@ Applied General Topology c© Universidad Politécnica de Valencia Volume 11, No. 2, 2010 pp. 89-93 Thin subsets of balleans Ievgen Lutsenko and Igor Protasov Abstract. A ballean is a set endowed with some family of balls in such a way that a ballean can be considered as an asymptotic counter- part of a uniform topological space. We characterize the ideal generated by the family of all thin subsets in an ordinal ballean, and apply this characterization to metric spaces and groups. 2000 AMS Classification: 54A25, 54E25, 05A18. Keywords: ballean, thin subsets, ideal. Let G be a group with the identity e. A subset A ⊆ G is called thin if |gA ∩ A| < ℵ0 for every g ∈ G, g 6= e. For thin subsets, its modifications, applications and references see [4]. We denote by TG the family of all thin subsets of G. Then the smallest ideal T ∗G (in the Boolean algebra of all subsets of G) containing TG is the family of all finite unions of thin subsets. Thus, to characterize T ∗G, we need some test which, for given A ⊆ G and m ∈ N, detect whether A can be represented as a union of 6 m thin subsets. Let (X, d) be a metric space. We say that a subset A ⊆ X is thin if, for every r ∈ R+, there exists a bounded subset Y ⊆ X such that A ∩ B(x, r) = {x} for every x ∈ A \ Y , where Bd(x, r) = {x ∈ Y : d(x, y) 6 r}. As in the group case, to characterize the ideal T ∗(X, d) generated by the family T (X, d) of all thin subsets of (X, d), we ask for a test recognizing if a subset A ⊆ X is a union of 6 m thin subsets. It is easy to see that a subset A ⊆ G is thin if and only if, for every finite subset F of G containing e, there exists a finite subset Y of G such that A ∩ Fg = {g} for every x ∈ A \ Y . Following [1], we say that Fg is a ball of radius F around g. From this point of view, the definitions of the thin subsets in groups and metric spaces are very similar syntactically. To formalize this similarity we use the ballean approach from [5]. A ballean is a set endowed with some family of 90 Ie. Lutsenko and I. Protasov its subsets which are called the balls. The property of the family of ball are postulated in such a way that the balleans can be considered as the counterparts of the uniform topological spaces (see Section 1 for precise definition). In Section 1 we define the thin subsets of a ballean and, for every ordinal ballean, characterize the ideal generated by the thin subsets. The group and metric spaces have the natural ballean structures. In Section 2 we apply the result from Section 1 to justify the following two tests. A subset A of a metric space X can be partitioned in 6 m thin subsets if and only if, for every r ∈ R+, there exists a bounded subset Y ⊆ X such that |A ∩ B(x, r)| 6 m for every x ∈ A \ Y . A subset A of a countable group G can be partitioned in 6 m thin subsets if and only if, for every finite subset F of G, there exists a finite subset Y of G such that |A ∩ Fx| 6 m for every x ∈ A \ Y . We do not know whether this test is effective for an uncountable group. 1. Ballean context A ball structure is a triple B = (X, P, B), where X, P are not-empty sets and, for every 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 and α ∈ P . The set X is called the support of B, P is called the set of radii. Given any x ∈ X, A ⊆ X, we put B∗(x, α) = {y ∈ X : x ∈ B(y, α)}, B(A, α) = ⋃ a∈A B(a, α). A ball structures B is called a ballean 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′); • for any α, β ∈ P , there exist γ ∈ P such that, for every x ∈ X, B(B(x, α), β) ⊆ B(x, γ). We note that a ballean can also be defined in terms of entourages of diagonal in X × X. In this case it is called a coarse structures [7]. A ballean B is called connected if, for any x, y ∈ X, there exists α ∈ P such that y ∈ B(x, y). All balleans under considerations are suppose to be connected. Replacing each ball B(x, α) to B(x, α) ∩ B∗(x, α), we may suppose that B∗(x, α) = B(x, α) for all x ∈ X, α ∈ P . A subset Y ⊆ X is called bounded if there exist x ∈ X and α ∈ P such that Y ⊆ B(x, α). We use a preordering 6 on the support X of B defined by the rule: α 6 β if and only if B(x, α) 6 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 ordinal if there exists a cofinal subset P ′ ⊆ P well ordered by 6. Let B = (X, P, B) be a ballean, m ∈ N. We say that a subset A ⊆ X is m-thin if, for every α ∈ P , there exists a bounded subset Yα ⊆ X such that |B(x, α) ∩ A| 6 m for every x ∈ A \ Yα. A 1-thin subset is called thin. Thus, Thin subsets of balleans 91 A is thin if, for every α ∈ P , there exists a bounded subset Yα of X such that B(x, α) ∩ A = {x} for every x ∈ A \ Yα. In the terminology of [5], the thin subsets are called pseudodiscrete. For pseudodiscreteness see also [2], [6]. We use the following notation: T (B) is the family of all thin subsets of X; Tm(B) is the family of all m-thin subsets of X;⋃ m T (B) is the family of all unions of 6 m thin subsets of X; T ∗(B) is the ideal generated by T (B). Clearly, T ∗(B) = ⋃ m∈N( ⋃ m T (B)). Lemma 1.1. For every ballean B, we have ⋃ m T (B) ⊆ Tm(B). Proof. Let A1, . . . , An be thin subsets of X. For every α ∈ P , we pick γ(α) ∈ P such that B(B(x, α), α) = B(x, γ(α)). For all α ∈ P and i ∈ {1, . . . , m}, we choose a bounded subset Yα(i) such that B(x, α) ∩ Ai = {x} for every x ∈ Ai \Yα(i), and put Yα = Yα(1)∪. . .∪Yα(m). We take an arbitrary element a ∈ (A1 ∪. . .∪Am)\Yα and suppose that |B(a, α)∩(A1 ∪. . .∪Am)| > m. Then there exists j ∈ {1, . . . , m} such that |Aj ∩B(a, α)| > 2. Let b, c ∈ Ai ∩B(a, α), b 6= c. Then c ∈ B(b, γ(α)) contradicting the choice of Yα(j). � The following theorem gives a characterization of T ∗(B) in the case of an ordinal ballean B. Theorem 1.2. For every ordinal ballean B and m ∈ N, we have Tm(B) =⋃ m T (B). Proof. In view of Lemma 1.1, it suffices to show that Tm(B) ⊆ ⋃ m T (B). Let A ∈ Tm(B). We may suppose that P is will ordered by 6. We construct inductively a family {Yα : α ∈ P} of bounded subsets of X such that |B(x, α)∩ A| for every x ∈ A\Yα and Yα ⊆ Yβ for all α 6 β. Then we consider a graph Γ with the set of vertices A and the set of edges E defined as follows: {x, y} ∈ E if and only if x 6= y and there exists α ∈ P such that x, y ∈ A\Yα and y ∈ B(x, α). We show that deg(x) 6 m − 1 for every x ∈ A, where deg(x) = |{y ∈ A : {x, y} ∈ E}|. We suppose the contrary and choose x ∈ A and distinct vertices y1, . . . , ym such that {x, yi} ∈ E for every i ∈ {1, . . . , m}. By the definition of E, for every i ∈ {1, . . . , m}, there exists αi ∈ P and a bounded subset Yαi of X such that yi ∈ B(x, αi) and x, yi ∈ A \ Yαi. Let α = max{α1, . . . , αm} and α = αj. Then y1, . . . , ym ∈ B(x, αj) and y1, . . . , ym ∈ A \ Yαj because Yαi ⊆ Yαj for all i ∈ {1, . . . , m}, so we get a contradiction with the choice of Yαj because |B(x, αj) ∩ A| 6 m. By [3, Corollary 12.2], the chromatic number of Γ does not exceed m. Hence A can be partitioned A = A1 ∪ . . . ∪Ak, κ 6 m so that, for every i ∈ {1, . . . , k} and x, y ∈ Ai, we have {x, y} /∈ E. We show that each subset Ai is thin. For every α ∈ P , we put Zα = B(Yα, α). Suppose that there exists x ∈ Ai \ Zα such that |B(x, α) ∩ Ai| > 1. Let y ∈ B(x, α)∩Ai and y 6= x. Since x /∈ Zα then y /∈ Yα. Thus x, y ∈ A\ Yαi and y ∈ B(x, α), so {x, y} ∈ E contradicting the choice of Ai. � 92 Ie. Lutsenko and I. Protasov 2. Applications Theorem 2.1. Let (X, d) be a metric space, m ∈ N. A subset A ⊆ X can be partitioned in 6 m thin subsets if and only if , for every r ∈ R+, there exists a bounded subset Y of X such that |B(x, r) ∩ A| 6 m for every x ∈ A \ Y . Proof. We consider (X, d) as the ballean B(X, d) = (X, R+, Bd). Clearly, B(X, d) is ordinal so we can apply Theorem 1.2. � Let G be a group, κ be an infinite cardinal, Fκ(G) = {F ⊆ G : |F | < κ, e ∈ F}. We consider the ballean Bκ(G) = (G, Fκ(G), B), where B(g, F) = Fg for all g ∈ G, F ∈ Fκ(G). If κ > |G|, Bκ(G) is bounded. For κ = |G|, Bκ(G) is ordinal. Indeed, let g0 = e, {gα : α < κ} be a numeration of G, Fα = {gβ : β 6 α}. Then the well ordered by ⊆ family F = {Fα : α < κ} is cofinal in F. We say that a subset A ⊂ G is κ-thin if |gA ∩ A| < κ for every g ∈ G, g 6= e. In the case κ = ℵ0, we get the thin subsets defined in the very beginning of the paper. Lemma 2.2. Let A be a subset of a group G. If A is thin in the ballean Bκ(G) then A is κ-thin. If A is κ-thin and κ is regular then A is thin in the ballean Bκ(G). Proof. Let A be thin in Bκ(G). For every g ∈ G, g 6= e, we put Fg = {e, g} and choose a bounded subset Yg in Bκ(G) such that B(x, F) ∩ A = {x} for every x ∈ A \ Yg. Then gx /∈ A for every x ∈ A \ Yg so gA ∩ A ⊆ Yg. Since Yg is bounded in Bκ(G) then |Yg| < κ and A is κ-thin. Let A be κ-thin, F ∈ Fκ(G). We put Y = ⋃ {gA ∩ A : g ∈ F \ {e}}. Since |gA ∩ A| < κ, |F | < κ and κ is regular, |Y | < κ so Y is bounded in Bκ(G). For every x ∈ A \ Y , we have Fx ∩ A = {x} hence A is thin in Bκ(G). � Theorem 2.3. Let G be a group of regular cardinality κ, m ∈ N. A subset A ⊆ G can be partitioned in 6 m κ-thin subsets if and only if, for every F ⊂ G, |F | < κ, there exists a subset Y ⊆ G such that |Y | < κ and |Fx ∩ A| 6 m for every x ∈ A \ Y . Proof. Since the ballean Bκ(G) is ordinal, in view of Lemma 2.2, we can apply Theorem 1.2. � Remark 2.4. A subset A of a group G is called almost thin if the set ∆(A) = {g ∈ G : gA ∩ A is infinite} is finite. By [4, Theorem 3.1], every almost thin subset of a group G can be partitioned in 3|∆(A)|−1 thin subsets, but the union of two thin subsets needs not to be almost thin [4, Theorem 3.2]. Thin subsets of balleans 93 References [1] A. Bella and V. Malykhin, On certain subsets of groups, Questions Answers Gen. Topol- ogy 17 (1999), 183–187. [2] M. Filali and I. V. Protasov, Spread of balleans, Appl. Gen. Topol. 9 (2008), 169–175. [3] F. Harary, Graph Theory, Addison-Wesley Publ. Comp., 1969. [4] Ie. Lutsenko and I. V. Protasov, Sparse, thin and other subsets of groups, Internat. J. of Algebra Comput. 19 (2009), 491–510. [5] I. Protasov and M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., Vol. 11, VNTL Publishers, Lviv, 2007. [6] O. Protasova, Pseudodiscrete balleans, Algebra Discrete Math. 4 (2006), 81–92. [7] J. Roe, Lecture on Coarse Geometry, AMS University Lecture Ser. 31 (2003). Received December 2009 Accepted September 2010 Ie. Lutsenko (ie.lutsenko@gmail.com) Departament of Cybernetics, Kyiv University, Volodimirska 64, Kyiv, 01033, Ukraine I. Protasov (i.v.protasov@gmail.com) Departament of Cybernetics, Kyiv University, Volodimirska 64, Kyiv, 01033, Ukraine Thin subsets of balleans. By Ie. Lutsenko and I. Protasov