Uniform ball structures

Uniform ball structures

A ball structure is a triple B = (X, P, B), where X, P are nonempty sets and, for all x ∈ X, α ∈ P, B(x, α) is a subset of X, x ∈ B(x, α), which is called a ball of radius α around x. We introduce the class of uniform ball structures as an asymptotic counterpart of the class of uniform topologica...

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Datum:2003
1. Verfasser: Protasov, I.V.
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Veröffentlicht: Інститут прикладної математики і механіки НАН України 2003
Schriftenreihe:Algebra and Discrete Mathematics
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Zitieren:Uniform ball structures / I.V. Protasov // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 93–102. — Бібліогр.: 2 назв. — англ.

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spelling irk-123456789-1552822019-06-17T01:30:36Z Uniform ball structures Protasov, I.V. A ball structure is a triple B = (X, P, B), where X, P are nonempty sets and, for all x ∈ X, α ∈ P, B(x, α) is a subset of X, x ∈ B(x, α), which is called a ball of radius α around x. We introduce the class of uniform ball structures as an asymptotic counterpart of the class of uniform topological spaces. We show that every uniform ball structure can be approximated by metrizable ball structures. We also define two types of ball structures closed to being metrizable, and describe the extremal elements in the classes of ball structures with fixed support X. 2003 Article Uniform ball structures / I.V. Protasov // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 93–102. — Бібліогр.: 2 назв. — англ. 1726-3255 2001 Mathematics Subject Classification: 03E99, 54A05, 54E15. http://dspace.nbuv.gov.ua/handle/123456789/155282 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description A ball structure is a triple B = (X, P, B), where X, P are nonempty sets and, for all x ∈ X, α ∈ P, B(x, α) is a subset of X, x ∈ B(x, α), which is called a ball of radius α around x. We introduce the class of uniform ball structures as an asymptotic counterpart of the class of uniform topological spaces. We show that every uniform ball structure can be approximated by metrizable ball structures. We also define two types of ball structures closed to being metrizable, and describe the extremal elements in the classes of ball structures with fixed support X.
format Article
author Protasov, I.V.
spellingShingle Protasov, I.V.
Uniform ball structures
Algebra and Discrete Mathematics
author_facet Protasov, I.V.
author_sort Protasov, I.V.
title Uniform ball structures
title_short Uniform ball structures
title_full Uniform ball structures
title_fullStr Uniform ball structures
title_full_unstemmed Uniform ball structures
title_sort uniform ball structures
publisher Інститут прикладної математики і механіки НАН України
publishDate 2003
url http://dspace.nbuv.gov.ua/handle/123456789/155282
citation_txt Uniform ball structures / I.V. Protasov // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 93–102. — Бібліогр.: 2 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT protasoviv uniformballstructures
first_indexed 2025-07-14T07:20:16Z
last_indexed 2025-07-14T07:20:16Z
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fulltext Jo ur na l A lg eb ra D is cr et e M at h.Algebra and Discrete Mathematics RESEARCH ARTICLE Number 2. (2003). pp. 93–102 c© Journal “Algebra and Discrete Mathematics” Uniform ball structures I. V. Protasov Communicated by V. M. Usenko Abstract. A ball structure is a triple B = (X,P,B), where X,P are nonempty sets and, for all x ∈ X, α ∈ P , B(x, α) is a sub- set of X,x ∈ B(x, α), which is called a ball of radius α around x. We introduce the class of uniform ball structures as an asymptotic counterpart of the class of uniform topological spaces. We show that every uniform ball structure can be approximated by metriz- able ball structures. We also define two types of ball structures closed to being metrizable, and describe the extremal elements in the classes of ball structures with fixed support X. Following [2], by ball structure we mean a triple B = (X, P, B), where X, P are nonempty sets and, for any x ∈ X, 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 . A set X is called a support of B, P is called a set of radiuses. Let B1 = (X1, P1, B1), B2 = (X2, P2, B2) be ball structures, f : X1 −→ X2 We say that f is a ≻-mapping if, for every β ∈ P2, there exists α ∈ P1 such that B2(f(x), β) ⊆ f(B1(x, α)) for every x ∈ X1. If there exists a surjective ≻-mapping f : X1 −→ X2, we write B1 ≻ B2. A mapping f : X1 −→ X2 is called a ≺-mapping if, for every α ∈ P1,there exists β ∈ P2 such that f(B1(x, α)) ⊆ B2(f(x), β) 2001 Mathematics Subject Classification: 03E99, 54A05, 54E15. Key words and phrases: ball structure, metrizability. Jo ur na l A lg eb ra D is cr et e M at h.94 Uniform ball structures for every x ∈ X. If there exists an injective ≺-mapping f : X1 −→ X2, we write B1 ≺ B2. A bijection f : X1 −→ X2 is called an isomorphism between B1 and B2 if f is a ≻-mapping and f is a ≺-mapping. Let B1 = (X1, P1, B1), B2 = (X2, P2, B2) be ball structures with common support X. We say that B1 ⊆ B2 if the identity mapping id: X −→ X is a ≺-mapping from B1 to B2. If B1 ⊆ B2 and B2 ⊆ B1, we write B1 = B2. A property P of ball structures is called a ball property if a ball structure B has a property P provided that B is isomorphic to some ball structure with property P. Now we describe four basic ball properties. Let B = (X, P, B) be a ball structure. For any x ∈ X, α ∈ P put B∗(x, y) = {y ∈ X : x ∈ B(y, α)}. A ball structure B ∗ = (X, P, B) is called dual to B. Note that B ∗∗ = B. A ball structure B = (X, P, B) is called symmetric if B = B ∗. A ball structure B = (X, P, B) is called multiplicative if, for any α, β ∈ P , there exists γ(α, β) ∈ P such that B(B(x, α), β) ⊆ B(x, γ(α, β)) for every x ∈ X. Here B(Y, α) = ⋃ y∈Y B(y, α), Y ⊆ X, α ∈ P. Let B = (X, P, B) be a ball structure, x, y ∈ X, We say that x, y are connected if there exists α ∈ P such that x ∈ B(y, α), y ∈ B(x, α). A subset Y ⊆ X is called connected if any two elements from Y are con- nected. A ball structure B is called connected if its support is connected. If B is symmetric and multiplicative, then connectivity is an equivalence on X, so X disintegrates into connected components. For an arbitrary ball structure B = (X, P, B) we define a preodering ≤ on the set P by the rule α ≤ 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 ≤ β. A cofinality cfB of B is the minimal cardinality of cofinal subsets of P . Let (X, d) be a metric space, R + = {x ∈ R : x ≥ 0}. Given any x ∈ X, r ∈ R +, put Bd(x, r) = {y ∈ X : d(x, y) ≤ r}. Jo ur na l A lg eb ra D is cr et e M at h.I. V. Protasov 95 A ball structure (X, R+, Bd) is denoted by B(X, d). We say that a ball structure B is metrizable if B is isomorphic to B(X, d) for some metric space (X, d). We shall use the following metrizability criterion [2]. A ball structure B is metrizable if and only if B is symmetric, multi- plicative, connected and cf B ≤ ℵ0. A ball structure is called uniform if it is symmetric and multiplicative. In §1 we define a wide spectrum of examples of uniform ball structures related to groups and filters. In §2 we introduce some ball operations which give new uniform ball structures from a pregiven family of uniform ball structures. It is well known [1], that every uniform topological space can be approximated by pseudometrizable spaces. In §3 we prove a ball analogue of such an approximation. In §3 − 4 we introduce two types of ball structures (inductively metrizable and submetrizable) close to being metrizable. In §5 we describe extremal by inclusion elements in the classes of ball structures with fixed support. §1 Examples Let G be an infinite group with the identity e, γ be an infinite cardinal, γ < |G|. Denote by ℑe(G, γ) the family of all subsets of G of cardinality < γ containing e. Given any g ∈ G, F ∈ ℑe(G), put Bl(g, F ) = Fg, Br(g, F ) = gF. The ball structures (G,ℑe(G, γ), Bl), (G,ℑe(G, γ), Br) will be denoted by Bl(G, γ), Br(G, γ). Note that the mapping g 7−→ g−1 is an isomorphism between Bl(G) and Br(G). In the case γ = ℵ0 we write Bl(G) and Br(G) instead of Bl(G, γ) and Br(G, γ). It easy to see that Bl(G)=Br(G) if and only if the set {x−1gx : x ∈ G} is finite for every g ∈ G. By metrizability criterion, Bl(G, γ) is metrizable if and only if γ = |G| and cf γ = ℵ0. In particular, Bl(G) is metrizable if and only if G is countable. Let X be a set and let ϕ be a filter on X. For any x ∈ X, F ∈ ϕ, put B(x, F ) = { X\F, if x /∈ F ; {x}, if x ∈ F ; and denote by B(X, ϕ) the ball structure (X, ϕ, B). Note that B(X, ϕ) is connected if and only if either ⋂ ϕ = ∅ or |X| = 1. Hence, B(X, ϕ) is Jo ur na l A lg eb ra D is cr et e M at h.96 Uniform ball structures metrizable if and only if either |X| = 1 or ⋂ ϕ = ∅ and ϕ has a countable base. Now we define a wide class of ball structures containing all ball struc- tures of filters and almost all ball structures of groups. Let X be a set and let P be a family of partitions of X. For any x, y ∈ X and P ∈ P, denote by B(x, P ) the set {y ∈ X : x, y are in the same cell of the partition P}. A ball structure (X,P, B) is denoted by B(X,P). Clearly, B(X,P) is symmetric. Given any P1, P2 ∈ P, we say that P2 is an enlargement of P1 if B(x, P1) ⊆ B(x, P2) for each x ∈ X. A ball structure B(X,P) is multiplicative if and only if, for any P1, P2 ∈ P, there exists P ∈ P such that P is an enlargement of P1 and P2. A ball structure B is called cellular if B is isomorphic to B(X,P) for some set X and some family P of partitions of X. Given any ball structure B = (X, P, B), x, y ∈ X and α ∈ P , we say that x, y are α-path connected if there exists a sequence x0, x1, ..., xn, x0 = x, xn = y such that xi+1 ∈ B(xi, α), xi ∈ B(xi+1, α) for every i ∈ {0, 1, ..., n − 1}. For any x ∈ X, α ∈ P , put B2(x, α) = {y ∈ X : x, y are α − path connected}. A ball structure B 2(X, P, B2) is called a cellularization of B. By [2], a ball structure B is cellular if and only if B = B 2. A metrizable ball structure B is cellular if and only if B is isomorphic to B(X, d) for some non-Archimedian metric space. Every ball structure B(X, ϕ) of a filter ϕ on X is cellular. A ball structure B(G, γ) of a group G is cellular if and only if either γ > ℵ0 or γ = ℵ0 and every finite subsets of G generates a finite subgroup. §2 Constructions Let {Bλ = (Xλ, P, Bλ) : λ ∈ I} be a family of ball structures with pairwise disjoint supports and common set of radiuses, X = ⋃ λ∈I Xλ. For every x ∈ X, x ∈ Xλ and every α ∈ P , put B(x, α) = Bλ(x, α). A ball structure B = (X, P, B) is called a disjoint union of the family {Bλ : λ ∈ I}. Every uniform ball structure is a disjoint union of its connected components. Let {Bλ = (X, Pλ, Bλ) : λ ∈ I} be a family of ball structures with common support. Suppose that, for any λ1, λ2 ∈ I, there exists λ ∈ I such that Bλ1 ⊆ Bλ, Bλ2 ⊆ Bλ. For every λ ∈ I, choose a copy P ′ λ = fλ(Pλ) of Pλ such that the family {P ′ λ : λ ∈ I} is disjoint. Put P = ⋃ λ∈I P ′ λ. For all x ∈ X, β ∈ P, β ∈ P ′ λ, put B(x, β) = Bλ(x, f−1 λ (β)). Jo ur na l A lg eb ra D is cr et e M at h.I. V. Protasov 97 A ball structure B = (X, P, B) is called an inductive limit of the family {Bλ : λ ∈ I}. Clearly, Bλ ⊆ B for every λ ∈ I. If every Bλ is uniform, B is uniform. Let B = (X, P, B) be a ball structure, Y ⊆ X. For any y ∈ Y , α ∈ P , put BY (y, α) = B(y, α) ⋂ Y . A ball structure BY = (Y, P, BY ) is called a substructure of B. If B is uniform, then BY is uniform. Let {Bλ = (Xλ, Pλ, Bλ) : λ ∈ I} be an arbitrary family of ball struc- tures. By box product of this family we mean a ball structure ∏ λ∈I Bλ = ( ∏ λ∈I Xλ, ∏ λ∈I Pλ, B), where B(x, p) = {y ∈ ∏ λ∈I Xλ : prλ(y) ∈ Bλ(prλ(x), prλ(p)), λ ∈ I} for all x ∈ ∏ λ∈I Xλ, p ∈ ∏ λ∈I Pλ. If every ball structure Bλ is uniform, then ∏ λ∈I Bλ is uniform. Note also that Bγ ≺ ∏ λ∈I Bλ, ∏ λ∈I Bλ ≻ Bγ for every γ ∈ I. Let B = (X, P, B) be a ball structure. A subset Y ⊆ X is called bounded if there exist x ∈ X, α ∈ P such that Y ⊆ B(x, α). We say that B is bounded if its support is bounded. Let B be a connected uniform ball structure, x0 ∈ X, Y ⊆ X. Then Y is bounded if and only if there exists α ∈ P such that Y ⊆ B(x0, α). A box product of an arbitrary family of bounded ball structures is bounded. It is metrizable if and only if every Bλ, λ ∈ I is metrizable and all but finitely many of them are bounded. We define also two modifications of box products. Let F be a family of all finite subsets of I. The first modification is ∨ ∏ λ∈I Bλ = ( ∏ λ∈I Xλ,ℑ× ∏ λ∈I Pλ, B̌), where B̌(x, (F, p)) = {y ∈ ∏ λ∈I Xλ : prλ(y) ∈ Bλ(prλ(x), prλ(p)) for every λ /∈ F}. Jo ur na l A lg eb ra D is cr et e M at h.98 Uniform ball structures The second modification is ∧ ∏ λ∈I Bλ = ( ∏ λ∈I Xλ,ℑ× ∏ λ∈I Pλ, B̂), where B̂(x, (F, p)) = {y ∈ ∏ λ∈I Xλ : prλ(y) ∈ Bλ(prλ(x), prλ(p)), λ ∈ F and prλ(x) = prλ(y), λ /∈ F}. Clearly, ∧ ∏ λ∈I Bλ ⊆ ∏ λ∈I Bλ ⊆ ∨ ∏ λ∈I Bλ. §3 Approximations A ball structure B is called pseudometrizable if B is a disjoint union of metrizable ball structures. Theorem 3.1. Every uniform ball structure B = (X, P, B) is an inductive limit of some family of pseudometrizable ball structures. Proof. We may suppose that B(X, α) = B∗(X, α) for all x ∈ X, α ∈ P . Denote by I the family of all subsets of P of the form {αn ∈ P : n ∈ ω} such that αn ≤ αn+1, n ∈ ω and, for any n, m ∈ ω, there exists k(n, m) ∈ ω such that B(B(x, n), m) ⊆ B(x, k(n, m)) for every x ∈ X. For every λ ∈ I, λ = {αn : n ∈ ω}, put Pλ = {αn : n ∈ ω}. By metrizability criterion, every connected component of Bλ = (X, Pλ, Bλ), Bλ(x, αn) = B(x, αn) is metrizable, so Bλ is pseu- dometrizable . It is easy to check that B is an inductive limit of the family {Bλ : λ ∈ I}. 2 A ball structure B is called inductively metrizable if B is an inductive limit of metrizable ball structures. Theorem 3.2. For every uniform ball structure B = (X, P, B) the following statements are equivalent (i) B is inductively metrizable; (ii) there exists a metric space (X, d) such that B(X, d) ⊆ B; (iii) there exists a subset P ′ ⊆ P, |P ′| ≤ ℵ0 and x0 ∈ X such that ⋃ α∈P ′ B(x0, α) = X. Proof. The implications (i) =⇒ (ii) =⇒ (iii) are trivial. Jo ur na l A lg eb ra D is cr et e M at h.I. V. Protasov 99 (iii) =⇒ (i). We may suppose that P ′ = {αn : n ∈ ω}, αn ≤ αn+1, n ∈ ω and, for any n, m ∈ ω there exists k(n, m) ∈ ω such that B(B(x, n), m) ⊆ B(x, k(n, m)) for every x ∈ X. Then B ′ = (X, P ′, B′), B′(x, α) = B(x, α), α ∈ P ′ is a metrizable ball structure. Consider the family ℑ of all metrizable ball structures on X such that B ′ ⊆ B ′′ for every B ′′ ∈ ℑ. Clearly, B is an inductive limit of ℑ. 2 By Theorem 3.2, a ball structure B(X, ϕ) of a filter ϕ on X, |X| > 1 is inductively metrizable if and only if there exists a countable subset ψ of ϕ such that ⋂ ψ = ∅. A ball structure B(G, γ) is inductively metrizable if and only if it is metrizable. §4 Submetrizability A uniform ball structure B = (X, P, B) is called submetrizable if there exists an unbounded metrizable ball structure B ′ = (X, P ′, B′) such that B ⊆ B ′. Let B = (X, P, B) be a ball structure, f : X −→ R. We say that f is a function of bounded oscilation (with respect to B) if, for every α ∈ P , there exists a natural number n(α) such that diam f(B(x, α)) ≤ n(α) for every x ∈ X, where diam A = sup{|a− b| : a, b ∈ A}. Clearly, every bounded function is of bounded oscilation. Theorem 4.1. For every uniform ball structure B = (X, P, B) the following statements are equivalent (i) B is submetrizable; (ii) there exists an unbounded function f : X −→ R of bounded os- cilation. Proof. (i) =⇒ (ii). Let (X, d) be an unbounded metric space such that B ⊆ B(X, d). Fix an arbitrary point x0 ∈ X and put f(x) = d(x, x0). Since (X, d) is unbounded, f is unbounded. If x, y ∈ X, then |f(x) − f(y)| = |d(x0, x) − d(x0, y)| ≤ d(x, y). Hence, f is of bounded oscilation on (X, d). Since B ⊆ B(X, d), f is of bounded oscilation with respect to B. (i) =⇒ (ii). For all x ∈ X, n ∈ ω, put B′(x, n) = {y ∈ X : |f(x) − f(y)| ≤ n}. Jo ur na l A lg eb ra D is cr et e M at h.100 Uniform ball structures Clearly, the ball structure B ′ = (X, ω, B′) is symmetric, multiplica- tive, connected and cfB ′ = ℵ0. Hence, B is metrizable. To show that B ⊆ B ′, fix an arbitrary α ∈ P , choose n(α) such that diam f(B(x, α)) ≤ n(α) for every x ∈ X. Then B(x, α) ⊆ B′(x, n(α)) for every x ∈ X. 2 A connected uniform ball structure B = (X, P, B) is called ordinal if there exists a cofinal well ordered by ≤ subset of P. Clearly, every metrizable ball structure is ordinal. Theorem 4.2. For every ordinal ball structure B = (X, P, B), the following statements are equivalent (i) B is metrizable; (ii) B is submetrizable; Proof. The implication (i) =⇒ (ii) is trivial. (ii) =⇒ (i). We may suppose that P is well ordered. Assume that B is not metrizable so cf P > ℵ0. By Theorem 4.1, there exists an unbounded function f : X −→ R of bounded oscilation. Choose a countable subset X ′ ⊆ X such that f(X ′) is unbounded in R. Since cf P > ℵ0, there exists X0 ∈ X, α ∈ P such that X ′ ⊆ B(x0, α). We get a contradiction to the definition of function of bounded oscilation. 2 The next results give us examples of nonmetrizable submetrizable ball structures. Theorem 4.3. If a group G has a normal subgroup H of countable index, then Bl(G) is submetrizable. Proof. Let ℑ be a family of all finite subsets of G containing the identity e of G. Given any g ∈ G, F ∈ ℑ put B′ l(g, FN) = FNg. Denote by ℑ′ the family of all subsets Y ⊆ G of the form Y = FN , F ∈ ℑ. Then B ′ = (G,ℑ′, B′) is metrizable ball structure and B ⊆ B ′. 2 Theorem 4.4. Let ϕ be a filter on an infinite set X, Y = ⋂ ϕ. Then B(X, ϕ) is submetrizable if and only if one of the following statements holds (i) Y is infinite; (ii) Y is finite and there exists a filter ψ on X with a countable base such that ϕ ⊆ ψ and Y /∈ ψ. Proof. If (i) holds, choose a countable subset {yn : n ∈ ω} of Y and put f(yn) = n, n ∈ ω and f(x) = 0 for every x ∈ X \ {yn : n ∈ ω}. Then f : X −→ R is an unbounded function of bounded oscilation. Apply Theorem 4.1. If (ii) holds, choose a base {Fn : n ∈ ω} of ψ such that Fn+1 ⊂ Fn for every n ∈ ω. For every x ∈ X, put Jo ur na l A lg eb ra D is cr et e M at h.I. V. Protasov 101 f(x) = { n, if x ∈ Fn\Fn+1; 0, otherwise. Since f is an unbounded function of bounded oscilation, we can apply Theorem 4.1. Assume that Y is finite and B(X, ϕ) is submetrizable. By Theorem 4.1, there exists an unbounded function f : X −→ R of bounded oscila- tion. For every n ∈ ω, put Xn = {x ∈ X : |f(x)| > n}. Let ψ be a filter on X with the base Xn : n ∈ ω. Take an arbitrary F ∈ ϕ. Since f is of bounded oscilation, f is bounded on X\F . If follows that Xm ⊆ F for some m ∈ ω, so ϕ ⊆ ψ. 2 Theorem 4.5. Let a ball structure B is a disjoint union of the fam- ily {Bλ = (Xλ, P, Bλ) : λ ∈ I} of uniform ball structures. Then B is submetrizable if and only if either I is infinite of there exists λ ∈ I such that Bλ is submetrizable. Proof. If I is infinite, choose a countable subset {λn : n ∈ ω} of I, put f |Xλn ≡ n, n ∈ ω and f |Xλ ≡ 0, λ /∈ {λn : n ∈ ω}. Apply Theorem 4.1. If Bλ is submetrizable, we fix an unbounded function f : Xγ −→ R of bounded oscilation and put f |Xγ ≡ 0 for every γ 6= λ. Apply Theorem 4.1. On the other hand, assume that I is finite and Bλ is not submetrizable for every λ ∈ I. By Theorem 4.1, B is not submetrizable. 2 §5 Extremalities Fix a set X and denote by B(X) the class of all ball structures with the support X. Clearly, every bounded ball structure is a maximal by inclusion ⊆ element of B(X), and any two bounded ball structures on X coincide. On the other hand, the discrete ball structure (X, {p}, B), B(x, p) = {x}, x ∈ X is a minimal by inclusion element of B(X). We can easily avoid these trivialities considering some natural subclasses of B(X). An unbounded uniform ball structure B is called prebounded if B ⊆ B ′, B ′ is unbounded and uniform, implies B ⊆ B ′. Theorem 5.1. For every unbounded uniform ball structure B = (X, P, B), there exists a prebounded ball structure B ′ such that B ⊆ B ′. Every prebounded ball structure is not submetrizable. Proof. The first statement follows directly from Zorn lemma and the construction of inductive limit. To prove the second statement, we take an unbounded metric space (X, d) and define a new metric d′ on X Jo ur na l A lg eb ra D is cr et e M at h.102 Uniform ball structures such that (X, d′) is unbounded and B(X, d) ⊂ B(X, d′). Fix an arbitrary element x0 ∈ X. For every x ∈ X, choose n ∈ ω such that n2 ≤ d(x) < (n + 1)2 and put f(x) = n. For every x ∈ X, m ∈ ω, denote B′(x, m) = {y ∈ X : |f(x) − f(y)| ≤ m}. Consider the ball structure B′ = (X, ω, B′). By metrizability criterion, B ′ is metrizable. Clearly, B ⊂ B ′. 2 Theorem 5.2. Let X be an infinite set and let ϕ = {F ⊆ X : X\F is finite}. Then B(X, ϕ) ⊆ B for every connected uniform ball structure B = (X, P, B). Proof. Take an arbitrary F ∈ ϕ. Since B is connected and uniform, the finite subset X\F is bounded. Choose α ∈ P such that X\F ⊆ B(x, α) for every x ∈ X\F . Then id B(x, F ) ⊆ B(x, α) for every x ∈ X. Hence, B(X, ϕ) ⊆ B. 2 Acknowledgements. The author express his sincere thanks to Michael Zarichnǐı for the question about asymptotic counterpart of uni- form topological spaces. References [1] Burbaki N. Obschaya topologiya. Ispol’zovanie veschestvennih chisel v obschej topologii, M., ”Nauka”, 1975, in russian. [2] Protasov I.V. Mertizable ball structures, Algebra and Discrete Math., 1 (2002). Contact information I. V. Protasov Department Cybernetics, Kyiv State Uni- versity, Volodimirska 64, Kyiv 01033, Ukraine E-Mail: kseniya@profit.net.ua Received by the editors: 31.01.2003.

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