such that A  B  C; (b) If AβC, then there is a B ∈ B such that B ⊆ C and AβB. When β is separated, then (X, β, B) is said to be a separated local proximity space. Note that (a) is equivalent to the following axiom: (a0 ) Let A ⊆ X and B ∈ B. If for every C ∈ B either AβC or (X \ C)βB, then AβB. Two local proximity spaces (X1 , β1 , B1 ) and (X2 , β2 , B2 ) are said to be isomorphic if there exists a bijection between X1 and X2 which preserves in both directions the bounded sets and proximity relations. Let (X, τ ) be a topological space and (X, β, B) be a local proximity space. We will say that (X, β, B) is a local proximity space on (X, τ ) iff τ (β) = τ . Example 2.1. ([14]) Let (X, β, B) be a separated local proximity space. Then the triple (RC(X), β, B∩ RC(X)) is a local contact algebra. Proposition 2.23. ([9]) Let X be a locally compact Hausdorff space. Then the triple (X, βX , BX ), where BX = {F ⊆ X : clX (F ) is compact in X} and, for A, B ⊆ X, AαX B iff clX (A) ∩ clX (B) 6= ∅, is a separated local proximity space on the space X. ¯ there exists a separated local proximity space (X, β, B) such that Proposition 2.24. For every LCA B B is LCA-isomorphic to a dense subalgebra of the LCA (RC(X), β, B ∩ RC(X)). Proof: This follows from Proposition 2.23, Example 2.1, [2, Fact 2.3] and [2, Theorem 5.4(a)].

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3. Equivalence Theorems Definition 3.1. A pair (B, X), where B is a CA and X ⊆ CLANS(B), is called a C-system. We will often regard C-systems (B, X) which satisfy some of the following conditions: (INJ) If a ∈ B and a 6= 1 then there exists a G ∈ X such that a 6∈ G; (OCE) If a, b ∈ B and aCb then there exists G ∈ X such that a, b ∈ G; (T1S) If G, H ∈ X and G ⊆ H then G = H; (T2S) If G, H ∈ X and G 6= H then there exist a 6∈ G and b 6∈ H such that a + b = 1. Lemma 3.1. Let (B, X) be a C-system. Then the function gX : B −→ Exp(X), defined by gX(a) = X ∩ gB (a), (see [2] for gB ), is an injection iff X satisfies condition (INJ). Proof: Let X satisfy condition (INJ) and let gX(a) = X. Suppose that a 6= 1. Then there exists a G ∈ X such that a 6∈ G. Thus G 6∈ gX(a), which is a contradiction. So, a = 1. Now, arguing as in the proof of [2, Lemma 5.1(iv)], we obtain that gX is an injection. The converse direction is obvious. u t Definition 3.2. Let (B, X) be a C-system. If X satisfies condition (INJ) (see Definition 3.1) then the pair (B, X) is called a T-system. Corollary 3.1. Let (B, X) be a T-system, a, b ∈ B and a 6= b. Then there exists a G ∈ X which contains exactly one of the elements a and b of B. Lemma 3.2. Let (B, X) be a C-system and X = CLANS(B). Then: (a) X is dense in the space (X, τB ) (see [2, Definition 5.1] for τB ) iff (B, X) is a T-system; (b) If X is open combinatorially embedded in the space (X, τB ) then X satisfies condition (OCE). Proof: (a) Let b ∈ B. Then, by [2, Lemma 5.7], X \ g(b) 6= ∅ (see [2] for g) iff b 6= 1. Hence, X is dense in X ⇐⇒ for every b 6= 1 there exists a G ∈ X ∩ (X \ g(b)) ⇐⇒ X satisfies condition (INJ). (b) Let a, b ∈ B and aCb. Put F = g(a) and G = g(b). By [2, Lemma 5.7(i1)] and [2, Lemma 5.3(ii)], F, G ∈ RC(X) and F ∩ G 6= ∅. Since X is dense in (X, τB ), U = X ∩ IntX (F ) and V = X ∩ IntX (G) are open non-empty subsets of X. Obviously, clX (U ) = F and clX (V ) = G. Hence, if we suppose that clX(U ) ∩ clX(V ) = ∅ then we will obtain that F ∩ G = ∅, which is a contradiction. Thus clX(U ) ∩ clX(V ) 6= ∅, i.e. F ∩ G ∩ X 6= ∅. Therefore, there exists a G ∈ X such that a, b ∈ G. u t Definition 3.3. Let (B, X) be a T-system. The pair (B, X) is called a T0 -system if X satisfies condition (OCE) (see Lemma 3.2). A T0 -system (resp., T-system) (B, X) is said to be: (a) a complete T0 -system (resp., complete T-system) if B is a complete Boolean algebra; (b) a connected T0 -system (resp., connected T-system) if B is a connected CA. We will now introduce some notations. Let (Z, T) be a topological space. The family {σz : z ∈ Z} (see [2] for σz ) will be denoted by XZ ; B Z will stay for the CA (RC(Z), CZ ); the map κZ : Z −→ XZ is defined by κZ (z) = σz .

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Example 3.1. 1. Let B be a CA. Then (B, CLANS(B)) is a T0 -system (see [2, Proposition 3.3(i)], Lemma 3.1 and [2, Lemma 5.7(i1)]). 2. Let Z be a topological space. Then, obviously, (B Z , XZ ) is a complete T0 -system (see Proposition [2, 4.1(ii)]). Lemma 3.3. Let (B, X) be a complete T0 -system. Then X is open combinatorially embedded in the space (CLANS(B), τB ) (see [2, Definition 5.1] for τB ). Proof: Put X = CLANS(B). By Lemma 3.2(a), X is dense in (X, τB ). Let U and V be open subsets of X and clX(U ) ∩ clX(V ) = ∅. Suppose that clX (U ) ∩ clX (V ) 6= ∅. Put F = clX (U ) and G = clX (V ). Then F, G ∈ RC(X). Hence, by [2, Lemma 5.7(i1)] and [2, Lemma 5.3(vi)], there exist a, b ∈ B such that aCb and g(a) = F , g(b) = G. Now the condition (OCE) implies that there exists a G ∈ X such that a, b ∈ G. Thus G ∈ X ∩ F ∩ G. Therefore, clX(U ) ∩ clX(V ) 6= ∅, which is a contradiction. So, clX (U ) ∩ clX (V ) = ∅. We have proved that X is open combinatorially embedded in (X, τB ). u t Proposition 3.1. Let (B, X) be a complete T0 -system. Then gX (see Lemma 3.1 for it) is a CAisomorphism between contact algebras B and (RC(X), CX). Proof: Let X = CLANS(B). By [2, Theorem 5.1(i)], g : B −→ (RC(X, τB ), CX ) is a CA-isomorphism. The map r : (RC(X), CX ) −→ (RC(X), CX) defined by r(F ) = F ∩ X is also a CA-isomorphism (by Lemma 3.3 and [2, Proposition 4.12]). Since gX = r ◦ g, we obtain that gX : B −→ (RC(X), CX) is a CA-isomorphism. u t Corollary 3.2. Let (B, X) be a complete connected T0 -system. Then the space (X, τB |X) is connected. Proof: It follows from Proposition 3.1 and [2, Fact 2.2].

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Lemma 3.4. Let B be a CA, X = CLANS(B) and X be a subspace of (X, τB ) (see [2, Definition 5.1] for τB ). Then: (a) (X, τB |X) is a T0 -space; (b) (X, τB |X) is a T1 -space iff X satisfies condition (T1S). Proof: (a) By [2, Lemma 5.7(i2)], the space (X, τB ) is a T0 -space. Hence, its subspace X is also a T0 -space. (b) We have that {X T ∩ g(b) : b ∈ B} is a closed base T of X. Let now X satisfy condition (T1S). Take an G ∈ X. Then G ∈ {X ∩ g(b) : b ∈ G}. Let H ∈ {X ∩ g(b) : b ∈ G}. Then G ⊆ H. Hence H = G. T So, G = {X ∩ g(b) : b ∈ G}. Therefore, {G} is a closed set in X. Thus XTis a T1 -space. {X ∩ g(b) : b ∈ G}. Since Conversely, let X be a TT 1 -space, G, H ∈ X and G ⊆ H. Then G = G ⊆ H, we obtain that H ∈ {X ∩ g(b) : b ∈ G}. Hence G = H. u t Lemma 3.5. Let B be a CA, X = CLANS(B) and X be a subspace of (X, τB ) (see [2, Definition 5.1] for τB ). Then: (a) (X, τB |X) is a T2 -space if X satisfies condition (T2S); (b) If (X, τB |X) is a T2 -space and X is dense in X then X satisfies condition (T2S).

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Proof: (a) Let G, H ∈ X and G 6= H. Then there exist a 6∈ G and b 6∈ H such that a + b = 1. Let U = X \ g(a) and V = X \ g(b). Then U and V are open in X, G ∈ U and H ∈ V . Further, by [2, Lemma 5.1(i)], U ∩ V = X \ (g(a) ∪ g(b)) = X \ g(a + b) = ∅. Hence, X is a T2 -space. (b) Let G, H ∈ X and G 6= H. Then there exist a, b ∈ B such that G ∈ X \ g(a) = U , H ∈ X \ g(b) = V and U ∩ V = ∅. Thus X = g(a + b) ∩ X. Then X = clX (X) ⊆ g(a + b) and hence, by [2, Lemma 5.7(i1)], a + b = 1. So, X satisfies condition (T2S). u t Definition 3.4. Let B = (B, C) be a CA. A T0 -system (B, X) is called: (i) a T1 -system if X satisfies condition (T1S) (see Lemma 3.4). (ii) a T2 -system if X satisfies condition (T2S) (see Lemma 3.5). Any such system is said to be complete if the Boolean algebra B is complete. Example 3.2. 1. Let B be a TCA. Then (B, MAX.CLANS(B)) is a T1 -system (see [2, Proposition 3.3(i)]). 2. Let Z be a semiregular T1 -space. Then, obviously, (B Z , XZ ) is a complete T1 -system (see [2, Proposition 4.1(ii)] and [2, Proposition 4.2(ii)]). 3. Let B be a RECA. Then (B, CO − ENDS(B)) is a T2 -system (see [2, Theorem 5.3(i)], Lemma 3.1 and [2, Lemma 5.7(v1)]). 4. Let Z be a semiregular T2 -space. Then (B Z , XZ ) is a complete T2 -system. Fact 3.1. Every T2 -system (B, X) is a T1 -system. Proof: Let G, H ∈ X and G ⊆ H. Suppose that G 6= H. Then, by (T2S), there exist a 6∈ G and b 6∈ H such that a + b = 1. Since G is a grill, we obtain that b ∈ G. Hence b ∈ H — a contradiction. Therefore, G = H. So, X satisfies condition (T1S). u t Lemma 3.6. Let B be a CA, X = CLANS(B), X be a subspace of (X, τB ) (see [2, Definition 5.1] for τB ) and X satisfies condition (INJ). Then: (a) If B is an ECA then X is a weakly regular T0 -space; (b) If X is a weakly regular space and (B, X) is a T0 -system then B is an ECA; (c) If X ⊆ CLUSTERS(B) then X is an N-regular T1 -space; (d) If X is an N-regular space and (B, X) is a complete T0 -system then X ⊆ CLUSTERS(B); (e) If X ⊆ CO − ENDS(B) then X is a regular T2 -space; (f) If (B, X) is a T0 -system and X is a regular space then the elements of X are co-ends in B. Proof: Since X satisfies condition (INJ), X is a dense subset of (X, τB ) (see Lemma 3.2(a)). Hence {X ∩ g(b) : b ∈ B} is a closed base of X contained in RC(X) (see [2, Lemma 5.3(ii)] and [2, Lemma 5.7(i1)]); thus X is a semiregular T0 -space (see Lemma 3.4). (a) Take a non-empty basic open set U = X \ g(a). Then, by Lemma 3.1, a 6= 1. Hence a∗ 6= 0. Since B is an ECA, there exists a b∗ ∈ B such that b∗ 6= 0 and b∗  a∗ (see [2, Lemma 2.1(iv)]). Put V = X \ g(b). Since b 6= 1 and X satisfies condition (INJ), V is a non-empty open subset of X. We

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have that clX(V ) = clX(X \ g(b)) = g(b∗ ) ∩ X (because X is dense in X). Since b∗ (−C)a, [2, Lemma 5.7(i1)] implies that g(b∗ ) ∩ g(a) = ∅. Then g(b∗ ) ∩ X ∩ g(a) = ∅. Hence clX(V ) ⊆ U . So, X is weakly regular. (b) Let a 6= 0. Then, by Lemma 3.1, gX(a∗ ) 6= X. So, X \ g(a∗ ) is a non-empty open subset of X. Since X is weakly regular, there exists a b ∈ B such that b∗ 6= 1 and clX(X \ g(b∗ ) ⊆ X \ g(a∗ ). Then X ∩ g(b) ∩ g(a∗ ) = ∅, because X is dense in X. Hence, by (OCE), b(−C)a∗ . So, b 6= 0 and b  a. Thus, by [2, Lemma 2.2(iv)], B is an ECA. (c) Let G ∈ X \ g(a∗ ). Then a∗ 6∈ G. Hence, by [2, Lemma 3.2(v)], there exists a b ∈ G such that b  a. Then [2, Lemma 5.7(i1)] implies that g(a∗ ) ∩ g(b) = ∅. Thus g(a∗ ) ∩ X ∩ g(b) = ∅. So, G ∈ gX(b) ⊆ X \ g(a∗ ). Therefore, X is an N-regular space. (d) Let G ∈ X and a∗ 6∈ G. Then G ∈ X \ g(a∗ ) = U . Since X is N-regular and B is complete, Proposition 3.1 implies that there exists b ∈ B such that G ∈ gX(b) ⊆ U . Thus X ∩ g(b) ∩ g(a∗ ) = ∅. Then, by (OCE), b(−C)a∗ . So, b ∈ G and b  a. Now [2, Lemma 3.2(v)] implies that G is a cluster in B. (e) Let G ∈ X\g(a). Then a 6∈ G. Hence there exists a b 6∈ G such that a  b. Thus a(−C)b∗ and [2, Lemma 5.7(i1)] implies that g(b∗ )∩g(a) = ∅. Then G ∈ X\g(b) ⊆ clX(X\g(b)) = X∩g(b∗ ) ⊆ X\g(a). Therefore X is regular. Since it is T0 , it is a Hausdorff space. (f) Let G ∈ X and a 6∈ G. Then G ∈ X \ g(a). Since X is regular, there exists a b ∈ B such that G ∈ X \ g(b) ⊆ clX(X \ g(b)) ⊆ X \ g(a). Hence X ∩ g(a) ∩ g(b∗ ) = ∅. Thus, by (OCE), a(−C)b∗ , i.e. a  b. Since b 6∈ G, we obtain that G is a co-end in B. u t Definition 3.5. A T0 -system (B, X) is called: (a) a WR-system if B is an ECA; (b) an NR-system if X ⊆ CLUSTERS(B); (c) an R-system if X ⊆ CO − ENDS(B). A WR-system (B, X) is called a WR1-system (resp., WR2-system) if (B, X) is a T1 -system (resp., T2 -system). An NR-system (B, X) is called a NR2-system if (B, X) is a T2 -system. Any such system is said to be complete if the Boolean algebra B is complete. Proposition 3.2. If (B, X) is an NR-system then it is a WR1-system. Proof: Let a 6= 0. Then a∗ 6= 1 and, by (INJ), there exists a G ∈ X such that a∗ 6∈ G. Since G is a cluster, [2, Lemma 3.2(v)] implies that there exists a b ∈ G such that b  a. As an element of G, b 6= 0; so, by [2, Lemma 2.2(iv)], B is an ECA. Hence (B, X) is a WR-system. Since every cluster is a maximal clan (see [2, Lemma 3.2(ii)]), (B, X) is a T1 -system. u t Proposition 3.3. If (B, X) is an R-system then it is an NR2-system. Proof: By [2, Lemma 3.2(vi)], every co-end is a cluster. Hence (B, X) is an NR-system. Let G, H ∈ X and G 6= H. Since G and H are maximal clans (see [2, Lemma 3.2(vi),(ii)]), there exists an a ∈ G \ H. Since a 6∈ H, there exists a b 6∈ H such that a  b. Thus a(−C)b∗ and, hence, b∗ 6∈ G. Therefore, (B, X) is a T2 -system. u t

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Proposition 3.4. If (Z, T) is a semiregular T0 -space then the mapping κZ : (Z, T) −→ (XZ , τB Z |XZ ) is a homeomorphism (see [2, Definition 5.1] for τB Z ). Proof: By [2, Proposition 4.2(i)], κZ is a bijection. Let F ∈ RC(Z). Then κZ (F ) = {σz : z ∈ F } = {σz : F ∈ σz } = XZ ∩ g(F ), where g : B Z −→ Exp(CLANS(B Z )) is defined as in [2]. Hence, by −1 [2, Definition 5.1], the map κ−1 Z is continuous (because Z is semiregular). Since κZ (XZ ∩ g(F )) = −1 κZ (κZ (F )) = F , the map κZ is continuous. Therefore, κZ is a homeomorphism. u t Example 3.3. 1. If B is an ECA, then (B, CLANS(B)) is a WR-system (by Example 3.1(1)) and the pair (B, MAX.CLANS(B)) is a WR1-system (see [2, Proposition 3.5], Example 3.2(1)). If B is an RECA then, by Example 3.2(3), the pair (B, CO − ENDS(B)) is a WR2-system. 2. If (Z, T) is a weakly regular T0 - (resp., T1 -; T2 -)space then (B Z , XZ ) is a complete WR-system (resp., WR1-system; WR2-system). This follows from the result of D¨untsch and Winter ([4]) cited in [2], from Example 3.1(2) and from Example 3.2(2)(4). 3. If (Z, T) is an N-regular T1 - (resp., T2 -)space then (B Z , XZ ) is a complete NR-system (resp., NR2-system). This follows from [2, Lemma 4.1] and from Example 3.2(2)(4). 4. If (Z, T) is a regular T2 -space then (B Z , XZ ) is a complete R-system. This follows from Example 3.2(4) and [2, Lemma 4.2]. 5. From Lemma 3.4(a)(b), Lemma 3.5(a) and Lemma 3.6(a)(c)(e) we get that if (B, X) is a Ti -system (i = 0, 1, 2) (resp., WR-system; NR-system; R-system) then (X, τB |X) is a Ti -space (i = 0, 1, 2) (resp., weakly regular T0 -space; N-regular T1 -space; regular T2 -space). Therefore, if (Z, T) is a semiregular T0 -space which is not a T1 -space then (B Z , XZ ) is a T0 -system which is not a T1 -system (indeed, it is a T0 -system (by Example 3.1); supposing that it is a T1 -system, we obtain that XZ is a T1 -space and hence, by Proposition 3.4, Z is a T1 -space — a contradiction). Similarly for all other properties. So, to find an example of a type of system which is not a system of another type, it is enough to find a semiregular space which has the corresponding first type property and does not have the corresponding second type property. Having this in mind, the space described in [2, Example 4.3] gives us examples of a T0 -system which is not a T1 -system (a simpler example is the space having three points x, y, z, whose topology is the family {∅, {x, y, z}, {x}, {y}, {x, y}} — it is semiregular, T0 and not T1 ) and of a WR-system which is not a WR1-system. Similarly, using the space described in [2, Example 4.2], we obtain an example of a WR2-system which is not an NR-system. With the help of the space described in [2, Example 4.1], we get that there exists an NR2-system which is not an R-system. It is well known that there exists a semiregular T1 -space which is not Hausdorff (see, e.g. [5]); hence there exists a T1 -system which is not a T2 -system. In [4], an example of a weakly regular T1 -space which is not T2 is described. Hence, there exists a WR1-system which is not a WR2-system. 6. If B is an NECA then (B, CLUSTERS(B)) is an NR-system (use [2, Lemma 5.7(iv1)] and Lemma 3.1). 7. If B is an RECA then (B, CO − ENDS(B)) is an R-system (by [2, Lemma 5.7(v1)] and Lemma 3.1). Proposition 3.5. Let (B, X) be a complete T0 -system. Then the map gX : B −→ B (X,τB |X ) (see [2, Definition 5.1] for τB and Lemma 3.1 for gX) is a CA-isomorphism and gX(G) = σG, for any G ∈ X

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(where, of course, gX(G) = {gX(a) : a ∈ G} and σG = {F ∈ RC(X, τB |X) : G ∈ F }). Hence gX induces the map κX : X −→ XX and κX is a bijection. Proof: Proposition 3.1 says that gX : B −→ (RC(X, τB |X), CX) is a CA-isomorphism. Using this fact we obtain that, for any G ∈ X, gX(G) = {gX(a) : a ∈ G} = {gX(a) : G ∈ gX(a)} = {F ∈ RC(X) : G ∈ u t F } = σG. Since gX is an injection, κX is a bijection. Definition 3.6. Let B = (B, C) and B 0 = (B 0 , C 0 ) be two CAs. Let (B, X) and (B 0 , X0 ) be two Csystems. A map ϕ : X −→ X0 is called an s-map if for any b ∈ B 0 there exists an Ab ⊆ B such that, for all G ∈ X, b ∈ ϕ(G) iff Ab ⊆ G. The s-maps will serve as morphisms between T0 -systems and Tych-systems (see Theorem 3.4); that’s why we will often write ϕ : (B, X) −→ (B 0 , X0 ) instead of ϕ : X −→ X0 . Fact 3.2. Let (B, X) be a C-system. Then the identity map idX : X −→ X is an s-map. The composition of two s-maps between C-systems is an s-map. Proof: The proof is straightforward.

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The next simple lemma is a slight generalization of Lemma 5.2(2)(3) from [4] and follows easily from it. Lemma 3.7. Let B be a CA, X be a subset of CLANS(B) and X = (X, τB |X) (see [2, Definition 5.1] for τB ). If M ⊆ X then: T T T (i) clX (M) = {G ∈ X : M ⊆ G}, where M = {H ⊆ B : H ∈ M}; (ii) M is a closed subset of X iff there exists an A ⊆ B such that M = {G ∈ X : A ⊆ G}. Lemma 3.8. Let (B, X) and (B 0 , X0 ) be C-systems and ϕ : X −→ X0 be a function. Then ϕ is a continuous map from (X, τB |X) to (X0 , τB 0 |X0 ) (see [2, Definition 5.1] for τB ) iff ϕ is an s-map from (B, X) to (B 0 , X0 ). Proof: (⇒) For every b ∈ B 0 , we have that ϕ−1 (gX0 (b)) is closed in X. Hence, by Lemma 3.7(ii), for every b ∈ B 0 , there exists an Ab ⊆ B such that ϕ−1 (gX0 (b)) = {G ∈ X : Ab ⊆ G}. Thus, for every G ∈ X, we have that b ∈ ϕ(G) iff Ab ⊆ G. Therefore, ϕ is an s-map. (⇐) Let b ∈ B 0 and F = gX0 (b). Since ϕ is an s-map, there T exists an Ab ⊆ B such that, for all G ∈ X, b ∈ ϕ(G) iff Ab ⊆ G. We will prove that ϕ−1 (F ) = {gX(a) : a ∈TAb }. Indeed, we have that G ∈ ϕ−1 (F ) iff ϕ(G) ∈ F iff ϕ(G) ∈ gX0 (b) iff b ∈ ϕ(G); further, G ∈ {gX(a) : a ∈ Ab } iff (G ∈ gX(a), for all a ∈ Ab ) iff (a ∈ G, for all a ∈ Ab ) iff Ab ⊆ G. So, ϕ is a continuous map. u t Proposition 3.6. Let (B, X) and (B 0 , X0 ) be C-systems and ϕ : X −→ T X0 be a function. Then ϕ : (B, X) −→ (B 0 , X0 ) is an s-map iff for every b ∈ B 0 and for every G ∈ X, {H ∈ X : b ∈ ϕ(H)} ⊆ G implies that b ∈ ϕ(G).

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Proof: (⇒) Let ϕ be an s-map. Then, by Lemma 3.8, ϕ is a continuous map from (X, τB |X) to (X0 , τB 0 |X0 ). Hence, for every T b ∈ B 0 , the set Mb = ϕ−1 (gX0 (b)) is closed in X. Then clX(Mb ) ⊆ Mb , i.e., T by Lemma 0 3.7(i), {G ∈ X : Mb ⊆ G} ⊆ Mb . Thus, for every b ∈ B and for any G ∈ X, we have that Mb ⊆ G implies G ∈ Mb . Since Mb = {H ∈ X : ϕ(H) ∈ gX0 (b)} = {H ∈ X : b ∈ ϕ(H)} and, in particular, for every G ∈ X, G ∈ Mb ⇐⇒ b ∈ ϕ(G), we complete the proof of this part. T (⇐) Let b ∈ B 0 . Put Ab = {H ∈ X : b ∈ ϕ(H)}. Then we have that, for every G ∈ X, Ab ⊆ G implies that b ∈ ϕ(G). If G ∈ X is such that b ∈ ϕ(G) then, by the definition of Ab , we obtain that Ab ⊆ G. Hence, b ∈ ϕ(G) implies that Ab ⊆ G. Thus, for every G ∈ X, b ∈ ϕ(G) ⇐⇒ Ab ⊆ G. Therefore, ϕ is an s-map. u t Proposition 3.7. Let (B, X) and (B 0 , X0 ) be C-systems and ϕ : X −→ X0 be a function. Then the following assertions are equivalent: (a) ϕ : (B, X) −→ (B 0 , X0 ) is an s-map; (b) for every b ∈ B 0 , there exists a stack Ab ⊆ B such that: (i) if a1 , a2 ∈ Ab then a1 Ca2 , and (ii) for any G ∈ X, Ab ⊆ G ⇔ b ∈ ϕ(G); (c) for every b ∈ B 0 , there exists a stack Ab ⊆ B such that, for any G ∈ X, Ab ⊆ G ⇔ b ∈ ϕ(G). Proof: (a)⇒(b) This follows from Proposition 3.6. The implications (b)⇒(c) and (c)⇒(a) are obvious.

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We will now introduce some notations. Let SK be the category of all semiregular T0 -spaces and all continuous maps between them. Let SFR (resp. SH; WRK; WRFR; WRH; NRFR; NRH; RH) be the full subcategory of the category SK whose objects are all semiregular T1 -spaces (resp., semiregular T2 -spaces; weakly regular T0 -spaces; weakly regular T1 -spaces; weakly regular Hausdorff spaces; Nregular T1 -spaces; N-regular T2 -spaces; regular T2 -spaces). (In this notations the letter “K” stands for “Kolmogoroff”, “FR” – for “Frechet”, “H” – for “Hausdorff”.) Let KS be the category of all complete T0 -systems and all s-maps between them. Let FRS (resp.HS; WRKS; WRFRS; WRHS; NRS; NRHS; RHS) be the full subcategory of the category KS whose objects are all complete T1 -systems (resp., T2 -systems; WR-systems; WR1-systems; WR2-systems; NRsystems; NR2-systems; R-systems). By SKC (resp., KSC) we will denote the full subcategory of SK (resp., KS) which has as objects all connected semiregular T0 -spaces (resp., all complete connected T0 -systems). Similarly, adding the letter C at the end of the names of all other categories defined above, we will get the names of their connected versions. Lemma 3.9. Two objects (B, X) and (B 0 , X0 ) of the category KS are isomorphic in KS iff there exists a CA-isomorphism f : B −→ B 0 such that, for every G ∈ X and every G0 ∈ X0 , f (G) ∈ X0 and f −1 (G0 ) ∈ X. Proof: Let ϕ : (B, X) −→ (B 0 , X0 ) be an isomorphism in the category KS or in any of its full subcategories. Then ϕ is an s-map and there exists an s-map ψ : (B 0 , X0 ) −→ (B, X) such that ϕ ◦ ψ = id(B 0 ,X0 ) and

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ψ ◦ ϕ = id(B,X) . Hence ϕ : X −→ X0 is a bijection. Thus, by Lemma 3.8, ϕ is a homeomorphism between (X, τB |X) and (X0 , τB 0 |X0 ). Let b ∈ B and let F = gX(b). Then, by Proposition 3.1, F ∈ RC(X) and hence ϕ(F ) ∈ RC(X0 ). Using again Proposition 3.1, we obtain that there exists a unique cb ∈ B 0 such that ϕ(F ) = gX0 (cb ). We put f (b) = cb . In this way, we define a map f : B −→ B 0 . Let us show that f is a Boolean isomorphism. We have that ϕ(gX(b)) = gX0 (f (b)).

(4)

Thus gX0 (f (0)) = ϕ(gX(0)) = ϕ(∅) = ∅. Then, by Lemma 3.1, f (0) = 0. Analogously, from gX0 (f (1)) = ϕ(gX(1)) = ϕ(X) = X0 we get that f (1) = 1. Further, gX0 (f (a + b)) = ϕ(gX(a + b)) = ϕ(gX(a) ∪ gX(b)) = ϕ(gX(a)) ∪ ϕ(gX(b)) = gX0 (f (a)) ∪ gX0 (f (b)) = gX0 (f (a) + f (b)) and hence, by Lemma 3.1, f (a + b) = f (a) + f (b). Finally, gX0 (f (b∗ )) = ϕ(gX(b∗ )) = ϕ((gX(b))∗ ) = ϕ(clX(X\gX(b))) = clX0 (ϕ(X\gX(b))) = clX0 (X0 \ϕ(gX(b))) = clX0 (X0 \gX0 (f (b))) = (gX0 (f (b)))∗ = gX0 ((f (b))∗ ) (see Proposition 3.1). Therefore, f (b∗ ) = (f (b))∗ . So, f is a Boolean homomorphism. Let f (b) = f (b0 ). Then ϕ(gX(b)) = ϕ(gX(b0 )). Since ϕ and gX are injections (see Lemma 3.1), we obtain that b = b0 . So, f is an injection. Let c ∈ B 0 . Then, by Proposition 3.1, gX0 (c) ∈ RC(X0 ). Since ϕ is a homeomorphism, we get that ϕ−1 (gX0 (c)) ∈ RC(X). Thus, by Proposition 3.1, there exists a (unique) b ∈ B such that ϕ−1 (gX0 (c)) = gX(b). Then, by (4), gX0 (f (b)) = ϕ(gX(b)) = gX0 (c). Hence f (b) = c. So, f is a Boolean isomorphism. Let G ∈ X. Then, by (4), f (G) = {f (b) : b ∈ G} = {f (b) : G ∈ gX(b)} = {f (b) : G ∈ −1 ϕ (gX0 (f (b)))} = {c ∈ B 0 : G ∈ ϕ−1 (gX0 (c))} = {c ∈ B 0 : ϕ(G) ∈ gX0 (c)} = {c ∈ B 0 : c ∈ ϕ(G)} = ϕ(G). Thus, for every G ∈ X, f (G) ∈ X0 . Let G0 ∈ X0 . Since ϕ is a surjection, there exists a G ∈ X such that ϕ(G) = G0 . Then f (G) = G0 and since f is a bijection, we obtain that f −1 (G0 ) = G, i.e. f −1 (G0 ) ∈ X. The map f is a CA-isomorphism because, by (OCE) and (4), aCb ⇐⇒ there exists a G ∈ X such that a, b ∈ G ⇐⇒ there exists a G ∈ X such that f (a), f (b) ∈ ϕ(G) ⇐⇒ f (a)C 0 f (b). Conversely, let there exist a CA-isomorphism f : B −→ B 0 such that, for every G ∈ X and every ∈ X0 , f (G) ∈ X0 and f −1 (G0 ) ∈ X. Then, as it is easy to see, f generates a bijection ϕ : X −→ X0 defined by the formula ϕ(G) = f (G), for every G ∈ X. Let us show that ϕ and ϕ−1 are s-maps between (B, X) and (B 0 , X0 ). Indeed, if b ∈ B 0 , we put Ab = {a}, where a is the unique element of B such that f (a) = b; then, for every G ∈ X, a ∈ G iff b = f (a) ∈ f (G) = ϕ(G). So, ϕ is an s-map. For showing that ϕ−1 is an s-map, we put, for every b ∈ B, Ab = {f (b)} and we get that, for all G0 ∈ X0 , b ∈ ϕ−1 (G0 ) iff f (b) ∈ G0 . So, ϕ−1 is also an s-map. Therefore, ϕ is an isomorphism in KS. u t G0

Corollary 3.3. Two objects (B, X) and (B 0 , X0 ) of the category KS are KS-isomorphic iff there exists a CA-isomorphism f : B −→ B 0 such that the correspondence G −→ f (G) is a bijection between the sets X and X0 (here, of course, f (G) = {f (b) : b ∈ G}). Theorem 3.1. The categories SK and KS are equivalent. Proof: We will define two (covariant) functors Φ : SK −→ KS and Ψ : KS −→ SK and will show that the compositions Φ ◦ Ψ and Ψ ◦ Φ are naturally isomorphic to the identity functors IdKS and IdSK respectively.

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Let (Z, T) be a semiregular T0 -space. We put Φ(Z, T) = (B Z , XZ ). By Example 3.1(2), Φ(Z, T) ∈ |KS|. Let f be an SK-morphism from (Z, T) to (Z 0 , T 0 ). Then we set ϕ = Φ(f ) : Φ(Z, T) −→ Φ(Z 0 , T 0 ) 0 to be the map from XZ to XZ 0 defined by the formula ϕ(σzZ ) = σfZ(z) , for every z ∈ Z. Let us show that ϕ is an s-map. Indeed, by Proposition 3.4, the maps κZ : Z −→ XZ and κZ 0 : Z 0 −→ XZ 0 are homeomorphisms. Since, obviously, ϕ = κZ 0 ◦ f ◦ κ−1 Z , we get that ϕ is a continuous map. Now, Lemma 3.8 implies that ϕ is an s-map. Therefore, Φ(f ) is an s-map and hence Φ(f ) ∈ KS(Φ(Z, T), Φ(Z 0 , T 0 )). So, Φ is well-defined on the objects and morphisms of the category SK. Obviously, Φ is a covariant functor. Let (B, X) be a complete T0 -system. Then we put Ψ(B, X) = (X, τB |X) (see [2, Definition 5.1] for τB ). Since (CLANS(B), τB ) is a semiregular T0 -space (by [2, Lemma 5.7(i2)]), Lemma 3.2(a) shows that Ψ(B, X) ∈ |SK|. Let ϕ be an element of KS((B, X), (B 0 , X0 )). Then ϕ is an s-map from X to X0 . We set f = Ψ(ϕ) : Ψ(B, X) −→ Ψ(B 0 , X0 ), where f (G) = ϕ(G), for all G ∈ X. Then, by Lemma 3.8, f is a continuous map. So, Ψ is well-defined on the objects and morphisms of the category KS. Obviously, Ψ is a covariant functor. Let (B, X) ∈ |KS|. Then Φ(Ψ(B, X)) = (B X, XX). By Proposition 3.1, we have that gX : B −→ B X is a CA-isomorphism. By Proposition 3.5, gX induces the map κX : X −→ XX (where κX(G) = X = g (G), for all G ∈ X) and κ is a bijection. Hence, by Lemma 3.9, κ : (B, X) −→ (Φ ◦ σG X X X Ψ)(B, X) is an isomorphism in the category KS. We will show that K : IdKS −→ Φ ◦ Ψ, defined by K(B, X) = κX, is a natural isomorphism. Indeed, let ϕ ∈ KS((B, X), (B 0 , X0 )). Then, for every X0 and ((Φ ◦ Ψ)(ϕ))(κ (G)) = ((Φ ◦ Ψ)(ϕ))(σ X ) = σ X0 . Hence G ∈ X we have that κX0 (ϕ(G)) = σϕ(G) X G ϕ(G) 0 0 (Φ ◦ Ψ)(ϕ) ◦ K(B, X) = K(B , X ) ◦ ϕ, i.e. K is a natural isomorphism. Let (Z, T) ∈ |SK|. Then Proposition 3.4 shows that the map κZ : (Z, T) −→ (Ψ ◦ Φ)(Z, T) is a homeomorphism. We will show that K 0 : IdSK −→ Ψ ◦ Φ, defined by K 0 (Z, T) = κZ , is a natural isomorphism. Indeed, let f ∈ SK((Z, T), (Z 0 , T 0 )). Then, for every z ∈ Z, we have that 0 0 κZ 0 (f (z)) = σfZ(z) and ((Ψ ◦ Φ)(f ))(κZ (z)) = σfZ(z) . Thus K 0 (Z 0 , T 0 ) ◦ f = ((Ψ ◦ Φ)(f )) ◦ K 0 (Z, T). Therefore, K 0 is a natural isomorphism. So, we have proved that SK and KS are equivalent categories. u t Theorem 3.2. The categories SKC and KSC are equivalent. Proof: It follows from Theorem 3.1, Corollary 3.2 and [2, Fact 2.2].

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Theorem 3.3. The following pairs of categories are equivalent: SFR and FRS, SH and HS, WRK and WRKS, WRFR and WRFRS, WRH and WRHS, NRFR and NRS, NRH and NRHS, RH and RHS. The same is true for their connected versions. Proof: It follows from Lemma 3.4(b), Lemma 3.5, Lemma 3.6, Example 3.2(2)(4), Example 3.3(2)(3)(4), Theorem 3.1 and Theorem 3.2. u t The next proposition follows in fact from [2, Lemma 3.3 and Lemma 3.5]; it can be derived also from some more general results from [3] (we are grateful to the referee for informing us for this).

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Proposition 3.8. Let B be a finite Boolean algebra and A be the set of all atoms of B. Then there exists a bijection between the set CB of all contact relations on B and the set RA of all reflexive and symmetric 1 binary relations on the set A. Hence, if n ∈ IN+ and |A| = n (i.e. |B| = 2n ) then |CB | = 2 2 n(n−1) . Proof: Let C ∈ CB . Then, for every a, b ∈ A, we put aρC b iff aCb. Then, obviously, ρC ∈ RA . Conversely, if ρ ∈ RA then, for every x, y ∈ B, we put xCρ y ⇐⇒ ∃a, b ∈ A such that a ≤ x, b ≤ y and aρb. It is easy to see that Cρ ∈ CB . Further, using the finiteness of B, it is not difficult to prove that for every ρ ∈ RA , ρ = ρCρ and that for every C ∈ CB , C = CρC . Hence the correspondence C −→ ρC is a u t bijection between the sets CB and RA . Example 3.4. C ONSTRUCTION OF ALL , UP TO HOMEOMORPHISM , FINITE SEMIREGULAR T0 - SPACES In [6], Evans, Harary and Lynn obtained the number k0 (n) of all different (but, possibly, homeomorphic) T0 topologies on a set with n elements, where n ∈ IN+ . For example, k0 (1) = 1, k0 (2) = 3, k0 (3) = 19, k0 (4) = 219, k0 (5) = 4231, k0 (6) = 130023, k0 (7) = 6129859. A bijective correspondence between all topologies on a finite set and special kinds of matrices was established in [8]. In [13], a homeomorphism classification of finite topological spaces is made. It is shown that there exists a bijective correspondence between the homeomorphism classes of finite topological spaces and certain classes of matrices. In what follows, we give a plan for an algorithm (and its explanation) for a direct construction of all, up to homeomorphism, finite semiregular T0 -spaces of rank n, where n ∈ IN+ (we say that a natural number n is a rank of a space (X, T) if |RC(X, T)| = 2n ). Let n ∈ IN+ . This will be the rank of the spaces which we will construct. All finite Boolean algebras are complete and atomic and hence they are isomorphic to the Boolean algebra Exp(A) for some set A. So, let us put A = {1, . . . , n} and let B be the Boolean algebra Exp(A) with the natural operations. Then A is the set of all atoms in B. As it is well known, every ultrafilter U in B is generated by a unique atom a in B (i.e. U = {b ∈ B : a ≤ b}), so that we will denote U by (a); hence, the ultrafilters S in B are (1), . . . , (n). Every grill G in B is a union of ultrafilters (by [2, Corollary 3.1]) and if G = {(aj ) : j = 1, . . . , m}, where m ∈ {1, . . . , n}, we will denote G by (a1 . . . am ) and we will say that a1 , . . . , am are the numbers of G. In this way, the grills can be regarded as combinations of the elements of A. In what follows, we will use the notations from Proposition 3.8. Our plan is the following: (1) for every ρ ∈ RA , we will find the elements of the set Yρ = CLANS(B, Cρ ); (2) we will show how one can obtain a set Ciso B of representatives of all CA-isomorphism classes of contact relations on B; (3) for every C ∈ Ciso , B we will find all T0 -systems on (B, C); (4) we will find a list SC of representatives of all KS-isomorphism classes of T0 -systems on (B, C); for every C ∈ Ciso B and for every ((B, C), X) ∈ SC , we will construct a topological space X((B,C),X) . Then, by Theorem 3.1 and Proposition 3.1, the set {X((B,C),X) : C ∈ Ciso B , ((B, C), X) ∈ SC } will be the required set of representatives of all homeomorphism classes of finite semiregular T0 -spaces. So, let ρ ∈ RA . Then, as it is easy to see, a grill G in B is a clan in the CA (B, Cρ ) iff for every two different numbers i and j of G we have that (i, j) ∈ ρ. In this way, we determine easily the elements of the set Yρ . Now, [2, Theorem 3.1] implies that CYρ = Cρ .

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Let ρ1 , ρ2 ∈ RA and ρ1 6= ρ2 . By [2, Theorem 3.1(b)], we have that (B, Cρ1 ) is CA-isomorphic to (B, Cρ2 ) iff there exists a Boolean isomorphism f of B onto B such that f (Yρ1 ) ⊆ Yρ2 and f −1 (Yρ2 ) ⊆ Yρ1 . Note that since B is complete and atomic, the Boolean isomorphisms of B onto B are completely determined by their restrictions on the set A; hence they are determined by the permutations of the elements of A. Hence, if π : A −→ A is a bijection (i.e, a permutation of A), we have just to check whether for every G ∈ Yρ1 and for every two different numbers i and j of G, we have that (π(i), π(j)) ∈ ρ2 . In a similar way we check the second inclusion. Now it is easy to construct a set Ciso B of representatives of all CA-isomorphism classes of contact relations on B. Let C ∈ Ciso B . We claim Sthat if X ⊆ YρC then ((B, C), X) is a T0 -system iff the following two conditions are satisfied: (a) {B \ G : G ∈ X} = B \ {1B } (i.e., X satisfies condition (INJ)), and (b) CLANS(B, CX) = YρC . Indeed, if ((B, C), X) is a T0 -system then the condition (a) is clearly satisfied and it implies that X satisfies condition (CR); since X satisfies condition (OCE), we get, using the definition of the relation CX, that C = CX; now, [2, Theorem 3.1(a)] shows that CLANS(B, CX) = CLANS(B, C) = YρC , i.e. the condition (b) is satisfied. Conversely, if the conditions (a) and (b) are satisfied then ((B, C), X) is, obviously, a T-system; using condition (b), we get, by [2, Theorem 3.1(a)], that CX = C; thus, from the definition of the relation CX we obtain that X satisfies condition (OCE); therefore, ((B, C), X) is a T0 -system. How can we check conditions (a) and (b)? The first is clear; for checking the second one, note that a grill G in B is a clan in (B, CX) iff any set consisting of two different numbers of G is a subset of the set of all numbers of some element of X. In this way, we obtain all T0 -systems on (B, C). Let ((B, C), X1 ) and ((B, C), X2 ) be two T0 -systems. In the above considerations, we clarified that X1 and X2 satisfy condition (CR) and CX1 = CX2 = C. Now, using Corollary 3.3 and [2, Theorem 3.1(b)], we get that ((B, C), X1 ) and ((B, C), X2 ) are KS-isomorphic ⇐⇒ there exists a CAisomorphism f : (B, C) −→ (B, C) such that the correspondence G −→ f (G) is a bijection between X1 and X2 ⇐⇒ there exists a Boolean isomorphism f of B onto B such that f (X1 ) ∪ f −1 (X2 ) ⊆ CLANS(B, C)(= YρC ) and the correspondence G −→ f (G) is a bijection between X1 and X2 . The last condition can be checked in the same way as we have checked above the condition f (Yρ1 ) ⊆ Yρ2 . After this procedure, we obtain a list SC of representatives of all KS-isomorphism classes of T0 -systems on (B, C). For every C ∈ Ciso B and every T0 -system ((B, C), X) ∈ SC , we construct the space (X, τ(B,C) |X) (recall that the topology τ(B,C) |X on X has as a closed base the family {gX(b) : b ∈ B}, where gX(b) = {G ∈ X : b ∈ G}). The space (X, τ(B,C) |X) is the required space X((B,C),X) . Note that for every space X = X((B,C),X) of the obtained list of spaces, we have that n ≤ |X| ≤ 2n − 1 (indeed, since the space X has rank equal to n, we have that |RC(X)| = 2n ; hence |X| ≥ n; the second inequality follows from the facts that X ⊆ ΓP(B) and |ΓP(B)| = 2n − 1). Taking a set A with three elements and setting C = Cl , we obtain easily, using the above algorithm, the following simple example of two non-homeomorphic semiregular T0 -spaces X and Y for which the contact algebras (RC(X), CX ) and (RC(Y ), CY ) are isomorphic: X = {x, y, z, u, v, w} and the open base of X is the family BX = {{x}, {u}, {v}, {x, u, w}, {x, z, v}, {y, u, v}}; Y = {x, y, z, u, v, w, t} and the open base of Y is the family BY = {{x}, {u}, {v}, {x, u, w}, {x, z, v}, {y, u, v, t}}. β Proposition 3.9. Let (B, X) be a T-system, where B = (B, C). Define a binary relation CX on B by setting:

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β for any a, b ∈ B, a(−CX )b iff there exists a continuous map α : X −→ I such that a ∈ G ∈ X implies α(G) = 0, and b ∈ H ∈ X implies α(H) = 1 (here I is regarded as a subspace of the real line with its natural topology, and on X the topology τB |X (see [2, Definition 5.1] for τB ) is taken). β β Then (B, CX ) is a CA. If (B, X) is a T0 -system then C ⊆ CX .

Proof: β ∗ Let us put, for any a, b ∈ B, a βX b (or simply a β b) if a(−CX )b . We will show that the relation β  satisfies conditions ( 1)-( 5) from [2, Definition 2.1]. For checking ( 1), set α(G) = 0, for every G ∈ X; then we obtain that 1 β 1. For ( 2), let x β y. Then there exists a continuous function α : X −→ I such that x ∈ G ∈ X implies α(G) = 0, and y ∗ ∈ H ∈ X implies α(H) = 1. Suppose that x 6≤ y. Then x∗ + y 6= 1. Hence, by (INJ), there exists a G ∈ X such that x∗ + y 6∈ G. Thus x.y ∗ ∈ G. Therefore, x ∈ G and y ∗ ∈ G. This implies that α(G) = 0 and α(G) = 1 — a contradiction. Hence, x ≤ y. Further, for ( 3), let x ≤ y β z ≤ t. Then there exists a continuous function α : X −→ I such that y ∈ G ∈ X implies α(G) = 0, and z ∗ ∈ H ∈ X implies α(H) = 1. If x ∈ G ∈ X then y ∈ G and hence α(G) = 0. If t∗ ∈ H ∈ X then z ∗ ∈ H and thus α(H) = 1. So, x β t. Now, for ( 4), let x β y. Then there exists a continuous function α : X −→ I such that x ∈ G ∈ X implies α(G) = 0, and y ∗ ∈ H ∈ X implies α(H) = 1. Set α1 = 1 − α. Then α1 is a continuous function from X to I and if y ∗ ∈ G ∈ X then α1 (G) = 0, if x ∈ H ∈ X then α1 (H) = 1. So, y ∗ β x∗ . Finally, for ( 5), let x β y and x β z. Then there exists a continuous function α1 : X −→ I such that x ∈ G ∈ X implies α1 (G) = 0, and y ∗ ∈ H ∈ X implies α1 (H) = 1. Also, there exists a continuous function α2 : X −→ I such that x ∈ G ∈ X implies α2 (G) = 0, and z ∗ ∈ H ∈ X implies α2 (H) = 1. Put α = max{α1 , α2 }. Then α is a continuous function from X to I and if x ∈ G ∈ X then α(G) = 0. β If y ∗ + z ∗ ∈ H ∈ X then y ∗ ∈ H or z ∗ ∈ H and hence α(H) = 1. So, x β y.z. Therefore, (B, CX ) is a CA. β Let (B, X) be a T0 -system and let aCb, where a, b ∈ B. Suppose that a(−CX )b. Then there exists a continuous function α : X −→ I such that a ∈ G ∈ X implies α(G) = 0, and b ∈ H ∈ X implies α(H) = 1. Since aCb, condition (OCE) implies that there exists a G ∈ X such that a, b ∈ G. Hence β β α(G) = 0 and α(G) = 1 — a contradiction. Hence, aCX b. Thus C ⊆ CX . u t Definition 3.7. Let B = (B, C) be an IECA, X ⊆ CLUSTERS(B) and X satisfies condition (INJ). Then the pair (B, X) is called a TI-system. A TI-system (B, X) is said to be complete if B is a complete CA. Example 3.5. Let (X, α) be an EF-proximity space and X = {σx : x ∈ X}, where, for every x ∈ X, σx = {F ∈ RC(X, τ (α)) : x ∈ F }. Then the pair ((RC(X, τ (α)), α|RC(X) ), X) is a complete TI-system. Indeed, it is easy to see that σx is a cluster in (RC(X, τ (α)), α|RC(X) ), for every x ∈ X; the condition (INJ) is obviously satisfied and, by Proposition 2.20, (RC(X), α|RC(X) ) is a complete IECA. β (see Proposition Definition 3.8. A complete TI-system ((B, C), X) is called a Tych-system if C = CX β 3.9 for CX). We will denote by TYCH the category of all Tychonoff spaces and all continuous maps between them, and by TYCHSYS the category of all Tych-systems and all s-maps between them (see Definition 3.6 for s-maps).

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Theorem 3.4. The categories TYCH and TYCHSYS are equivalent. Proof: We will define two functors Φ : TYCH −→ TYCHSYS and Ψ : TYCHSYS −→ TYCH and we will show that their compositions are naturally isomorphic to the corresponding identity functors. Let (X, T) be a Tychonoff space and δβ be the EF-proximity on the space X defined, for any A, B ⊆ X, by A(−δβ )B iff there exists a continuous function f : X −→ I such that f (A) = 0 and f (B) = 1 (see, e.g., [5] or [10] for δβ ). Put B = (B, C) = (RC(X, T), δβ |RC(X) ). Then, by Example 3.5, (B, X), where X = {σx : x ∈ X}, is a complete TI-system. It is easy to see that the function jX : (X, T) −→ (X, τB |X) (see [2, Definition 5.1] for τB ), defined by jX (x) = σx , is a homeomorphism. Indeed, jX is, −1 obviously, a bijection; jX is a continuous map because jX (X ∩ g(F )) = F , for every F ∈ RC(X), and jX is an open map because jX (X \ F ) = X ∩ Int(g(F ∗ )), for every F ∈ RC(X). Further, since, for β every F ∈ RC(X), F ∈ G ∈ X ⇐⇒ G ∈ g(F ) ∩ X ⇐⇒ G ∈ jX (F ), we obtain that C = CX . So, (B, X) is a Tych-system and we put Φ(X, T) = (B, X). Let f : X −→ X 0 be a continuous map between Tychonoff spaces X and X 0 . Let Φ(X) = (B, X) and Φ(X 0 ) = (B 0 , X0 ). Then, as it was proved above, the maps jX : X −→ X and jX 0 : X 0 −→ X0 −1 are homeomorphisms. Hence the map ϕf : X −→ X0 , defined by the formula ϕf = jX 0 ◦ f ◦ jX , is a continuous map. Thus, by Lemma 3.8, ϕf is an s-map between Tych-systems (B, X) and (B 0 , X0 ). Now we put Φ(f ) = ϕf . Then Φ(f ) ∈ TYCHSYS(Φ(X), Φ(X 0 )). So, Φ is defined also on the morphisms of the category TYCH. It is easy to see that Φ : TYCH −→ TYCHSYS is a functor. Let (B, X) be a Tych-system. Then we set Ψ(B, X) = (X, τB |X). Put, for short, T = τB |X. Since (X, T) is a subspace of the compact Hausdorff space (CLUSTERS(B), τB ) (see [2, Theorem 5.2(i)]), (X, T) is a Tychonoff space, i.e. it is an object of the category TYCH. Let ϕ : (B, X) −→ (B 0 , X0 ) be a morphism in the category TYCHSYS. Then ϕ is an s-map and Lemma 3.8 implies that fϕ : (X, τB |X) −→ (X0 , τB 0 |X0 ), defined by fϕ (G) = ϕ(G), for every G ∈ X, is a continuous map. So, it is correct to set Ψ(ϕ) = fϕ . Then Ψ(ϕ) ∈ TYCH(Ψ(B, X), Ψ(B 0 , X0 )). Hence Ψ is defined also on the morphisms of the category TYCHSYS. It is easy to see that Ψ : TYCHSYS −→ TYCH is a functor. Let (X, T) be a Tychonoff space, B = (B, C) = (RC(X, T), δβ |RC(X) ) and X = {σx : x ∈ X}. Then Ψ(Φ(X)) = Ψ(B, X) = (X, τB |X). Hence the function jX defined above is a homeomorphism between X and Ψ(Φ(X)). We will show that j : IdTYCH −→ Ψ ◦ Φ, defined by j(X) = jX , is −1 . Hence a natural isomorphism. Indeed, let f ∈ TYCH(X, X 0 ). Then Ψ(Φ(f )) = jX 0 ◦ f ◦ jX 0 Ψ(Φ(f )) ◦ j(X) = j(X ) ◦ f , i.e. j is a natural isomorphism. Let (B, X) be a Tych-system. Put X = X and T = τB |X. Then we have that Ψ(B, X) = (X, T) and that Φ(Ψ(B, X)) = ((RC(X, T), δβ |RC(X) ), XX ), where XX = {σx : x ∈ X} = ({σG : G ∈ X}). Set (B X , C X ) = (RC(X, T), δβ |RC(X) ) and B X = (B X , C X ). Then, as it was proved above, the function jX : (X, T) −→ (XX , τB X |XX ) is a homeomorphism. Using Lemma 3.8, we obtain that the map JX : (B, X) −→ (B X , XX ), defined by JX (G) = jX (G) for every G ∈ X, is an isomorphism in the category TYCHSYS. Now, it is easy to see that J : IdTYCHSYS −→ Φ ◦ Ψ, defined by J(B, X) = JX, is a natural isomorphism. Therefore, the categories TYCH and TYCHSYS are equivalent. u t

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Now we will prove an equivalence theorem for the category LC of all locally compact Hausdorff spaces and all continuous maps between them. ¯ = (B, C, IB), we will denote by X ¯ the We will first introduce some notations. For every LCA B B ¯ set ABC(B) (see [2, Definition 5.2] for ABC). Definition 3.9. We will define a category LCS. Its objects are all complete LCAs. For any two objects ¯ to B ¯1 are all s-maps from ¯ = (B, C, IB) and B ¯1 = (B1 , C1 , IB1 ) of LCS, the morphisms from B B ((B, αC ), XB¯ ) to ((B1 , αC1 ), XB¯1 ) (note that ((B, αC ), XB¯ ) and ((B1 , αC1 ), XB¯1 ) are C-systems). It is easy to see that LCS is really a category. ¯ = (B, C, IB) and B ¯1 = (B1 , C1 , IB1 ) are LCA-isomorphic (see [2, DefiniLemma 3.10. Two LCAs B tion 2.5]) iff they are LCS-isomorphic. Proof: We put B = (B, αC ) and B 1 = (B1 , αC1 ) (see [2, Theorem 2.1] for αC ). ¯ −→ B ¯1 be an LCA-isomorphism. Then it is easy to see that the map fα : B −→ B 1 , Let f : B defined by fα (b) = f (b) for every b ∈ B, is a CA-isomorphism. Thus, for every G ∈ XB¯ and every G1 ∈ XB¯1 we have that fα (G) ∈ XB¯1 and fα−1 (G1 ) ∈ XB¯ . Now, exactly as in the second part of the proof of Lemma 3.9, we obtain that the map ϕ : (B, XB¯ ) −→ (B 1 , XB¯1 ) (i.e. ϕ : XB¯ −→ XB¯1 ), defined by ϕ(G) = fα (G)(= f (G)) for every G ∈ XB¯ , is a bijection, is an s-map and its inverse is also an s-map. ¯ and B ¯1 are LCS-isomorphic. Hence B ¯ and B ¯1 are LCS-isomorphic. Then there exists a bijection ϕ : X ¯ −→ Conversely, suppose that B B −1 XB¯1 such that it and its inverse ϕ are s-maps between (B, XB¯ ) and (B 1 , XB¯1 ). By [2, Theorem 5.4(a)], ¯ −→ (RC(X ¯ , τB |X ¯ ), CX ¯ , CompRC(X ¯ )) is an LCA-isomorphism and the same the map gXB¯ : B B B B B is true for the map gXB¯ . Hence, using these facts instead of Proposition 3.1 and Lemma 3.1, we can 1 define exactly as in the first part of the proof of Lemma 3.9 a map f : B −→ B1 by the formula ϕ(gXB¯ (b)) = gXB¯ (f (b)). 1

(5)

As in the proof of Lemma 3.9, we can show that f is a Boolean isomorphism, using the fact that gXB¯ and gXB¯ are Boolean isomorphisms. Also, as there, we can prove that f (G) = ϕ(G), for every G ∈ XB¯ . 1 Let b ∈ B. Then, by (5), by [2, Theorem 5.4(a)] and since ϕ is a homeomorphism between (XB¯ , τB |XB¯ ) and (XB¯1 , τB 1 |XB¯ ) (by Lemma 3.8), we have that b ∈ IB ⇐⇒ gXB¯ (b) is compact 1 ⇐⇒ ϕ(gXB¯ (b)) is compact ⇐⇒ gXB¯ (f (b)) is compact ⇐⇒ f (b) ∈ IB1 . Hence f −1 (IB1 ) = IB. 1 Further, let a, b ∈ B and aCb. Then, by (BC2) (see [2, Definition 2.5]), there exists an element c of IB such that aC(b.c). Then aαC (b.c) and since (B, αC ) is an IECA (by [2, Theorem 2.1]), Corollary [2, 3.4] implies that there exists a cluster G in the CA (B, αC ) such that a, b.c ∈ G. Then G ∈ XB¯ (because b.c ∈ G ∩ IB). Now, as in the proof of Lemma 3.9, we obtain that f (a), f (b.c) ∈ ϕ(G). Hence f (a)αC1 f (b.c). As it was shown above, f (b.c) ∈ IB1 (because b.c ∈ IB). Hence, by the definition of αC1 , f (a)C1 f (b.c), i.e. f (a)C1 (f (b).f (c)). Thus f (a)C1 f (b). Let now f (a)C1 f (b). Then we obtain, ¯ to B ¯1 . analogously, that aCb. Therefore, f is an LCA-isomorphism from B u t Theorem 3.5. The categories LC and LCS are equivalent.

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Proof: Let us define a functor Λa : LCS −→ LC. On the objects of the category LCS it is defined as in [2, ¯ and B ¯1 be two objects of the category LCS and ϕ ∈ LCS(B, ¯ B ¯1 ). We put Theorem 5.4(b)]. Let B ¯ ¯ Λa (ϕ) = ϕ. Then, by Lemma 3.8, Λa (ϕ) is a continuous map from Λa (B) to Λa (B1 ), i.e. it is a morphism in the category LC. It is easy to see that Λa is really a functor. Further, let us define a functor Λs : LC −→ LCS. On the objects of the category LC it is defined as in [2, Theorem 5.4(b)]. Let L and L1 be two objects of the category LC and f : L −→ L1 be a continuous map. Then put Λs (f ) = ϕf , where ϕf : ABC(Λs (L)) −→ ABC(Λs (L1 )) is defined by ϕf (σx ) = σf (x) , for every x ∈ L (see [2, Lemma 5.8] for an explanation of this formula). Let us show that Λs (f ) ∈ LCS(Λs (L), Λs (L1 )). Indeed, using the notations from [2, Theorem 5.4(b)], we have that λL1 ◦f = Λa (ϕf )◦λL . Since λL and λL1 are homeomorphisms, we obtain that Λa (ϕf ) : Λa (Λs (L)) −→ Λa (Λs (L1 )) is a continuous map. Thus, by Lemma 3.8, Λs (f )(= ϕf ) is an LCS-morphism. Now it is easy to see that Λs is really a functor. Let L be an object of LC. Then the function λL : L −→ Λa (Λs (L)), defined by the formula λ(x) = σx , for every x ∈ L is a homeomorphism (see [2, Theorem 5.4(b)]). We will show that λ : IdLC −→ Λa ◦ Λs , defined by λ(L) = λL , for every L ∈ |LC|, is a natural isomorphism. Indeed, let L, L1 ∈ |LC| and f ∈ LC(L, L1 ). Then, as it follows from the preceding two paragraphs, λ(L1 ) ◦ f = Λa (Λs (f )) ◦ λ(L). Thus, λ is a natural isomorphism. ¯ ∈ |LCS|. Then, by [2, Theorem 5.4(b)], the function γ ¯ : B ¯ −→ Λs (Λa (B)), ¯ defined by Let B B γB¯ (b) = {G ∈ XB¯ : b ∈ G}, is a complete LCA-isomorphism. Hence, by Lemma 3.10, there exists an ¯ to Λs (Λa (B)), ¯ defined by γ˜ ¯ (H) = {γ ¯ (b) : b ∈ H}, for every H ∈ X ¯ . LCS-isomorphism γ˜B¯ from B B B B It is easy to see that γ˜B¯ (H) = {F ∈ RC(XB¯ ) : H ∈ F } = σH. We will show that γ : IdLCS −→ Λs ◦ ¯ = γ˜ ¯ , for every B ¯ ∈ |LCS|, is a natural isomorphism. Indeed, let B, ¯ B ¯1 ∈ |LCS| Λa , defined by γ(B) B ¯ B ¯1 ). Then ϕ : X ¯ −→ X ¯ , (Λs ◦ Λa )(ϕ) : ABC(Λs (X ¯ )) −→ ABC(Λs (X ¯ )) and and ϕ ∈ LCS(B, B B1 B B1 ¯1 ) ◦ ϕ = (Λs ◦ Λa )(ϕ) ◦ γ(B). ¯ Thus, γ is ((Λs ◦ Λa )(ϕ))(σH) = σϕ(H) , for every H ∈ XB¯ . Hence γ(B a natural isomorphism. u t Definition 3.10. We will define a category CHS. Its objects are all complete IECAs. For any two objects B = (B, C) and B 1 = (B1 , C1 ) of CHS, the morphisms from B to B 1 are all s-maps from (B, CLUSTERS(B)) to (B 1 , CLUSTERS(B 1 )) (see Definition 3.6 for the notion of s-map). It is easy to see that CHS is really a category. Theorem 3.6. The category CH of all compact Hausdorff spaces and all continuous maps between them is equivalent to the category CHS. Proof: Let us put, for every IECA B, XB = CLUSTERS(B). ¯ = (B, C, B) As it was mentioned above, every IECA B = (B, C) can be identified with the LCA B (i.e. with IB = B). Then αC ≡ C (see [2, Theorem 2.1] for αC ) and XB¯ = XB . Hence CHS is in fact a full subcategory of the category LCS. Since, for every IECA B, we have, by [2, Theorem 5.2(i)], X X that (XB , τB B ) (see [2, Definition 5.1] for τB B ) is a compact Hausdorff space, our theorem follows from Theorem 3.5. u t

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Definition 3.11. We will define a category CSKS. Its objects are all complete CAs. For any two objects B = (B, C) and B 1 = (B1 , C1 ) of CSKS, the morphisms from (B, C) to (B1 , C1 ) are all s-maps from (B, CLANS(B)) to (B 1 , CLANS(B 1 )) (see Definition 3.6 for the notion of s-map). It is easy to see that CSKS is really a category. Theorem 3.7. The category CSK of all C-semiregular spaces and all continuous maps between them is equivalent to the category CSKS. The same is true for their connected versions. Proof: By Example 3.1(1), for every CA B, the pair (B, CLANS(B)) is a T0 -system. Hence, identifying any complete CA B with the complete T0 -system (B, CLANS(B)), we obtain that CSKS is a full subcategory of the category KS. Now, Theorem 3.1 and [2, Theorem 5.1(I(ii))] (and their proofs) imply that the category CSK is equivalent to the category CSKS. The result for the connected versions follows from Theorem 3.2. u t Definition 3.12. We will define a category CMSS (respectively, CWRS; CNRFS; OCERS). Its objects are all complete TCAs (resp., complete ECAs; complete NECAs; complete RECAs). For any two objects B = (B, C) and B 1 = (B1 , C1 ) of CMSS (resp., CWRS; CNRFS; OCERS), the morphisms from B to B 1 are all s-maps from (B, MAX.CLANS(B)) (resp., from (B, MAX.CLANS(B)); from (B, CLUSTERS(B)); from the pair (B, CO − ENDS(B))) to the pair (B 1 , MAX.CLANS(B 1 )) (resp. (B 1 , MAX.CLANS(B 1 )); (B 1 , CLUSTERS(B 1 )); (B 1 , CO − ENDS(B 1 ))) (see Definition 3.6 for the notion of s-map). It is easy to see that CMSS (resp., CWRS; CNRFS; OCERS) is really a category. Proposition 3.10. Two objects B and B 0 of the category CHS (resp., of the category CSKS; CMSS; CWRS; CNRFS; OCERS) are isomorphic iff they are CA-isomorphic. Proof: Let B, B 0 ∈ |CHS|. Then, using Lemma 3.9, we obtain that the pairs (B, CLUSTERS(B)) and (B 0 , CLUSTERS(B 0 )) are KS-isomorphic (and, hence, the objects B and B 0 are CHS-isomorphic) iff there exists a CA-isomorphism f : B −→ B 0 . (Indeed, for every cluster G in B and every cluster G0 in B 0 , we have that f (G) is a cluster in B 0 and f −1 (G0 ) is a cluster in B.) The proof for all other categories is analogous. u t We will now introduce some notations. We will denote by CMS (resp., by CWR; by CNRF; by OCER) the category of all CM-semiregular (resp., all C-weakly regular; all CN-regular T1 ; all OCEregular) spaces and all continuous maps between them. Theorem 3.8. The category CMS (resp., CWR; CNRF; OCER) is equivalent to the category CMSS (respectively, CWRS; CNRFS; OCERS). The same is true for their connected versions. Proof: The proof is analogous to that of Theorem 3.7, using Example 3.2(1), Example 3.3(1)(6)(7) and [2, Theorem 5.1(II(ii),III(ii))], Theorem 3.3 (and their proofs). u t

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4. Examples I. D¨untsch and M. Winter have posed in [4] the following question: Question. Are there ECAs which do not have a regular T1 representation space? We can answer the weaker version of this question when the embedding is dense: Is there an ECA which cannot be densely embedded in the contact algebra (RC(X), CX ), where X is a regular T1 space? The next example shows that there are even such RCC models. Example 4.1. There exists a complete RCC model B = (B, C) (i.e. a complete connected ECA) for which there is no regular T1 -space (X, τ ) such that B can be densely embedded in (RC(X), CX ). Proof: We will even show that there is a complete connected NECA B for which there is no regular T1 -space (X, τ ) such that B can be densely embedded in (RC(X), CX ). We will use here the notations from [2, Example 4.1] and its proof. Let K be the square in IR2 (together with its interior in IR2 ) with center in the origin O = (0, 0) and sides with length 2. Regard the subspace Y = K ∪ {O0 } of the “Double Origin Topology” space X (see the proof of [2, Example 4.1]). Then Y is a connected Hausdorff N-regular space and Y is not regular. Put B = (B, C) = (RC(Y ), CY ). Then B is a complete connected NECA (see [2, Example 2.1, Proposition 4.7, Fact 2.2]). Suppose that there exists a regular T1 -space Z and a dense embedding m : (B, C) −→ (RC(Z), CZ ). Then m is a Boolean monomorphism and hence, by [2, Proposition 5.1], we will have that m(B) = RC(Z). Thus B will be isomorphic to (RC(Z), CZ ). Then, by [2, Proposition 4.10], B will satisfy Regularity Axiom. We will prove that this is not true, obtaining, in this way, a contradiction. So, we will show that there exist two elements a and b of B such that aCb but there is no co-end in B containing them. Take two equilateral triangles a and b (together with their interiors in IR2 ) which lie in the set {O} ∪ {(x, y) ∈ IR2 : y > 0}, have sides with length 1/2 and the origin O=(0,0) is their unique common point. Suppose that there exists a co-end σ in B containing a and b. Put ξ = c(σ)(= {d ∈ B : d∗ 6∈ σ}). Y is a co-cluster in B. We Then ξ is an end in B and σ = c(ξ). By [2, Lemma 4.1, Proposition 4.1(i)], νO Y. will show that ξ ⊆ νO Y . Then O 6∈ Int (F ) and F ∈ ξ. Since ξ is a round Suppose that there exists an F ∈ ξ \ νO Y filter, there exist G, H, T, S ∈ ξ such that S  T  H  G  F . Since G  F , we obtain that G ∩ clY (Y \ F ) = ∅ and hence G ⊆ IntY (F ). Hence O 6∈ G. Thus O 6∈ T . Let us show that O0 6∈ T . Suppose that O0 ∈ T . Since T ⊆ IntY (H), we obtain that O0 ∈ IntY (H). Hence there exists an n0 ∈ IN+ such that Vn−0 ⊆ H. Then clY (Vn−0 ) ⊆ H ⊆ IntY (G). Let n ∈ IN+ . Then there exists a point xn ∈ clY (Vn−0 ) ∩ clY (Vn+ ) and, since xn ∈ IntY (G), there exists a neighborhood U of xn such that U ⊆ G. Since, obviously, U ∩ Vn+ 6= ∅, we obtain that G ∩ Vn+ 6= ∅. This is true for every n ∈ IN+ . Thus, since G is closed in Y , we get that O ∈ G, which is a contradiction. Therefore, O0 6∈ T . We have proved that O, O0 6∈ T . Then there exists an n1 ∈ IN+ such that (Vn+1 ∪Vn−1 )∩T = ∅. Hence IntY (T )∩(clY (Vn+1 )∪clY (Vn−1 )) = ∅. Since S ⊆ IntY (T ), we obtain that S ∩(clY (Vn+1 )∪clY (Vn−1 )) = ∅.

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Let Un1 = {(x, y) ∈ IR2 : x2 + y 2 < 1/n21 }. Then S ⊆ K \ Un1 = D. Since the topology on D coincides with that induced by IR2 , we get that D is compact. Hence S is compact. Since ξ is a filter in B, S.ξ = {W.S : W ∈ ξ}T is a family of closed subsets the finite intersection property and T of S which has T is a subfamily of ξ. Thus S.ξ 6= ∅ and therefore ξ 6= ∅. Let x ∈ ξ. Then x ∈ S ⊆ Y \ {O, O0 }. Since Y is a Hausdorff space and Y is locally compact at the point x, we get easily that the space Y is regular at the point x. Then [2, Lemma 4.2] and [2, Proposition 4.1(i),(ii)] imply that νxY is an end in B. Let us show that ξ ⊆ νxY . Suppose that there exists an A ∈ ξ \ νxY . Since ξ is a round filter in B, there exists an A1 ∈ ξ such that A1  A. Then A1 ⊆ IntY (A). Since T A 6∈ νxY , we have that x 6∈ IntY (A) and hence x 6∈ A1 . This is a contradiction because A1 ∈ ξ and x ∈ ξ. So, ξ ⊆ νxY . Since νxY and ξ are ends and hence they are minimal E-filters (see [2, Lemma 3.2(vii),(iii)]), we obtain that ξ = νxY . Then σ = c(ξ) = c(νxY ) = σxY (see [2, Proposition 4.1(i)] and [2, Fact 3.1(iii)]). Since a, b ∈ σ and a ∩ b = {O}, we obtain that x = O, which is a contradiction since x ∈ S ⊆ K \ {O}. Y. So, ξ ⊆ νO Y is a minimal E-filter in B because it is a co-cluster in B. Thus ξ = ν Y . By [2, Lemma 3.2(iii)], νO O Y ) = σ Y . Since Y is not regular at the point O, we obtain that σ Y is not a Therefore σ = c(ξ) = c(νO O O co-end (see [2, Lemma 4.2]), i.e. σ is not a co-end, which is a contradiction. Hence, there is no co-end in B containing both a and b. Thus B does not satisfy Regularity Axiom. Therefore, there is no regular u t T1 -space (X, τ ) such that B can be densely embedded in (RC(X), CX ). Example 4.2. There exists a compact semiregular T0 -space X which is not C-semiregular. Proof: Let X = IR ∪ {x0 }, where IR is the real line and x0 6∈ IR. We will define a neighborhood system for a topology T on X. Let X be the unique neighborhood of the point x0 and let the basic neighborhoods of the points x of IR be the open intervals in IR of the form (x− n1 , x+ n1 ), where n ∈ IN+ . Then, obviously, (X, T) is a compact semiregular T0 -space, IR is a dense open subset of X, the induced topology on IR is the usual one and RC(X) = {clX (U ) : U ⊆ IR, U ∈ T}. It is clear that if A ⊆ X then x0 ∈ clX (A). Thus, if x, y ∈ IR then σ = σx ∪ σy is a clan in the CA (RC(X), CX ). Therefore σ is a clan in the CA (RC(X), CX ) which is not a point-clan. This means that X is not a C-semiregular space. u t Remark 4.1. Let us note that if X is a compact semiregular T0 -space which is not C-semiregular (like that from Example 4.2) and if we set B = (RC(X), CX ) then, by [2, Theorem 5.1], there exists a Csemiregular space Y such that B is isomorphic to the CA (RC(Y ), CY ). Hence the contact algebras (RC(X), CX ) and (RC(Y ), CY ) are isomorphic but the spaces X and Y are not homeomorphic. Acknowledgements. We are very grateful to the referee for his/her helpful remarks, suggestions and comments.

References [1] Ad´amek, J., Herrlich, H., Strecker, G.: Abstract and Concrete Categories, Wiley Interscience, 1990. [2] Dimov, G., Vakarelov, D.: Contact algebras and region-based theory of space: A proximity approach – I, Fund. Informaticae, 74, 2006, 209–249.

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[3] D¨untsch, I., Winter, M.: Remarks on lattices of contact relations, Research Report, Department of Computer Science, Brock University, 2006. [4] D¨untsch, I., Winter, M.: A representation theorem for Boolean contact algebras, Theoretical Computer Science (B), 347, 2005, 498–512. [5] Engelking, R.: General Topology, PWN, 1977. [6] Evans, J. W., Harary, F., Lynn, M. S.: On the computer enumeration of finite topologies, Comm. Assoc. Comput. Mach., 10, 1967, 295–298. [7] Harris, D.: Regular-closed spaces and proximities, Pacific J. Math., 34, 1970, 675–686. [8] Krishnamurthy, V.: On the number of topologies on a finite set, Amer. Math. Monthly, 73, 1966, 154–157. [9] Leader, S.: Local proximity spaces, Math. Annalen, 169, 1967, 275–281. [10] Naimpally, S. A., Warrack, B. D.: Proximity Spaces, Cambridge University Press, 1970. [11] Porter, J. R., Votaw, C.: S(α) spaces and regular Hausdorff extensions, Pacific J. Math., 45, 1973, 327–345. [12] Sikorski, R.: Boolean Algebras, Springer-Verlag, 1964. [13] Stong, R. E.: Finite topological spaces, Trans. Amer. Math. Soc., 123, 1966, 325–340. [14] Vakarelov, D., Dimov, G., D¨untsch, I., Bennett, B.: A proximity approach to some region-based theories of space, J. Applied Non-Classical Logics, 12, 2002, 527–559. ˇ [15] Cech, E.: Topological Spaces, Interscience, 1966.

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Appendix Index LE-proximity, 252 (LE1)-(LE4), 253 A ≺ B, 253 (SYM), 253 LO-proximity, 253 separated LE-proximity, 253 (SEP), 253 Efremoviˇc proximity, 253 (EFR), 253 EF-proximity, 253 Aα , 253 τ (α), 253 compatible topology and proximity, 253 proximity on a topological space, 253 (CSYM), 253 CA-proximity, 253 CA-proximity space, 253 proximity model, 254 e-filter, 254 minimal e-filter, 254 TCA-proximity, 254 (TCAP), 254 TCA-proximity space, 254 αw , 255 ECA-proximity, 255 (ECAP), 255 ECA-proximity space, 255 (ECAP0 ), 255 N-proximity, 256 (NP), 256 N-proximity space, 256 regular co-cluster, 256 NCA-proximity, 257 NCA-proximity space, 257 basic proximity, 258 R-proximity, 258 (RP), 258 LR-proximity, 258 round filter, 258 round end, 258 c-proximity, 258 basic proximity space, 258 R-proximity space, 258 LR-proximity space, 258 c-proximity space, 258 RCA-proximity, 259 RCA-proximity space, 259 local proximity space, 261

C-system, 262 (INJ), 262 (OCE), 262 (T1S), 262 (T2S), 262 gX , 262 T-system, 262 T0 -system, 262 complete T0 -system, 262 complete T-system, 262 connected T0 -system, 262 connected T-system, 262 XZ , 262 B Z , 262 κZ , 262 T1 -system, 264 T2 -system, 264 complete Ti -system, i = 1, 2, 264 WR-system, 265 NR-system, 265 R-system, 265 WR1-system, 265 WR2-system, 265 NR2-system, 265 complete WR-system, etc., 265 s-map, 267 SK, 268 SFR, 268 SH, 268 WRK, 268 WRFR, 268 WRH, 268 NRFR, 268 NRH, 268 RH, 268 KS, 268 FRS, 268 HS, 268 WRKS, 268 WRFRS, 268 WRHS, 268 NRS, 268 NRHS, 268 RHS, 268 SKC, 268 KSC, 268 rank of a finite topological space, 271 β CX , 272 βX , 273 β , 273 TI-system, 273 complete TI-system, 273

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Tych-system, 273 TYCH, 273 TYCHSYS, 273 LC, 275 XB¯ , 275 LCS, 275 CHS, 276 CH, 276 CSKS, 277

CSK, 277 CMSS, 277 CWRS, 277 CNRFS, 277 OCERS, 277 CMS, 277 CWR, 277 CNRF, 277 OCER, 277