ProtaAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 8, No. 2, 2007 pp. 283-291 Cellularity and density of balleans I. V. Protasov Abstract. A ballean is a set X endowed with some family F of balls in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. Then we define the asymp- totic counterparts for dense and open subsets, introduce two cardinal invariants (density and cellularity) of balleans and prove some results concerning relationship between these invariants. We conclude the pa- per with applications of obtained partitions of left topological group in dense subsets. 2000 AMS Classification: 54A25, 54E25, 05A18. Keywords: ballean, large and thick subsets, density, cellularity. 1. Introduction Every infinite group G can be partitioned in |G|-many subsets dense in every totally bounded group topology on G. In [5] this statement was extracted from the following combinatorial claim. For every infinite group G there exists a disjoint family F of cardinality |G| such that, for every F ∈ F and every finite subset K of G, there exists g ∈ F such that Kg ⊆ F . Each subset F ∈ F looks like a set with non-empty interior in some structure dual to uniform topological space. To explain this duality we need some definitions and notations. 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). 284 I. V. Protasov 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′); • 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. A ball structure is said to be a ballean if it is upper symmetric and upper multiplicative. In entourage form the balleans arouse in coarse geometry [10] under name coarse structures and independently in combinatorics [6] under name uniform ball structures. Now we define the mappings which play the parts of uniformly continuous and uniformly open mappings on the ballean stage. 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 mapping f : X1 → X2 is called ≻-mapping if, for every β ∈ P2, there exists α ∈ P1 such that, for every x ∈ X1 B2(f(x),β) ⊆ f(B1(x,α)). If f : X1 → X2 is a bijection such that f is a ≺-mapping and f is a ≻- mapping, we say that f is an asymorphism and B1, B2 are asymorphic. Given an arbitrary ballean B = B(X,P,B), we can replace every ball B(x,α) by B∗(x,α) ∩ B(x,α) and get an asymorphic ballean in which B∗(x,α) = Cellularity and density of balleans 285 B(x,α). In what follows we shall assume that B∗(x,α) = B(x,a) for all x ∈ X, α ∈ P . We need also some classification of subsets of X for a ballean B = (X,P,B). Given a subset A ⊆ X, we say that A is • large if there exists α ∈ P such that X = B(A,α); • small if X \ B(A,α) is large for every α ∈ P ; • thick if, for every α ∈ P there exists a ∈ A such that B(a,α) ⊆ A. For some special balleans these types of subsets were introduced in [1] and [2]. We note also that large, small and thick subsets of a ballean may be considered as asymptotic duplicates of dense, nowhere dense and subsets with non-empty interior of uniform spaces. Following this (non-formal) duality between uniform spaces and balleans, we define the density d(B) and cellularity c(B) as d(B) = min{|L| : L ⊆ X,L is large}, c(B) = sup{|F| : F is a disjoint family of thick subsets of X}. As in the case of uniform spaces, density of a ballean is much more easy to calculate or evaluate than its cellularity, so our main goal is to find some relationships between d(B) and c(B). 2. Observations (1) Let B = (X,P,B) be a ballean, T be a thick subset of X and L be a large subset of X. Then there exists α ∈ P such that X = B(L,α) and B(x,α) ⊆ T for some x ∈ T , so L ∩ T 6= ∅. Since every large subset meets every thick subset, we have c(B) 6 d(B). (2) Given α ∈ P and Y ⊆ X, we say that Y is α-discrete if the family {B(y,α) : y ∈ Y } is pairwise disjoint. By Zorn Lemma, every α- discrete subset Y of X is contained in some maximal (by inclusion) α-discrete subset Z of X. If y ∈ X then B(y,α) ∩ B(Z,α) 6= ∅. We choose β ∈ P such that B(B(x,α),α) ⊆ B(x,β) for every x ∈ X. Then y ∈ B(Z,β) and Z is large. On the other hand, let L be a large subset of X, X = B(L,α) and let Z be a maximal α-disjoint subset of Y . Then |Z| 6 |L| and Z is large. Hence, d(B) can be defined as the minimal cardinality of maximal α-disjoint subsets of X where α runs over P . (3) Let (X,d) be a metric space. Given any x ∈ X, n ∈ ω, we put Bd(x,n) = {y ∈ X : d(x,y) 6 n} and say that B(X,d) = (X,ω,Bd) is a metric ballean. A ballean B is called metrizable if B is asymorphic to some metric ballean. To characterize metrizable balleans we need two definitions. A ballean B = (X,P,B) is called connected if, for any x,y ∈ X, there exists α ∈ P such that y ∈ B(x,α). 286 I. V. Protasov We define the preordering 6 on 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 β ∈ P ′ such that α 6 β. The cofinality cf(B) is the minimal cardinality of the cofinal subsets of P . By [6, Theorem 9.1], a ballean B is metrizable if and only if B is connected and cf(B) 6 ℵ0. For approximation of arbitrary balleans via metrizable balleans see [7]. (4) A connected ballean B = (X,P,B) is called ordinal if there exists a well-ordered by ≤ cofinal subset of P . Replacing P to its minimal cofinal subset, we get the asymorphic ballean. Hence, we can write B as (X,β,B), where β is a regular cardinal (considered as a set of ordinals). We note that every metrizable ballean is ordinal, and metric balleans are the main subject of asymptotic topology [3]. (5) A subset Y ⊆ X is called bounded if there exist y ∈ Y and α ∈ P such that Y ⊆ B(y,α). A ballean is called bounded if its support is bounded. Clearly, d(B) = c(B) = 1 for every bounded ballean B. 3. Results Theorem 3.1. For every ordinal ballean B, c(B) = d(B) and there exists a disjoint family F of cardinality d(B) consisting of thick subsets of X. Proof. Let B = (X,ρ,B), κ = d(B) and cf(κ) be the cofinality of κ. If B is bounded, we use observation 5, so we assume that B is unbounded. We fix some element x0 ∈ X and consider four cases. Case ρ < cfκ. We prove the following auxiliary statement. For every α < ρ, there exist β, α < β < ρ and an α-discrete subset Yα of X such that x0 ∈ Yα, B(Yα,α) ⊆ B(x0,β) and |Yα| = κ. Let Z be a maximal α-discrete subset of X such that x0 ∈ Z. By observation 2, |Z| > κ. For every λ < ρ, we put Zλ = Z ∩ B(x0,λ). Since Z = ⋃ λ<ρ Zλ, |Z| > κ and ρ < cf(κ), there exists µ < ρ such that |Zµ| > κ. We choose β < ρ such that α < β and B(B(x0,µ),α) ⊆ B(x0,β). Then B(Zµ,α) ⊆ B(x0,β) and we can choose a subset Yα ⊆ Zµ such that x0 ∈ Y and |Yα| = κ. Using the auxiliary statement and regularity of ρ, we can define inductively a mapping f : ρ → ρ and a family {Yf (α) : α < ρ} of subsets of X such that, for every α < ρ, f(α) > α, Yf (α) is f(α)-discrete, |Yf (α)| = κ and B(Yf (α),f(α)) ⊆ B(x0,f(α + 1)) \ B(x0,f(α)). For every α < ρ, we enumerate Yf (α) = {y(f(α),λ) : λ < κ} and, for every λ < κ, put Tλ = ⋃ α<ρ B(y(f(α),λ),f(α)). Clearly, every subset Tλ is thick and the family {Tλ : λ < κ} is disjoint. Cellularity and density of balleans 287 Case cf(κ) 6 ρ < κ. We prove the following auxiliary statement. For any α < ρ and k′ < κ, there exist β, α < β < ρ and an α-discrete subset Yα of X such that x0 ∈ Yα, B(Yα,α) ⊆ B(x0,β) and |Yα| > κ ′. Let Z be a maximal α-discrete subset of X such that x0 ∈ Z. By observation 2, |Z| > κ. Let I be a cofinal subset of κ such that |I| = ρ. For every λ ∈ I, we put Zλ = Z ∩ B(x0,λ). Clearly, Z = ⋃ λ∈I Zλ. If |Zλ| 6 κ ′ for every λ ∈ I, then |Z| 6 κ′|I| = κ′ρ < κ. Hence, there exists µ ∈ I such that |Zµ| > κ ′. We choose β < ρ such that α < β and B(B(x0,µ),α) ⊆ B(x0,β). Then B(Zµ,α) ⊆ B(x0,β) and we put Yα = Zµ. Let ϕ : ρ → κ be an injective mapping such that ϕ(ρ) is cofinal in κ and α < β < ρ implies ϕ(α) < ϕ(β) < κ. Using the auxiliary statement and regularity of κ, we can define inductively a mapping f : ρ → ρ and a family {Yf (α) : α < ρ} of subsets of X such that • (i) f(ρ) is cofinal in ρ and α < β < ρ implies f(α) < f(β) < ρ; • (ii) Yf (α) is f(α)-discrete; • (iii) B(Yf (α),f(α)) ⊆ B(x0,f(α + 1)) \ B(x0,f(α)); • (iv) |Yf (α)| = ϕ(α). For every α < ρ, we enumerate Yf (α) = {y(f(α),λ) : λ < ϕ(α)}. Then for any α and γ such that ϕ(α) 6 γ < ϕ(α + 1), we put Tγ = ⋃ { B(f(β),γ) : α + 1 < β < ρ } . By (i), Tγ is thick. By (ii) and (iii), the family F = { Tγ : ϕ(α) 6 γ < ϕ(α + 1),α < ρ } is disjoint. Since ϕ(ρ) is cofinal in κ, by (iv), we have |F| = κ. Case ρ = κ. Using the assumption, we can construct inductively the subset {yα : α < κ} of X such that the family {B(yα,α) : α < κ} is disjoint. Then we partition κ = ⋃ λ<κ Iλ into κ cofinal subsets and, for every λ < κ, put Tλ = ⋃ { B(yα,α) : α ∈ Iλ } . Clearly, every subset Tλ is thick and the family {Tλ : λ < κ} is disjoint. Case ρ > κ. We show that this variant is impossible. Suppose the contrary. Let Z be a large subset of X such that |Z| = κ and X = B(Z,α). For every z ∈ Z, we pick α(z) < ρ such that B(z,α) ⊆ B(x0,α(z)). Since ρ is regular and κ < ρ, there exists β < ρ such that β > α(z) for every z ∈ Z. Then B(z,α) ⊆ B(x0,β) for every z ∈ Z, so X = B(x0,β) and B is bounded. � Corollary 3.2. For every metrizable ballean B, c(B) = d(B) and there exists a disjoint family F of cardinality d(B) consisting of thick subsets of X. Theorem 3.3. Let B = (X,P,B) be a ballean, |X| = κ and let |P | 6 κ. Then c(B) = d(B) = κ and there exists a disjoint family F of cardinality κ consisting of thick subsets of X provided that one of the following conditions is satisfied: • (i) there exists κ′ < κ such that B(x,α) 6 κ′ for all x ∈ X and α ∈ P; • (ii) |B(x,α)| < κ for all x ∈ X, α ∈ P and κ is regular. 288 I. V. Protasov Proof. Let L be a large subset of X, α ∈ P and X = B(L,α). Then each of the assumptions (i) and (ii) gives |L| = κ so d(B) = κ. (i) Let Z be a subset of X such that |Z| < κ, α ∈ P . Let Y be a maximal (by inclusion) α-discrete subset of X such that B(Z,α) ∩ B(Y,α) = ∅. If x ∈ X then B(x,α) ∩ B(Z ∪ Y,α) 6= ∅. Hence, Z ∪ Y is large and |Y | = κ. We fix a bijection f : P × κ → κ and note that, for every α ∈ P , the set f(α,κ) is cofinal in κ (as a set of ordinals). We define also two mappings ϕ : κ → P and ψ : κ → κ be the rule: if f(α,λ) = γ then ϕ(γ) = α, ψ(γ) = λ. We take an arbitrary ϕ(0)-discrete subset Y0 of X such that |Y0| = ψ(0). Assume that, for some γ < κ, we have defined the family {Yλ : λ < γ} of subsets of X such that each subset Yλ is ϕ(λ)-discrete, |Yλ| = ψ(λ) and the family {B(Yλ,ϕ(λ)) : λ < γ} is disjoint. Put Z = ⋃ λ<γ B(Yλ,ψ(λ)). In view of above paragraph there exists a ϕ(γ)-discrete subset Yγ such that |Yγ| = ψ(γ) and Z ∩ B(Yγ,ϕ(γ)) = ∅. After κ steps we get the family {Yγ : γ < κ}. For every γ < κ, we enumerate Yγ = {y(λ,γ) : λ < |Yγ|} and put T0 = ⋃ γ<κ B(y(0,γ),ϕ(γ)). Since ϕ is surjective, T0 is thick. Assume that, for some δ < κ, we have defined disjoint family {Tµ : µ < δ} of thick subsets of X. Put T = ⋃ µ<δ Tµ. To define Tδ we denote I = {γ : γ < κ,Yγ \T 6= ∅} and put Tδ = ⋃ γ∈I B(y(δ,γ),ϕ(γ)). After κ steps we put F = {Tδ : δ < γ}. Since f(α,κ) is cofinal in κ for every α ∈ P , every subset Tδ is thick. (ii) Let γ < κ and {Yλ : λ < γ} be a family of subsets of X such that |Yλ| < κ for every λ < γ. Let {pλ : λ < γ} be a subset of P . We put Z = ⋃ λ<γ B(Yλ,λ). By (ii), |Z| < κ. Hence, for any α ∈ P and κ′ < κ, we can take an α-disjoint subset Y(γ,α) of X such that B(Y (γ,α),α) ∩Z = ∅ and |Y (γ,α)| > κ ′. Using this remark, we can construct the family F as in (i). � 4. Examples We show that, for every infinite cardinal κ, there exists a metric space X such that d(B(X)) = κ. Example 4.1. Let I be a (non-directed) graph with the set of vertices ω and the set of edges {(i, i+ 1) : i ∈ ω}. We consider the set {Iγ : γ < κ} of copies of I, identify the terminal vertices of these copies and denote by Γ the resulting graph. Let X = V (Γ) be the set of vertices of Γ. We endow X with path metric: the distance between two vertices u,v ∈ X is the length of the shortest path between u and v. If L is a large subset of X, then L∩V (Iγ ) is infinite for every γ < κ, so |L| = κ and d(B(X)) = κ. The next two examples show that cellularity of a ballean could be much more smaller than its density. Cellularity and density of balleans 289 Example 4.2. Let X be a set and ϕ be a filter on X. For any x ∈ X and F ∈ ϕ, we put Bϕ(x,F) = { {x}, if x ∈ F ; X \ F , if x /∈ F and consider the ballean B(X,ϕ) = (X,ϕ,Bϕ). A ballean B = (X,P,B) is called pseudodiscrete if, for every α ∈ P , there exists a bounded subset V of X such that B(x,α) = {x} for every x ∈ X \ V . By [8], a ballean B is pseudodiscrete if and only if there exists a filter ϕ on X such that B is asymorphic to B(X,ϕ). Now let X be infinite and ∩ϕ = ∅. Then B(X,ϕ) is an unbounded connected ballean. A subset L ⊆ X is large if and only if L ∈ ϕ, so d(B) = min{|F | : F ∈ ϕ}. On the other hand, let T be a thick subset of X, F ∈ ϕ. We take x ∈ X such that Bϕ(x,F) ⊆ X. Then either x ∈ F or X \ F ⊆ T . It follows that T is cofinal with respect to ϕ, i.e.: F ∩ T 6= ∅ for every F ∈ ϕ. Hence, if ϕ is an ultrafilter then any two thick subsets of X have non-empty intersection and c(B(X,ϕ)) = 1. Example 4.3. Let X be an infinite set of regular cardinality κ. Denote by F the family of all subsets of X of cardinality < κ. Let P be a set of all mappings f : X → F such that, for every x ∈ X, we have x ∈ f(x) and ∣ ∣ { y ∈ X : x ∈ f(y) } ∣ ∣ < κ. Given any x ∈ X and f ∈ P , we put B(x,f) = f(x) and consider the ball structure B = (X,P,B). Since B∗(x,f) = {y ∈ X : x ∈ f(y)}, B is upper symmetric. Since κ is regular, B is upper multiplicative. Hence, B is a ballean. Clearly, B is connected and unbounded. If L is a large subset of X, by regularity of κ, we have |L| = κ. If A is a subset of X and |A| = κ, we fix an arbitrary bijection h : A → X and put f(x) = { {x}, if x /∈ A; {x,h(x)}, if x ∈ A Then f ∈ P and B(A,f) = X. Hence, a subset L of X is large if and only if |L| = κ, so d(B) = κ. If A ⊆ X and |X \ A| = κ, by observation 1 and above paragraph, A is not thick. It means that any two thick subsets of X are not disjoint and c(B) = 1. Now we compare cellularity and density with another cardinal invariant of balleans, namely resolvability, defined in [9]. Given a ballean B = (X,P,B) and a cardinal κ, we say that B is κ- resolvable if X can be partitioned in κ-many large subsets. The resolvability of B is the cardinal r(B) = sup{κ : B is κ-resolvable}. If B is a ballean from Example 4.1, by [9, Theorem 2.3], r(B) = ℵ0. By Corollary 3.2, d(B) = c(B) = κ. If B is a ballean from Example 4.2, defined by free ultrafilter, then c(B) = r(B) = 1, but d(B) = min{|F | : F ∈ ϕ}. 290 I. V. Protasov If B is a ballean from Example 4.3, then r(B) = d(B) = κ, c(B) = 1. The above remarks show that there are no direct correlations between re- solvability on one hand and density or cellularity on the other hand. 5. Applications Let G be a group with the identity e endowed with some topology. Then G is called left topological if all the left shifts x 7→ gx, g ∈ G are continuous. Let G be an infinite left topological group, |G| = κ, γ be an infinite cardinal such that γ 6 κ. We say that G is • totally bounded if, for every nieghbourhood U of e, there exists a finite subset F of G such that G = FU; • γ-bounded if, for every neighbourhood U of e, there exists a subset F of G such that |F | < γ and G = FU; • weakly bounded if, for every neighbourhood U of e, there exists a subset F of G such that |F | < κ and G = FU. In this terminology, totally bounded groups are ℵ0-bounded and weakly bounded groups are κ-bounded. We denote by Fγ the family of all subsets F of X such that e ∈ F , F = F −1 and |F | < γ. Given any g ∈ G and F ∈ Fγ , we put B(g,F) = Fg and denote by B(G,γ) the ballean (G,Fγ,B). If γ = ℵ0 then a subset L is large (with respect to B(G,ℵ0)) if and only if G = FL for some finite subset F of G. By Theorem 3.2 (case (ii) for κ = ℵ0 and case (i) for κ > ℵ0), there exists a disjoint family F of cardinality κ consisting of thick subsets. By observation 1, every subset F ∈ F meets every large subset L. It follows that F ∩ gU 6= ∅ for every g ∈ G and every neighbourhood U of e in every totally bounded topology τ on G, so F is dense in τ. Hence, G can be partitioned to κ subsets dense in each totally bounded topology. If γ < κ, the same arguments applying to B(G,γ) and Theorem 3.2 (i) prove that G can be partitioned to κ subsets dense in every γ-bounded topology on G. if γ = κ and κ is regular, we apply either Theorem 3.1 or Theorem 3.2 (ii) and conclude that G can be partitioned in κ-many subsets dense in every weakly bounded topology on G. What about γ = κ and κ is singular? This is old (and unsolved) problem posed by the author [4, Problem 13.45] in the following weak form. Problem 5.1. Every infinite group G of regular cardinality κ can be partitioned G = A1 ∪A2 so that FA1 6= G and FA2 6= G for every subset F of G such that |F | < κ. Is the same true for groups of singular cardinalities? 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Roe, Lectures on Coarse Geometry, AMS University Lecture Series, 31, 2003. Received May 2006 Accepted February 2007 Igor Protasov (islab@unicyb.kiev.ua) Department of Cybernetics, Kyiv University, Volodimirska 64, Kyiv 01033, Ukraine