ProtasovAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 9, No. 2, 2008 pp. 189-195 Asymptotic proximities I. V. Protasov Abstract. A ballean is a set endowed with some family of subsets which are called the balls. The properties of the family of balls are pos- tulated in such a way that the balleans can be considered as a natural asymptotic counterparts of the uniform topological spaces. We intro- duce and study an asymptotic proximity as a counterpart of proximity relation for uniform topological space. 2000 AMS Classification: 54E05, 54E15. Keywords: ballean, determining covering, proximity. 1. Introduction and preliminaries A ball structure is a triple B = (X, P, B) where X, P are non-empty sets and, for any x ∈ X and α ∈ P , B(x, α) is a subset of X which is called a ball of radius α around x. It is supposed that x ∈ B(x, α) for all x ∈ X, α ∈ P . The set X is called the support of B, P is called the set of radii. Given any x ∈ X, A ⊆ X, α ∈ P , we put B ∗(x, α) = {y ∈ X : x ∈ B(y, α)}, B(A, α) = ⋃ a∈A B(a, α). A ball structure is called • lower symmetric if, for any α, β ∈ P , there exist α′, β′ ∈ P such that, for every x ∈ X, B ∗(x, α′) ⊆ B(x, α), B(x, β′) ⊆ B∗(x, β); • upper symmetric if, for any α, β ∈ P , there exist α′, β′ ∈ P such that, for every x ∈ X, B(x, α) ⊆ B∗(x, α′), B∗(x, β) ⊆ B(x, β′); 190 I. V. Protasov • lower multiplicative if, for any α, β ∈ P , there exists γ ∈ P such that, for every x ∈ X, B(B(x, γ), γ) ⊆ B(x, α) ∩ B(x, β); • upper multiplicative if, for any α, β ∈ P , there exists γ ∈ P such that, for every x ∈ X, B(B(x, α), β) ⊆ B(x, γ). Let B = (X, P, B) be a lower symmetric and lower multiplicative ball struc- ture. Then the family { ⋃ x∈X B(x, α) × B(x, α) : α ∈ P } is a base of entourages for some (uniquely determined) uniformity on X. On the other hand, if U ⊆ X × X is a uniformity on X, then the ball structure (X, U, B) is lower symmetric and lower multiplicative, where B(x, U ) = {y ∈ X : (x, y) ∈ U}. Thus, the lower symmetric and lower multiplicative ball structures can be identified with the uniform topological spaces. We say that a ball structure is a ballean if B is upper symmetric and upper multiplicative. A structure on X, equivalent to a ballean, can also be defined in terminology of entourages. In this case it is called a coarse structure [5]. For motivations to study balleans see [1],[4],[5]. Let B1 = (X1, P1, B1) and B2 = (X2, P2, B2) be balleans. A mapping f : X1 → X2 is called a ≺-mapping if, for every α ∈ P1, there exists β ∈ P2 such that, for every x ∈ X1, f (B1(x, α)) ⊆ B2(f (x), β). A bijection f : X1 −→ X2 is called an asymorphism between B1 and B2 if f and f −1 are ≺-mappings. Let B1, B2 be balleans with common support X. We say that B1 ≺ B2 if the identity mapping id:X → X is a ≺-mapping of B1 to B2. If B1 ≺ B2 and B2 ≺ B1, we say that B1 and B2 coincide and write B1 = B2. Let B = (X, P, B) be a ballean. A subset Y ⊆ X is called bounded if there exist α ∈ P such that Y ⊆ B(x, α) for some x ∈ Y . A family F of subsets of X is called uniformly bounded if there exists α ∈ P such that F ⊆ B(x, α) for all F ∈ F, x ∈ F . We use the following observation: the ballean B1 and B2 with common support coincide if and only if every family of subsets of X uniformly bounded in B1 is uniformly bounded in B2 and vise versa. For an arbitrary ballean B = (X, P, B) we define preordering 6 on the set P by the rule: α 6 β if and only if B(x, α) ⊆ B(x, β) for every x ∈ X. A subset P ′ ⊆ P is called cofinal if, for every α ∈ P , there exists α′ ∈ P ′ such that α 6 α′. A ballean B is called connected if, for any x, y ∈ X, there exists α ∈ P such that y ∈ B(x, α). A connected ballean B is called ordinal if there exists a well-ordered by 6 subset P ′ of P . Asymptotic proximities 191 Every metric space (X, d) determines the metric ballean (X, R+, Bd) where Bd(x, r) = {y ∈ X : d(x, y) ≤ r}. A ballean is called metrizable if it is asymor- phic to some metric ballean. By [4, Theorem 9.1], a ballean B = (X, P, B) is metrizable if and only if B is connected and P has a countable cofinal subset. Clearly, every metrizable ballean is ordinal. We begin the proper exposition with characterization (section 2) of families of coverings of a set X which determine a ballean on X. Then we introduce and study (section 3) an asymptotic proximity as an equivalence relation σ on the family P(X) of all subsets of a set X such that Y ⊆ Z ⊆ Y ′ and Y σY ′ imply Y σZ. Every proximity σ determines some ballean B(σ) on X. Given a ballean B = (X, P, B), we say that the subsets Y, Z of X are close if there exists α ∈ P such that Y ⊆ B(Z, α), Z ⊆ B(Y, α). The closeness relation is a prototype for the asymptotic proximity. We show (Theorem 3.1) that, given an asymptotic proximity σ on P(X), the closeness σ′ defined by B(σ) is finner then σ. On the other hand (Theorem 3.4), if B = (X, P, B) is a ballean and σ is a closeness on P(X) determined by B, then σ = σ′ where σ′ is closeness determined by B(σ). In Section 4 we examine the question whether the closeness on P(X) arising from a ballean B = (X, P, B) determines B. In general case this is not so, but our main result (Theorem 4.2) gives a positive answer in the case of ordinal (in particular, metrizable) balleans. 2. Determining coverings Let X be a set, F be a family of subsets of X, Y ⊆ X. We put st(Y, F) = ⋃ {F ∈ F : Y ⋂ F 6= ∅}. Given any x ∈ X, we write st(x, F) instead of st({x}, F). For two families F, F′ of subsets of X, we put st(F, F′) = {st(F, F′) : F ∈ F}. A family F of subsets of Xis called hereditary if, for any subsets F, F ′ of X such that F ∈ F and F ′ ⊆ F , we have F ′ ⊆ F. A family F of subsets of X is called a covering if ⋃ F = X. We say that a family {Fα, α ∈ P } of hereditary coverings of X is star stable if, for any α, β ∈ P , there exist γ ∈ P such that st(Fα, Fβ ) ⊆ Fγ . Let {Fα : α ∈ P } be a family of star stable coverings of X. We consider a ball structure B = (X, P, B), where B(x, α) = st(x, Fα), and show that B is a ballean. Given any x ∈ X and α ∈ P , we have B(x, α) = {y ∈ X : y ∈ st(x, Fα)}, B ∗(x, α) = {y ∈ X : x ∈ st(y, Fα)}. Since y ∈ st(x, Fα) if and only if x ∈ st(y, Fα), then B ∗(x, α) = B(x, α), so B is upper symmetric. 192 I. V. Protasov Given any x ∈ X and α, β ∈ P , we choose α′ ∈ P and γ ∈ P such that st(Fα, Fα) ⊆ Fα′ and st(Fα′ , Fβ ) ⊆ Fγ . Then we have B(B(x, α), β) = st(st(x, Fα), Fβ) ⊆ st(x, Fγ ) = B(x, γ), so B is upper multiplicative. We note that a subset Y of X is bounded in B if and only if Y ∈ Fα for some α ∈ P . A family F of subsets of X is bounded in B if and only if there exists α ∈ P such that F ⊆ Fα. Thus we have shown that every star stable family of coverings of X deter- mines some ballean on X. On the other hand, let B = (X, P, B) be an arbitrary ballean on X. For every α ∈ P , we put Fα = {F ⊆ X : F ⊆ B(x, α) f or some x ∈ X}. Then the ballean on X determined by the star stable family {Fα : α ∈ P } of coverings of X coincides with B. 3. Proximities and closeness Let X be a set, P(X) be a family of all subsets of X. Let σ be an equivalence on P(X) such that, for all Y, Y ′, Z ∈ P(X), Y ⊆ Z ⊆ Y ′, Y σY ′ =⇒ Y σZ. We say that σ is (an asymptotic) proximity and describe a way in which σ defines some ballean B(σ) on X. We call a family F of subsets of X to be non-expanding with respect to σ if, for every subset Y of X, we have Y σ(Y ⋃ st(Y, F)). We note that every subfamily of non-expanding family is non-expanding. Let F1, F2 be non-expanding with respect to σ families of subsets of X. We show that the family st(F1, F2) is also non-expanding with respect to σ. We fix an arbitrary subset Y of X and put F′2 = {F ′ ∈ F2 : Y ⋂ F ′ 6= ∅}. Since F′2 is non-expanding, we have Y σ(Y ⋃ ⋃ F′2). We put Z = Y ⋃ ⋃ F′2 and F′1 = {F ∈ F1 : F ⋂ F ′ 6= ∅ f or some F ′ ∈ F′2}. Since F′1 is non-expanding, we have Zσ(Z ⋃ ⋃ F′1). Asymptotic proximities 193 We put T = Z ⋃ ⋃ F′1. Since F2 is non-expanding, we have T σ(T ⋃ ( ⋃ {F ∈ F2 : F ⋂ T 6= ∅})). We put H = T ⋃ ( ⋃ {F ∈ F2 : F ⋂ T 6= ∅}). Then Y σH and Y ⊆ H. By the construction of H, we have Y ⊆ Y ⋃ ( ⋃ {S ∈ st(F1, F2) : S ⋂ Y 6= ∅}) ⊆ H. Since σ is a proximity, we conclude Y σ(Y ⋃ ( ⋃ {S ∈ st(F1, F2) : S ⋂ Y 6= ∅})). In particular, we proved that the family of all non-expanding (with respect to σ) hereditary covering of X is star stable. Following Section 2, we define B(σ) by means this family of coverings. We note that a subset Y of X is bounded in B(σ) if and only if the family {Y } is non-expanding, equivalently, {y}σY for every y ∈ Y . A family F of subsets of X is uniformly bounded in B(σ) if and only if F is non-expanding. Let B = (X, P, B) be a ballean. We consider a relation σ on P(X) defined by the rule: Y σZ if and only if there exists α ∈ P such that Y ⊆ B(Z, α), Z ⊆ B(Y, α). It is easy to see that σ is a proximity; we call it a closeness defined by B. We note that Y, Z are close if and only if there exists a uniformly bounded covering F of X such that ⋃ {F ∈ F : F ⋂ Y 6= ∅} = ⋃ {F ∈ F : F ⋂ Z 6= ∅}. Theorem 3.1. Let X be a set, σ be a proximity on P(X), σ′ be a closeness defined by B(σ). Then σ′ ⊆ σ. Proof. We remind that a family F of subsets of X is uniformly bounded in B(σ) if and only if F is non-expanding with respect to σ. Let Y, Z ∈ P(X) and Y σ′Z. Then there exists a non-expanding (with respect to σ) family F of subsets of X such that Y ⊆ ⋃ F, Z ⊆ ⋃ F and Y ⋂ F 6= ∅, Z ⋂ F 6= ∅ for every F ∈ F. It follows that Y σ( ⋃ F) and Zσ( ⋃ F), so Y σZ. 2 � The following two examples show that the proximity σ from Theorem 3.1 could be much more coarse than σ′. Example 3.2. Let X be an infinite set. We define an equivalence σ on P(X) by the rule: Y σZ if and only if either Y, Z are finite, or Y, Z are infinite. Then a subset Y of X is bounded in B(σ) if and only if Y is finite; a family F of subsets of X is uniformly bounded in B(σ) if and only if each subset F ∈ F is finite and, for every x ∈ X, the set {F ∈ F : x ∈ F } is finite. We show that Y σ′Z if and only if either Y, Z are finite, or Y, Z are infinite and |Y | = |Z|. We should only check that if Y, Z are infinite and |Y | = |Z| then Y σ′Z. To this end we fix some bijection f : Y −→ Z, and put F = {{y, f (y)} : y ∈ Y }. Then F is uniformly bounded in B(σ), Y σ′( ⋃ F) and Zσ′( ⋃ F), so Y σ′Z. Now if X is uncountable than σ is coarser than σ′. 2 194 I. V. Protasov Example 3.3. Let X be a well-ordered set. We define an equivalence σ on P(X) by the rule: Y σZ if and only if minY = minZ. Then a subset Y is bounded in B(σ) if and only if Y is a singleton. It follows that Y σ′Z if and only if Y = Z. 2 Theorem 3.4. Let B = (X, P, B) be a ballean, σ be a closeness defined by B, σ′ be a closeness defined by B(σ). Then σ = σ′. Proof. By Theorem 3.1, σ ⊆ σ′. To see that σ ⊆ σ′ it suffices to note that every uniformly bounded in B family of subsets of X is non-expanding with respect to σ. � 4. Does closeness determine a ballean? Let B1 and B2 be balleans with common support X, σ1 and σ2 be closeness on P(X) defined by B1 and B2. Is B1 = B2 provided that σ1 = σ2? We give a negative answer to this general question, but prove one partial statement (Theorem 4.2) in positive direction. Example 4.1. Let X be a countable set. We consider two families ϕ1, ϕ2 of coverings of X. A family ϕ1 is defined by the rule: F ∈ ϕ1 if and only if every subset F ∈ F is finite, and the set {F ∈ F : x ∈ F } is finite for every x ∈ X. A family ϕ2 is defined by the rule: F ∈ ϕ2 if and only if there exists a natural number n such that |F | ≤ n for every F ∈ F, and there exists a natural number m such that |{F ∈ F : x ∈ F }| ≤ m for every x ∈ X. Clearly, the families ϕ1 and ϕ2 are star-stable. Let B1 and B2 be balleans on X determined by ϕ1 and ϕ2. Using arguments from Example 3.2, it is easy to see that B1 and B2 define the same closeness σ: Y σZ if and only if either Y, Z are finite, or Y, Z are infinite. Then we take a partition {Fn : n ∈ ω} of X such that |Fn| = n for every n ∈ ω. Clearly, F is uniformly bounded in B1, but F is not uniformly bounded in B2. It follows that B1 is stronger than B2. It is worth to mark that Example 4.1 gives a ballean B with the closeness σ such that B 6= B(σ). To see this, we put B = B2 and note that B(σ) = B1. Theorem 4.2. Let B1 = (X1, P1, B1) and B2 = (X2, P2, B2) be ordinal balleans with common support and the same closeness. Then B1 = B2. Proof. We assume on the contrary that, say, B2 ≺ B1 does not hold, and choose β ∈ P2 such that, for every α ∈ P1, there exists x(α) ∈ X such that B2(x(α), β) * B1(x(α), α). We may suppose that P1 is well-ordered. In the proof of Theorem 2.1 from [3] we constructed inductively a subset Y = {y(α) : α ∈ P1} of X such that the family {B1(y(α), α) : α ∈ P1} is disjoint and, for every α′ ∈ P , B2(y(α ′), β) * ⋃ {B1(y(α), α) : α ∈ P1}. We put Z = B2(Y, β). Then Y, Z are close in B2, but Y, Z are not close in B1, whence a contradiction. � Asymptotic proximities 195 References [1] A. Dranishnikov, Asymptotic topology, Russian Math. Surveys 55 (2000), 71–116. [2] R. Engelking, General Topology, PWN, Warszava, 1985. [3] M. Filali and I. V. Protasov, S lowly oscillating function on locally compact groups, Applied General Topology 6, no. 1 (2005) 67-77. [4] I. V. Protasov and T. Banakh, Ball Structures and Colorings of Groups and Graphs, Math. Stud. Monogr. Ser. V.11, 2003. [5] J. Roe, Lectures on Coarse Geometry, AMS University Lecture Series, 31 (2003). Received March 2007 Accepted December 2007 I. V. Protasov (protasov@unicyb.kiev.ua) Department of Cybernetics, Kyiv National University, Volodimirska 64, Kyiv 01033, UKRAINE.