A Study in Grzegorczyk Point-Free Topology Part I: Separation and Grzegorczyk Structures

This is the first, out of two papers, devoted to Andrzej Grzegorczyk’s point-free system of topology from Grzegorczyk (Synthese 12(2–3):228–235, 1960. https://doi.org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation (the dual notion of connection). Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze (quasi-)separation structures and Grzegorczyk structures, and establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk’s corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces.


Introduction
Andrzej Grzegorczyk's paper "Axiomatizability of geometry without points" [10] is devoted to construction of points and topological spaces thereof. 1 The presentation is based on a theory of mereological fields, whose primitive notions are spatial body and containment of one body in another (see Section 2). These two are enriched with the binary relation of being separated characterized by additional postulates.
(i) if x y then there exists the difference of x and y, i.e., the largest region x − y which is part of x and is incompatible with y (x and y do not have a common part); (ii) x and y have the least upper bound which is their mereological sum x y.
Note that, by (i), for all x and y from R we obtain: (iii) if x and y have at least one common part, then they have the greatest lower bound x y.
These conditions correspond directly to the theory from [9], composed of the axioms M1-M6, as denoted in the paper. The above-mentioned strengthenings we have in mind concern existence of mereological sums of infinite sets and existence of the unity. For example, the proofs of propositions 5.9 and 6.6, and of Corollary 6.7 witness applications of a bounded version of sum existence axiom from page 28 (w∃sum) for infinite sets (finite sets always have sums due to accepted axioms), and some of the representation theorems in the second part of the paper will require its stronger version (∃sum), which we introduce on page 10. We proceed along the similar line of thought with proofs concerning separations structures, i.e., we prove as much as we can with only three axioms put upon separation relation, since these are enough to demonstrate interesting results.
The main achievement of the first part is analysis of the notion of representative of a point (intuitively: the collection of regions shrinking to a unique location in space), with the proofs of facts, that under some standard topological interpretation of regions, parthood and separation (see (df ][)): • in the class of Urysohn spaces with countable chain condition, to every representative of point of a space S corresponds a unique point of S, • the class of regular open sets of any first-countable Tychonoff space satisfies all axioms of Grzegorczyk's from [10]; this, in a nutshell, means that any first-countable Tychonoff space has enough point representatives in the sense that the place of contact of regions of the space is represented by at least one pre-point in the sense of Grzegorczyk's (in consequence it is also represented by a point, but this will be the object of our study in the second part).
Terminology and properties of topological spaces we make use of in the paper are presented in the "Appendix" on page 39.

Basic Properties of Parthood
We assume that R = ∅ is a set of regions and ⊆ R × R is a parthood relation which is reflexive, antisymmetric and transitive. These postulates correspond directly to the axioms M1-M3 from [9, p. 91].
By means of we introduce three auxiliary relations: of being proper part, of overlapping and of being exterior to, respectively: x y :⇐⇒ x y ∧ x = y , (df ) x y :⇐⇒ ∃ z∈R (z x ∧ z y), (df ) x y :⇐⇒ ¬ x y.
(df ) In the case x y (resp. x y; x y) we say that x is proper part of y (resp. x overlaps y; x is exterior to y). By definitions, and are symmetric; so if x y (resp. x y), then we also say that x and y overlap (resp. are exterior to each other). Moreover, is reflexive and is irreflexive, and, of course, is irreflexive, transitive, and asymmetric. The next axiom: says that if x is not part of y then there is part z of x which not only is exterior to y but is also the largest among all parts of x which are exterior to y. By means of simple logical transformations and (df ) one may show that (∃−) is equivalent to the axiom M4 from [9, p. 91]. So for all regions x and y such that x y, the axiom (∃−) postulates existence of unique region which can be treated as the difference of x and y (or the relative complement of y with respect to x), and will be denoted by 'x − y'. Moreover, the region x − y is equal to the least upper bound of the set {u ∈ R | u x ∧ u y}. Thus, for any x, y ∈ R such that x y we put: 4 4 The Greek letter 'ι' stands for the standard description operator. The expression (ι x) ϕ(x) is read "the only object x which satisfies the condition ϕ(x)". To use 'ι' we first have to ensure both existence and uniqueness of the object that satisfies ϕ, i.e., we have: In an obvious way, from (∃−) we obtain that the partial order is separative: The formula (sep ) traditionally bears the name of Strong Supplementation Principle. From (sep ) we obtain the so-called Weak Supplementation Principle: If R has at least two members, then the set R has at least two members which are exterior to each other 5 : From this we obtain that non-trivial structures do not have zero-element, i.e.: Our last basic axiom has the following form: (∃sup 2 ) (∃sup 2 ) is a direct counterpart of Grzegorczyk's M5 from [9] (where it was formulated only by means of primitive notions) and it says that any regions x and y have the least upper bound. Due to the absence of zero there cannot exist unrestricted infimum operation. Yet still-thanks to the following lemma-we can define a partial operation of mereological product. Lemma 2.1. ([17]) The following condition: Proof. Indeed, if x y then we put z := x. If x y and x y, then we put Moreover, assume towards contradiction that for some u: u x, u y, and u x−(x−y). Hence, by (∃−), for some v we have: v u and v x−(x−y).
So v x and v y; and also v y. Hence v x − y; so v y.
(∃inf 2 ) says that any overlapping regions x and y have the greatest lower bound. This condition (again, formulated in the primitive terms only) is found in [9] as the axiom M6, and due to Lemma 2.1, it is redundant. Therefore, naming the structures satisfying M1-M6 Grzegorczyk mereologies we see that the class MF defined by antisymmetry, transitivity, (∃−), and (∃sup 2 ) is nothing but the class of all Grzegorczyk mereologies.
The classical Leśniewski mereology is based on the notion of mereological sum (not that of supremum). We now show that the theory of the class MF is closely related to the aforementioned notion.

Mereological Sums
We define a relation sum ⊆ R × P(R) by means of the following formula: 6 and say that z is a mereological sum of all members of X in case z sum X. By reflexivity we have that sum ⊆ R × P + (R), i.e.: It is known that (cf. e.g. [15][16][17]) antisymmetry, transitivity and (sep ) guarantee the uniqueness of mereological sum: and we obtain: From the same conditions it follows that (see [8,15,17]): Thus, for any mereological field R, we have that: the field is non-trivial iff the relation sum is equal to the relation of being the least upper bound.
In light of (2.1), on the base of the remaining axioms, our last axiom (∃sup 2 ) is equivalent to: So, via (∃sup 2 ), we postulate existence of mereological sums for arbitrary pairs of regions. Moreover, by (df −), for all regions x and y such that x y we obtain: x − y sum {z ∈ R | z x ∧ z y}.

Binary Operations of Sum and Product
By (∃sup 2 ), (∃sum 2 ), and (2.1) for all x, y ∈ R we can put: For all regions x, y and z we have (cf. e.g. [17]): For all overlapping regions x and y, the infimum of the set {x, y} will be denoted by 'x y' and will be called the mereological product of x and y. Formally, for all x, y ∈ R such that x y we put: For all overlapping regions x and y we have (cf. e.g. [17]): (2.6)

The Unity
We call the unity (sometimes the space) the maximum region in a mereological field M, if such a maximum exists (and in such case it will be denoted by '1'). For example: . For any set S the structure P + (S), ⊆ is a mereological field with the unity S. But if S is infinite, then P + fin (S), ⊆ is a mereological field without unity. It is also interesting that P + fin (S) ∪ {S}, ⊆ / ∈ MF (cf. Remark 2.1).
Convention. If K is a class of structures and ϕ 1 , . . . , ϕ n are conditions formulated in their language, then: K + ϕ 1 + · · · + ϕ n is a subclass of K which consists of all structures from K which additionally satisfy all ϕ 1 , . . . , ϕ n . Moreover, if among structures from K there are such that have unity, K1 is a subclass of K restricted to its elements with the unity.
According to the above convention, if (∃1) is the condition which postulates existence of unity, MF1 := MF + (∃1) and thanks to Model 2.1 we have: MF1 MF.
It is worth observing that structures from MF\MF1 are not only infinite, but also have a property postulated by Whitehead [25]: ∈ MF and for any x ∈ R we put R x := {y ∈ R | y x}. Then the structure R x, | R x is a mereological field with the unity x.
Proof. By (df ) the subset R x is closed under . By (2.5), R x is closed under for any two overlapping members of R x. Moreover, if y x, z x, and y z, then y − z y x.

Mereological Complement
Let R, be any mereological field with the unity 1. Then for any x ∈ R we have: Hence, by (WSP), since all members of R overlap 1, we have: For any x ∈ R such that x = 1 we can define: The object −x will be called the mereological complement of x. The operation of complement has the following properties (cf. e.g. [17]):

Mereological Fields and Boolean Lattices
There is a strong kinship between mereological fields with unity and Boolean lattices (i.e. lattices that are bounded, complemented and distributive), expressed in the following theorems.  (ii) Let R 0 , +, ·, -, 0 , 1 be the non-trivial Boolean algebra obtained from the Boolean lattice R 0 , 0 , 0 , 1 . Then for all x, y ∈ R 0 we have: In consequence, if in the set R 0 we define three operations +, ·, and -, using equations from (ii), then from the mereological field R, , 1 we obtain a non-trivial Boolean algebra R 0 , +, ·, -, 0 , 1 , the same that we obtain from the Boolean lattice R 0 , 0 , 0 , 1 .
Indeed, assume towards contradiction that R 1 , 1 is a mereological field. Then 1 is the unity of R 1 , 1 , 1 = , 1 | R = , and for any x ∈ R: (ii) Notice that R, cannot be created from any mereological field R , with the unity 1 by deleting this unity. Indeed, assume towards contradiction that for R , we have R := R \{1} and := | R . Then = | R and 1 − x ∈ R, for any x ∈ R. Hence

Mereological Structures
A mereological structure is any separative poset R, that satisfies the following condition: (∃sum) Since (u sum ) holds in all separative posets, so we also have: Since every nonempty subset of the domain R has the unique sum, we can introduce a unary operation on P + (R) of being the mereological sum of all members of a given non-empty set: All mereological structures satisfy (df −) and (2.1) (cf. e.g. [8,15]). So for any X ∈ P + (R) we have X = sup X. Moreover: Again, as in case of mereological fields, there is a strong dependence between mereological structures and complete Boolean lattices, expressed in theorems which we obtain, respectively, from Theorems 2.4-2.6 replacing the term 'mereological field with the unity' by the term 'mereological structure' and adding to the term 'Boolean lattice (algebra)' the word 'complete'. 7   Thus, mereological structures may be called complete mereological fields.
Remark 2.2. Biacino and Gerla [2] interpret the term 'mereological field' as "the structure obtained from a complete Boolean algebra B by deleting the zero-element, i.e., R = B − {0}" (p. 432). Therefore, their mereological fields are mereological structures, i.e., our complete mereological fields (cf. Theorem 2.8).  8 Then the pair rO, ⊆ is a complete Boolean lattice with the zero-element ∅, the unity S, and such that for all U, V ∈ rO: U ⊆ V iff Cl U ⊆ Cl V (see e.g. [13]). So, in the light of Theorem 2.8, the pair rO + , ⊆ is a mereological structure with := ⊆, where for all U, V ∈ rO + we have:

Atoms in Mereological Fields
Let M = R, ∈ MF. Due to absence of zero-element we have a "natural" notion of atom, according to which an atom is a member of R that has no proper parts. Let At M be the set of all atoms of M. We have that: We say that M is atomic iff for any x ∈ R there exists a ∈ At M such that a x.
We say that M is atomistic iff for every is trivial, then R = {1} = At M and 1 sum {1} and 1 = sup {1}. So, by the above lemma and (2.1), we obtain: M is atomistic iff M is atomic.
If M has the unity 1 then: M is atomic iff the non-trivial Boolean lattice R 0 , 0 , 0 , 1 is atomic. Existence of atoms is independent from all axioms listed above.
A subset of R is an antichain iff its any two distinct elements are exterior to each other. We say that a structure R, has the countable chain condition (abbrv.: c.c.c.) iff every antichain of its regions is countable.
Lemma 2.12. If M is infinite, then: 1. Either M is atomic and has infinitely many atoms, or for some x ∈ R the set R x is infinite.

M has some infinite antichain.
Proof. Suppose that R is infinite. Ad 1. If for any x ∈ R the set R x is finite, then M is atomic and At M is infinite, in light of Lemma 2.11. Ad 2. If M is atomic and has infinitely many atoms then At M is an infinite antichain. Otherwise, by the previous point, for some x ∈ R the set R x is infinite. Then, by Lemma 2.3 and Theorem 2.5, respectively, R x, | R x , x is a mereological field with the unity x and (R x) 0 , | 0 R x , 0 , x is a Boolean lattice to which we can apply Proposition 3.4 from Koppelberg [13] and obtain an infinite antichain in M.
We say that x ∈ R is atomless iff there is no atom a such that a x. M is atomless iff all its elements are atomless (iff At M = ∅).

Filters and Ultrafilters in Mereological Fields
A non-empty subset F of R is a filter in M ∈ MF iff F fulfills the following two conditions: • if x, y ∈ F , then both x y and x y ∈ F ; • if x ∈ F and x y, then y ∈ F . If M has the unity 1 then, obviously, 1 belongs to all filters in M.
We say that a filter F is an ultrafilter in M iff F is a maximal filter in We say that a non-empty subset X of R has finite intersection property (abbrv.: f.i.p.) iff for all x 1 , . . . , x n ∈ X (n > 0) there exists the product x 1 · · · x n . If X has f.i.p., then X generates the filter F X : is called a principal filter generated by x. Moreover, F a ∈ Ult(M), for any a ∈ At M . We also have: We now give general conditions for ultrafilters in structures from MF: 10 Proposition 2.15. For any filter F in M ∈ MF the following conditions are equivalent: x ∈ R and assume that for any y ∈ F we have y x. Thus F ∪ {x} has f.i.p. and generates the filter

Definition and Basic Properties
Let R be any non-empty set and and )( be binary relation in R. A triple R, , )( is a quasi-separation structure iff it satisfies the following conditions: So is a parthood relation, )( will be called a relation of being separated and in the case x )( y we say that x is separated from y or that x and y are separated, since the relation )( is symmetric, by (S2). The condition (S1) says that the relations and )( are disjoint. Thus, from (S1) we obtain that the relation )( is irreflexive, i.e.: (irr )( ) Moreover, the relation )( is included in the relation , i.e.: Indeed, assume towards contradiction that x )( y and x y, i.e., for some z both z x and z y. Then, by (S2) and (S3): z )( y and z )( x. So z y, by (S1). Finally, by (2.2) and (S3), we obtain: The class of all quasi-separation structures is defined as: The reason behind introducing )( is that the relation does not differentiate between two kinds of situations that may hold between regions. The first kind involves regions that are separated, the second one such that are externally tangent to each other (see Figure 1). Clearly, it must be the case that )( ⊆ ; cf. (I )( ). The notion of external tangency can be thus expressed by the following difference: \)(. So the motivation for introducing )( could be justified as follows: find a binary relation in R, which will share the essential properties of and will differentiate between regions that are external but are not tangent to each other and those that are both external and tangent (of course in the case there exist such regions in some structure R, , )( ).
A quasi-separation structure R, , )( has the unity iff R, ∈ MF1. In general, we say that a quasi-separation structure R, , )( is complete iff R, ∈ MS. Let qSep c be the class of all structures from qSep which are complete. Since MS MF1 MF, the inclusions qSep c qSep1 qSep hold by Proposition 3.1. Let R, , )( belongs to qSep1. Since all regions overlap the unity 1, then by (I )( ) we have: 2) Moreover, if x y, then x = 1 = y and x −y, by (2.8) and the properties of the operation of complement from pp. 8-9. By (S3) therefore we have: 11

The Relation of Connection of Regions
Let R = R, , )( be any quasi-separation structure. We introduce the following auxiliary binary relation C in R: which is called a relation of being connected ; in the case x C y we say that regions x and y are connected. Of course, by (df C) and, respectively, (S1)-(S3), (irr )( ), (I )( ), (3.1) the following conditions hold: The conditions (r C ), (C2), (C1), and (I C ) say, respectively, that the relation C is reflexive, symmetrical, and it includes the relations and . For any quasi-separation structure R, , )( with the unity 1, by (3.2), we obtain: ∀ x∈R x C 1.
Proposition 3.2. Any quasi-separation structure R, , )( is definitionally equivalent to the quasi-connection structure R, , C via (df C) and the following formula: 11 If there exists region u such that u and −u are separated, then the space 1 is not coherent, since 1 = u −u and u )( −u. Generally, we say that a region x is coherent iff x is not the sum of any separated regions, i.e., ¬ ∃y,z∈R(x = y z ∧ y )( z). The term and its definition come from [20]; see also [7]. For some complete G-structures there are regions that are separated from their complements (see e.g. Proposition 6.8).

Non-tangential Inclusion of Regions
From the point of view of developing topology on the basis of quasi-separation structures we need the binary relation of non-tangential inclusion between regions (i.e., if x y then we say that x is non-tangentially included in y or that x is a non-tangential part of y). Intuitively, we want to express the situation in which a region x is part of a region y and is separated from its complement (see Figure 2). 12 The following formula concerns such a situation in any quasi-separation structure R = R, , )( : i.e., x is non-tangentially included in y iff all regions exterior to y are separated from x. Of course, from (df ), (df ), and (df C) we have: i.e., x is non-tangentially included in y iff all regions connected to x also overlap y. The relation is included in the relation , i.e., we have: Indeed, assume that x y and x y. Then x − y y and x − y x, by (MF). Hence we obtain a contradiction: x − y )( x and x − y C x, by (df ) and (C1), respectively.
Moreover, the relation has the following properties: Indeed, for (3.5): If ∀ u∈R u y, i.e., if y is the unity, then there is no z ∈ R such that z y. For (3.6): Suppose that x y and there is u ∈ R such that u y. Then for z := u − y we have z y, by (MF). Hence z )( x, by (df ). Now notice that from (I ) and parthood antisymmetry we obtain: Moreover, we have the following two conditions: Thus, by (I ) and one of (3.7) and (3.8), we have: Now we prove: Proof. Ad 1. By (df ) and (I )( ): Finally, suppose that ⊆ . Then, since x x, so x x. Hence, by (df ), we have ∀ z∈R (z x ⇒ z )( x)), i.e., ⊆ )( . Ad 2. If R has the unity 1, then R × 1 ⊆ , by (3.5). Ad 3. If )( = ∅ and R does not have the unity, then for any y ∈ R there is z ∈ R such that z y. So = ∅, by (df ). Thus, if )( = ∅ and = ∅, then R has the unity 1. If x y and y = 1, then for some z ∈ R we have z y. So, by (df ), we obtain a contradiction: If R, , )( has the unity 1, then (3.5) and (3.6) have the following forms: We also have the following characterization of the relation : Indeed, for "⇒" suppose that x y and y = 1. Then −y y. So x )( −y, by (df ) and (S2). "⇐" If y = 1, then we use (3.5 ). Now suppose that y = 1, x )( −y, and z y. Then z −y. So z )( x, by (S3). Thus, x y. Finally, by Proposition 3.3 and (3.9), we obtain: Proposition 3.4. For any quasi-separation structure R, , )( with the unity 1: Proof. Ad 1. Let ⊆ . Then is reflexive, by Proposition 3.3. If y = 1 then y )( −y, by (df ), since y −y. Conversely, let for any y = 1: y )( −y. Suppose that x y. Then x )( −y, by (S2) and (S3). Hence x y, by (3.9).

Separation Structures
A quasi-separation structure R, , )( will be called a separation structure iff it satisfies both implications converse to (S3) and (3.1), i.e.: We define: A separation structure R, , )( is complete (resp. has the unity) iff R, ∈ MS (resp. R, ∈ MF1). Let Sep c be the class of complete separation structures. We have Sep c Sep1 Sep.
Model 4.1. Let R, be any mereological structure whose domain contains exactly seven elements, that is R, is obtained from the eight element atomic Boolean lattice by deleting zero. At R has three members a, b, c. For all x, y ∈ R we put: x )( y :⇐⇒ x, y ∈ At R ∧ x = y. We have ∅ = )( At R × At R and it is easy to check that the conditions (S1)-(S3) are satisfied. But this model does not satisfy (S4), since c )( a and c )( b, but c C a b. Moreover, (S5) is not satisfied either.
The conditions (S4) and (S5) are absent from Grzegorczyk's axiomatization, but they are consequences of (MF), (S1)-(S3) and his own axiom (G), that we call Grzegorczyk axiom (see Theorem 6.4). For this reason, we establish some properties of separation structures which will be useful later in examination of Grzegorczyk structures.
Finally, by application of (3.1), (3.3), (S4), and (C4), we get: Remark 4.1. In the literature there is no standard definition of a separation (resp. connection) structure. The axioms chosen by us may be considered as "natural" properties of separation (resp. contact) derived from basic geometrical intuitions concerning the space. Moreover, the axioms are either postulates or theorems of theories which are known as some standard approaches to the problem in the literature (see e.g. [1,2,7,20,24]).
Proposition 3.1 says that for any mereological field R, the triple R, , ∅ is a quasi-separation structure, where )( := ∅. The situation is different in case of non-trivial separation structures. Proof. If )( = ∅, then for all x, y ∈ R the condition ∀ z∈R (z )( y ⇔ z )( x) is trivially true. So x = y, by (S5), i.e., the structure is trivial. By (I )( ) and (I C ), respectively, we have suitable inclusions.
It is not difficult to notice that the relation shares all the properties expressed in the axioms (MF), (S1)-(S5). More interesting quasi-separation (resp. separation) structures are obtained from topological spaces by means of the following well-known method, which will be useful in various constructions further in the paper.
Let T = S, O be a topological space. Then rO + , ⊆ ∈ MS, by Lemma 2.9. In rO + we define the separation relation ][ by: Ad 2. By 1. and (3.9) for all U, V ∈ rO we obtain: ), but − Cl −V := Int(S\ Cl(Int(S\U )) = Int(S\(S\ Int(S\ − U ))) = Int Int(S\ Int(S\U )) = Int(S\ Int(S\U )) = Int(S\ Int(S\U )) = Int Cl U = U . Thus, we obtain: The complete structure rO + , ⊆, ][ will be called the quasi-separation structure associated with T and we write: qsepT . We have qsepT = qsepT sr , for any topological space T (see Remark 4.2). 14 Düntsch and Winter [4] prove this for regular closed sets, for which (S4) reduces to: C ][ D ⇐⇒ C ∩ D = ∅. As it can be seen in the proof of ours, the transition to regular open sets is not immediate and requires some effort. 15 Biacino and Gerla [2] state this fact without proof (as Theorem 3.3) unnecessarily requiring that T be a Hausdorff space. Moreover, by Proposition 4.3, if T is weakly regular, then qsepT belongs to Sep c and so it will be called the separation structure associated with T ; so we write: sepT . Notice that in this case we have sepT = sepT sr .

Representatives of Points and Points in Quasi-separation Structures
Let R = R, , )( be a quasi-separation structure. Every non-empty subset X of R which satisfies the following three conditions: will be called a representative of a point in R, or a pre-point of R, for short. Let Q R be the family of all pre-points of R. Let us analyze a couple of examples in order to grasp the geometrical meaning hidden in the definition of pre-points. For their formal description we may assume that our underlying structure is the Cartesian space R 2 , regions are its regular open non-empty subsets and non-tangential inclusion is interpreted as in Proposition 4.3. In Figure 3 a descending set X of cross-like regions is not a pre-point, since x and y overlap all regions in X, but are not connected with each other. So X does not meet the condition (r3). What X represents is rather a pair of perpendicular lines than a point.
Assuming completeness, in Figure 4 a descending set S of "unbounded" regions is not a pre-point, since both x := i∈ω x i and y := i∈ω y i (where ω is the set of all natural numbers) overlap all regions in S, but are not connected with each other. In consequence, S does not meet the condition (r3). Intuitively, the intention of the third condition is to eliminate points in infinity, and as we will show later, this works in the case of quasi-separation structures associated with a certain class of topological spaces.
If we treat the whole sheet of paper as the space, then the set of rectangular regions in Figure 5 is not a pre-point, since the relation ordering the rectangles is not non-tangential inclusion (only parthood). So this set does not meet the condition (r1).
In Figure 6 we consider pairs of circles with the same diameter as regions (which are not coherent). The set depicted above is not a pre-point. The regions x and y overlap all regions in this set but are separated. So the set does not meet the condition (r3). In this case the intention is to eliminate those sets of regions that represent more than one location in space.
Since the role of pre-points is to represent points, what are the points themselves? These are, in a given quasi-separation structure, all filters generated by pre-points. 16 16 We characterize the notion of point in quasi-separation structures similarly to Biacino and Gerla [2]. Grzegorczyk By a point in R we will mean any filter in the mereological field R, generated by some pre-point in R. Let Pt R be the set of all points in R. Thus, for any X ∈ P(R): We will denote elements of Pt R by means of small Gothic letters. The situation depicted in Figure 7 justifies the definition of point. If we agreed to treat pre-points as points then we would have the situation in which two pre-points representing the same location in space were different points. In other words, we would (usually) have more than one point in the same location, in extreme cases even uncountably many of them.
More formally, there is a class of topological spaces (e.g. first-countable Tychonoff spaces having the countable chain condition; which includes Euclidean spaces) such that for any space T = S, O from this class and for any point p ∈ S there are distinct pre-points Q 1 and Q 2 in qsepT (where Q 1 and Q 2 are non-empty subsets of rO + ) which correspond to p in the following sense: Q 1 = {p} = Q 2 . But in such case we obtain: F Q 1 = F Q 1 , i.e., pre-points Q 1 and Q 2 generate the same point (cf. Proposition 5.2 and Theorem 5.13).
Already for quasi-separation structures we get the following interesting facts: Proposition 5.2 (cf. [10]). Let R = R, , )( ∈ qSep. Then: 1. For all Q 1 , Q 2 ∈ Q R : if for all x ∈ Q 1 and y ∈ Q 2 we have x y, then Q 1 and Q 2 generate the same point, i.e., F Q 1 = F Q 1 .
2. For all p, q ∈ Pt R : if for all x ∈ p and y ∈ q we have x y, then p = q.
Proof. Ad 1. Assume towards contradiction that for all x ∈ Q 1 and y ∈ Q 2 we have x y, but F Q 1 = F Q 1 . Then either for some x 0 ∈ F Q 1 we have x 0 / ∈ F Q 2 , or for some y 0 ∈ F Q 2 we have y 0 / ∈ F Q 1 . In the first case for any y ∈ F Q 2 , y x 0 . Note that for some u ∈ Q 1 we have u x 0 . So, by (r2), for some by  (3.7). Moreover, by assumption, for any z ∈ Q 2 we have z x 1 . Let y be an arbitrary member of F Q 2 . Then for any z ∈ Q 2 , z y − x 0 (indeed, either y − x 0 y z or both y − x 0 y, z y, and z x 0 , so z − x 0 y − x 0 ). Hence, by (r3), x 1 C y − x 0 . Thus, we obtain a contradiction. Namely, since The second case is proved in an analogous way. Ad 2. Directly by 1. and definition of Pt R .
Thanks to Proposition 5.2(2), by definition of filters, we also have: In this way we obtain the next: Proof. Let Q ∈ Q R and p ∈ Pt R , i.e., p = F Q p := {y ∈ R | ∃ z∈Q p z y}, for some Q p ∈ Q R . Assume Q ⊆ p. Let x ∈ F Q , i.e., there is y 0 ∈ Q such that y 0 x. Since Q ⊆ p = F Q p , so there is z 0 ∈ Q p such that z 0 y 0 . Hence z 0 x. So x ∈ F Q p . Thus, we obtain F Q ⊆ F Q p . Hence, by Corollary 5.3, Generally, the theory of quasi-separation structures is too weak to prove that Q R = ∅. For example, the seven-element structure from Model 4.1 has no pre-points. However, in light of Lemma 5.5 below, in order to show separation structures without pre-points we must resort to atomless structures. Existence of such structures is a consequence of Proposition 5.9 (see also Proposition 5.10).
For G-structures existence of pre-points is guaranteed axiomatically and entails that Pt R = ∅. In Theorem 6.10 we prove that for any first-countable Tychonoff space T the separation structure sepT is a G-structure. Points of G-structures will be the object of our study in the second installment to this paper, for now we move on to formal description of representatives of points and their properties.
The following lemmas will be used in the sequel. Firstly, in the light of Proposition 4.2, for any mereological field R, , we have the separation structure R, , with )( := and C = and = (see Proposition 3.3). Let R = R, , )( ∈ qSep. For given X, Y ∈ P(R) we say that X is coinitial with Y iff ∀ y∈Y ∃ x∈X x y. Lemma 5.6. Let R ∈ qSep and Q ∈ Q R . All subsets of Q which are coinitial with Q also belong to Q R . In consequence, for any Proof. Suppose that X ⊆ Q and X is coinitial with Q ∈ Q R . Then (r1) is immediate, since Q satisfies (r1). For (r2) take x ∈ X. Q satisfies (r2) and x ∈ Q, so there is y ∈ Q such that y x. Since X is coinitial with Q, there is z ∈ X such that z y. But then z x, by (3.8). For (r3) assume that regions y and z are given such that ∀ u∈X (u y ∧ u z). Let v ∈ Q. Again, we use the assumption that X is coinitial with Q and take some x 0 ∈ X for which x 0 v. Moreover, x 0 y and x 0 z. So, by (MF), we get that v y and v z. Thus, x C y, by (r3) for Q, since v was arbitrary.
For the sake of the presentation, as we are interested in descending chains (with respect to ) being the representatives of points, we define the set X of regions to be well-ordered iff X is linearly ordered by and such that its every non-empty subset has the largest element with respect to (thus it may be said that X is dually well-ordered with respect to parthood).
For a limit non-zero ordinal λ, x α | α < λ is a transfinite sequence of regions indexed by elements of λ. For a given ordinal α, if there is an ordinal β such that α = 2 · β (where the dot is the ordinal multiplication), then α is even ordinal number, otherwise it is odd. For a given limit ordinal λ, E λ is the set of all even ordinals below λ and O λ is the set of all odd ordinals below λ. In the sequel we will use the standard set-theoretical result: Lemma 5.7 ([11] Exercise on p. 68 and Counting Theorem on p. 80). Every linearly ordered set L, ≤ has a coinitial well-ordered subset W, ≤ with ≤ := ≤ ∩ (W × W ). Since every well-ordered set is order-isomorphic to an ordinal α, the set W, ≤ may be arranged into a sequence w β | β < α such that for all β 1 , β 2 < δ, if β 1 < β 2 , then w β 1 ≤ w β 2 and w β 1 = w β 2 .
Thanks to this we can prove: Lemma 5.8. For any Q ∈ Q R there is Q ∈ Q R such that Q ⊆ Q, Q is coinitial with Q, and Q is well-ordered by . Moreover, for any x ∈ Q there is y ∈ Q such that y x.
Proof. By (r1), since ⊆ , Q is linearly ordered by . Therefore, by Lemma 5.7, there is Q ⊆ Q which is well-ordered by and coinitial with Q. Hence, Q ∈ Q R , by Lemma 5.6. Now let x ∈ Q. Since Q is coinitial with Q, there is z ∈ Q such that z x. By (r2) for Q , there is y ∈ Q such that y z. Hence y x, by (3.7).
Proposition 5.9. For each separation structure R = R, , satisfying the following weakened version of (∃sum): 17 no atomless region from R belongs to Q R .
Proof. Since )( = , we have that C = , = , and is reflexive. Suppose that R satisfies (w∃sum) and assume towards contradiction that x ∈ R is an atomless region which belongs to some pre-point Q in R. We consider two cases.
First, assume that ( †) Q has the minimal element y with respect to . Then y / ∈ At R , y x, and for some u we have u / ∈ Q and u y. Hence y − u y and y − u u, so y u and y y − u. By ( †), for any z ∈ Q we have z u and z y − u, which contradicts (r3), since u )( y − u. Second, assume that Q does not have the minimal element with respect to and consider the set Q x which belongs to Q R by Lemma 5.6. Moreover, by Lemma 5.8, there is a transfinite sequence y α | α < λ of regions from Q x such that y 0 = x, y α+1 y α , and the set Q := {y α | α < λ} is coinitial with Q x and belongs to Q R . For all α < λ there exists z α := y α − y α+1 and z α | α < λ is an antichain. We divide it into z α | α ∈ E λ and z β | β ∈ O λ . Both sequences are bounded by x. Thus, by (w∃sum), there are z 1 and z 2 such that z 1 := {z α | α ∈ E λ } and z 2 := {(z β | β ∈ O λ }. By construction, z 1 z 2 , i.e., z 1 )( z 2 . Of course, for any y α ∈ Q both y α z 1 and y α z 2 . So, by (r3) for Q , we obtain a contradiction: z 1 C z 2 .
Directly by the above proposition we have: For any topological space T = S, O , qsepT is the quasi-separation structure associated with T . In qsepT representatives of points are non-empty subfamilies of rO + , which satisfy (r1)-(r3). Note that, by (df ][) and Proposition 4.3, for any family X ∈ P + (O + ) the conditions (r1)-(r3) have the following form: These conditions can also be used for any non-empty family X of non-empty subsets of the set S.
Finally, if B p ⊆ rO + , then B p is a pre-point in sepT , since rB p ⊆ rO + .
Proposition 5.12. Let T = S, O be a T 1 -space such that for any point p there is a base B p at p satisfying (R1). Then: 1. T is a Hausdorff space, so T is weakly regular.

If T is second-countable, then T is perfectly normal.
3. qsepT belongs to Sep c (and therefore we refer to this structure as sepT ).  (2), for any p ∈ S, the family rB p is a base at p which satisfies (R1). The rest by Lemma 5.11.

For any p ∈ S the family rB
Generally it does not have to be the case that to every point of a given topological space T corresponds some pre-point Q of qsepT which uniquely determines this point. To be more precise, the following is not always true: for any point p of T there is a pre-point Q in qsepT such that Q = {p}. In the extreme case, we may take any set S with at least two points with the anti-discrete topology

If in addition T is second-countable, then T is perfectly normal and has
c.c.c., so in consequence sepT satisfies (IA) and has c.c.c. as a structure.
Proof. Ad 1. Let p ∈ S. By Lemma A.4 for some continuous function f : S → [0, 1] we have that f (p) = 0 and the family B p : Theorem 5.13 does not resolve the problem of uniqueness of points determined by intersections of pre-points, i.e., we would like to know as well for which class of spaces topologically interpreted pre-points are unambiguous in "pin-pointing unique locations in space", in the sense that for every prepoint there is exactly one point of the space which there is within every regular open set from the pre-point. In Theorem 5.17 we prove that all separation structures based on Urysohn spaces with c.c.c. have this property. In consequence we have that in the class of separation structures for firstcountable Tychonoff spaces with c.c.c. 19 every pre-point determines a unique point of the space, and to every point p corresponds a pre-point Q such that Q = {p}. This, in particular, shows that Grzegorczyk's definitions of pre-points and points are "correct" in the sense, that the aforementioned properties hold in the class of separations structures for Euclidean spaces (see Corollary 5.18), which is of course the subclass of the former class of structures.
In the proof of Lemma 5.15 we use the standard fact from set theory: Lemma 5.14. If λ > 0 is a countable limit ordinal, then cf(λ) = ω. Lemma 5.15. Let T = S, O be a topological space having c.c.c. and U be an infinite chain of open sets satisfying (R1). Then there is a monotone ωsequence of elements of U which is coinitial with U and such that Cl U n+1 U n , for any n ∈ ω.
Proof. In U we define: Then U , ≤ is a linearly ordered set in light of (R1). By Lemma 5.7, U , ≤ has a coinitial well-ordered subset V , ≤ , with ≤ := ≤ ∩ (V × V ), and we can arrange it into a sequence U α | α < λ , where for any α < λ we have: Let V α := U α \ Cl U α+2 . The sequence V α | α < λ is an antichain of non-empty open sets of T , and so it is countable. Hence λ is countable and cf(λ) = ω, by Lemma 5.14. Thus, there is a monotone function f : ω → λ such that f [ω] is unbounded in λ, and in consequence for any α ∈ λ there is n ∈ ω such that α < f(n). Hence U α ⊆ U f (n) . This shows that U f (n) | n < ω is coinitial with U α | α < λ , and so with U as well.
Now we take every second element from U f (n) | n < ω , i.e., we put A 0 := U f (0) and A n+1 := U f (2n) . This guarantees that Cl A n+1 A n and A n | n ∈ ω is a monotone ω-sequence which coinitial with U .
The condition (R3) can be expanded onto the whole family O + : U . Finally we prove that U = ∅. Assume U is empty. Then ( * * ): U must be infinite, since otherwise, by ( * ), it would have to contain the minimal set V , and then U = V = ∅. So, by ( * ), ( * * ), and c.c.c., from Lemma 5.15 follows existence of a monotone sequence U n | n < ω of sets from U which is coinitial with U and such that Cl U n+1 U n , for any n ∈ ω. For all n ∈ ω we put V n := U n \ Cl U n+1 ∈ O + , A := n∈ω V 4n , and B := n∈ω V 4n+2 . By construction, for any U ∈ U we have Notice that ( †): Cl A = n∈ω Cl V 4n and Cl B = n∈ω Cl V 4n+2 . Indeed, first, n∈ω Cl V 4n ⊆ Cl A. Second, for any n ∈ ω we have: Assume towards contradiction that p ∈ Cl A\ n∈ω Cl V 4n . Then p ∈ n>0 Cl U n . Yet the sequence Cl U n | 0 < n < ω is coinitial with the sequence U n | n < ω and therefore also with U . Hence we obtain a contradiction: p ∈ U . In a similar way we prove that Cl B = n∈ω Cl V 4n+2 .
By ( †) there are m, k ∈ ω such that the difference between them is at least two, say k m + 2, and Thanks to Lemma  Proof. Just note that for any non-empty subfamily X of P + (S): if X satisfies (R3), then X satisfies (R3 • ). Indeed, for any Let n > 0 and E (R n ) be the standard topology on the Cartesian product R n of the set of real numbers R. By the topological Euclidean n-space we mean the space E n := R n , E (R n ) . Let rE (R n ) be the family of all regular open sets of E n and let rE + (R n ) := rE (R n )\{∅}.

Definition and Basic Properties
Let R = R, , )( be a quasi-separation structure. The Grzegorczyk axiom says that for all connected x, y ∈ R there exists Q ∈ Q R such that: (g1) either x y or there is z ∈ Q such that z x y, (g2) for any z ∈ Q we have z x and z y.
Moreover we have: ∧ z y)) .  7). In all G-structures the following two facts hold.
Proposition 6.2. Every region has at least one non-tangential part.
Proof. By (6.2), for every x ∈ R there is a Q ∈ Q R with z ∈ Q such that z x. Q has the property (r2), so for some u ∈ Q we have u z, and thus u x, by (3.7).
Proposition 6.3. Every region is the mereological sum of its non-tangential parts.
Proof. Fix x ∈ R and put S := {z ∈ R | z x}. First, in light of (I ), for any z ∈ S we have z x. Second, let y x. Then, by Proposition 6.2, for some u we have u y. So u x, by (3.7). Hence u ∈ S and u y, by (I ).
Thus, x sum S.

Grzegorczyk Structures versus Separation Structures
Theorem 6.4 below is similar to results that can be found in [2, pp. 435-436]. The differences lie in two facts: in [2] the counterparts of the theorem were proven after the notion of point had been introduced, and different spaces were taken into account. Proof. Let R = R, , )( be any G-structure. Ad (S5): Suppose that x y. Then in R there exists x − y. By reflexivity of and (6.4) there are Q ∈ Q R and z ∈ Q such that z x − y. Hence, by (df ) and (S2), we have z )( y, since y x − y. On the other hand z x. Hence z C x, by (C1).
Ad (S4): Suppose that z C x y. Then, by (G), there exists Q ∈ Q R such that z x y ∨ ∃ u∈Q u z (x y) and ∀ u∈Q (u z ∧ u x y).
Indeed, suppose that Q Q x . Then for some u 0 ∈ Q we have u 0 x. Hence u 0 y, by (2.3), since u 0 x y. So u 0 ∈ Q y . Let now u be arbitrary member of Q. In the case when u = u 0 , u ∈ Q y . Otherwise, by (r1), either u 0 u or u u 0 . So, by (I ), either (a) u 0 u or (b) u u 0 . In (a) we have u y, since u 0 u and u 0 y. So u ∈ Q y . In (b) we have u x, since u u 0 and u 0 x. Hence u y, since u x y. So again u ∈ Q y , and Q = Q y . Now let u be an arbitrary member of Q. Suppose that Q x = Q. Then u x, u z and z C x, by (r3). Similarly, if Q y = Q then z C y. Thus, either z C x or z C y, as required.
We will show now that the class of separation structures is broader than that of G-structures; i.e., G Sep. By Proposition 5.10 and (6.1) we obtain: Proposition 6.5. If an atomless separation structure with )( := satisfies (w∃sum), then it is not a G-structure.
Thus, since the structure rE + (R n ), ⊆, is an atomless complete separation structure, it must be an element of Sep which is not in G.
More generally, by Proposition 5.9, we have: Proposition 6.6. If a separation structure with )( := satisfies (w∃sum) and has at least one atomless element, then it is not a G-structure.
Proof. Assume towards contradiction that R = R, , belongs to G and x ∈ R is an atomless region. Then, by (6.2), for some Q ∈ Q R there is z ∈ Q such that z x. Yet then z must be atomless, which contradicts Proposition 5.9.
In consequence, from Theorem 6.4 and Propositions 4.1 and 6.6 we have: Corollary 6.7. Let R = R, , )( be a non-trivial G-structure. Then: 1. Both )( = ∅ and C = R × R. 2. If R satisfies (w∃sum) and has at least one atomless element, then Therefore for all members of the class G+(w∃sum) (and the more so of G c ) the counterpart of Proposition 4.2 holds only for atomic structures. Thus, in (standard) G-structures (see Theorem 6.11) separation is different from disjointness: )( = . However, it is not excluded that in some G-structures we have )( = . It will hold true, for example, for all atomic G-structures of the form R, , .
Proposition 6.8. Every atomic separation structure R = R, , is a Gstructure.
Proof. First, we show that R satisfies (G ). Let x, y ∈ R be such that x y. Then in R there are the product x y and a ∈ At R such that a x y. But {a} ∈ Q R , by Lemma 5.5. Second, since C = , the condition 'x C y ∧ x y' is false for all x, y ∈ R. Hence (G ) also holds.
The following general result will be used in the sequel: Theorem 6.9. Let T = S, O be a T 1 -space such that for any point p ∈ S there is a base B p at p satisfying (R1). Then sepT belongs to G c .
Proof. By Proposition 5.12, for any p ∈ S, the family rB p := {Int Cl B | B ∈ B p } is a base at p and it is a pre-point in sepT .
For (G ): Suppose that for U, V ∈ rO + we have U V , i.e., U ∩ V = ∅. Let p ∈ U ∩ V ∈ rO. Then for some Z ∈ rB p we have Z ⊆ U ∩ V , i.e., Z U V .
For (G ): Suppose that for U, V ∈ rO + we have U C V , i.e., Cl U ∩ Cl V = ∅. Let p ∈ Cl U ∩ Cl V . Then for any A ∈ rB p : A ∩ U = ∅ = Z ∩ V ; i.e., Z U and Z V . By Theorems 6.9 and 5.13 we obtain: Theorem 6.10. If T is a first-countable Tychonoff space, then sepT belongs to G c .
Of course, for any n > 0, the finitely dimensional topological Euclidean space E n (see p. 33) is a Tychonoff space. Thus, by Theorem 6.10, we have: Corollary 6.11. For any n > 0, the structure sepE n belongs to G c , satisfies (IA), and has c.c.c.
The following is another explanation of this conclusion. Thanks to Corollary 5.18, for any p ∈ R n the family B p of all open balls with center at p is a base at p, which is a pre-point in sepE n , and thus satisfies (R1). So we can refert Theorem 6.9.
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A. Appendix: Definitions and Facts from Topology
The set of all real numbers is denoted by R, ω (resp. ω + ) denotes the set of all (resp. all positive) natural numbers.
Let T := S, O be a topological space. Let C be the family of all closed sets of T , and let Int and Cl be standard interior and closure operations of T . For any V ∈ O we have V ⊆ Int Cl V and Cl V = Cl Int Cl V . For a given p ∈ S we put O p := {U ∈ O | p ∈ U }.
In the standard way we define T 1 , T 2 (of Hausdorff ), T 2 1 2 (or Urysohn), T 3 (or regular ), T 3 1 2 (or completely regular, or a Tychonoff ), T 4 (or normal ) and perfectly normal spaces (we include T 1 in the definitions of T 3 -T 4 ).
We define a base B of T in the standard way. For any point p, a family B p ∈ P(O) is called a base for T at p iff B p ⊆ O p and for any U ∈ O p there exists V ∈ B p such that V ⊆ U . If B is a base for T then for any p ∈ S the family B p := {U ∈ F | p ∈ U } is a base at p. On the other hand, if for any p ∈ S a base B p at p is given, then the union p∈S B p is a base for T . The space T is semiregular iff T has a base consisting of regular open sets. 22 We say that T is weakly regular iff T is semiregular and for any U ∈ O + there is V ∈ O + such that Cl V ⊆ U iff T is semiregular and for any U ∈ O + there is V ∈ rO + such that Cl V ⊆ U . Not all semiregular Urysohn spaces are regular, nor all connected. Semiregular Hausdorff spaces are weakly regular, but it is known that all regular spaces are weakly regular: Proof. Let p ∈ S and {B n | n ∈ ω + } be a countable base at p ∈ S such that B n = S, for all n ∈ ω + . For all n ∈ ω + we put C n := S\B n ; so ∅ = C n ∈ C and p / ∈ C n . Then for all n ∈ ω + , we have a continuous mapping f n : S → [0, 1] such that f n (p) = 0 and f n [C n ] = {1}. We define the continuous function f : S → [0, 1] such that f (q) := ∞ n=1 f n (q) 2 n , for any q ∈ S. The collection of all sets U n := f −1 [[0, 1 2 n )] (with n ∈ ω + ) is a base at p. To see this, we take V ∈ O p . Then, by assumption, for some k ∈ ω we have B k ⊆ V . But U k ∩ C k = ∅, because for any q ∈ C k we have f (q) > 1 2 k , and therefore U k ⊆ B k ⊆ V .
Let us remind that antichains of open sets of a given topological space are families of pairwise disjoint open sets (algebraically speaking, these are antichains in the lattice of open sets of the space). We say that a given topological space has the countable chain condition (abbrv.: c.c.c.) iff every antichain of its open sets is countable. If T = S, O is semiregular then T sr = T , since rO is a base for T .