We establish some basic theorems in dimension theory and absolute extensor
theory in the coarse category of metric spaces. Some of the statements in this
category can be translated in general topology language by applying the Higson
corona functor. The relation of problems and results of this `Asymptotic
Topology' to Novikov and similar conjectures is discussed.Comment: 38 pages, AMSTe

Dranishnikov, A. N.

English

arXiv

summary:We show that, under CH, the corona of a countable ultrametric space is homeomorphic to $\omega^*$. As a corollary, we get the same statements for the Higson's corona of a proper ultrametric space and the space of ends of a countable locally finite group

Protasov, I. V.

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeIgor V. ProtasovCoronas of ultrametric spacesCommentationes Mathematicae Universitatis Carolinae, Vol. 52 (2011), No. 2, 303--307Persistent URL: http://dml.cz/dmlcz/141502Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2011Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 52,2 (2011) 303–307 303Coronas of ultrametric spacesI.V. ProtasovAbstract. We show that, under CH, the corona of a countable ultrametric space ishomeomorphic to ω∗. As a corollary, we get the same statements for the Higson’scorona of a proper ultrametric space and the space of ends of a countable locallyfinite group.Keywords: Stone-Čech compactification, ultrametric space, corona, Higson’s co-rona, space of endsClassification: 54D35, 54D80, 54G05Let (X, ρ) be a metric space, x0 ∈ X , Xd be a set X endowed with the discretetopology, βXd be the Stone-Čech compactification of Xd. We identify βXd withthe set of all ultrafilters on X and denote by X# the set of all ultrafilters onX whose members are unbounded subsets of X . A subset A is bounded if thereexists n ∈ ω such that A ⊆ B(x0, n) where B(x0, n) = {x ∈ X : ρ(x0, x) ≤ n}.In what follows, all metric spaces are supposed to be unbounded, so X# 6= ∅.Clearly, X# is closed in βXd.Given any r, q ∈ X#, we say that r, q are parallel (and write r ‖ q) if thereexists n ∈ ω such that, for every R ∈ r, we have B(R, n) ∈ q where B(R, n) =⋃x∈R B(x, n). By [5, Lemma 4.1], ‖ is an equivalence on X#. We denote by ∼ thesmallest (by inclusion) closed (in X# ×X#) equivalence on X# such that ‖⊆∼.By [2, Theorem 3.2.11], the quotient X#/ ∼ is a compact Hausdorff space. It iscalled the corona of X and is denoted by X̌ . To clarify the virtual equivalence ∼,we use the slowly oscillating functions.A function h : (X, ρ) → [0, 1] is called slowly oscillating if, for any n ∈ ω andε > 0, there exists a bounded subset V of X such that, for every x ∈ X \ V ,diamh(B(x, n)) < ε,where diamA = sup{|x− y| : x, y ∈ A}.By [6, Proposition 1], p ∼ q if and only if hβ(p) = hβ(q) for every slowlyoscillating function h : (X, ρ) → [0, 1], where hβ is an extension of h to βXd. If Xis ultrametric we may use only the slowly oscillating functions taking values 0, 1[5, Lemma 4.3]. Recall that (X, ρ) is ultrametric if ρ(x, y) ≤ max{ρ(x, z), ρ(y, z)}for all x, y, z ∈ X .A metric space X is called proper if every ball B(X,n) is compact. In this caseX̌ is homeomorphic to the Higson’s corona νX of X (see [1, §6] and [6, p. 154]).304 I.V. ProtasovLet (X1, ρ1), (X2, ρ2) be metric spaces. A bijection f : X1 → X2 is called anasymorphism if, for any n ∈ ω, there exists m ∈ ω such that, for all x1, x2 ∈ X1and y1, y2 ∈ X2,ρ1(x1, x2) ≤ n ⇒ ρ2(f(x1), f(x2)) ≤ m,ρ2(y1, y2) ≤ n ⇒ ρ1(f−1(y1), f−1(y2)) ≤ m.A subset Y of a metric space (X, ρ) is called large if there exists n ∈ ω such thatB(Y, n) = X . The metric spaces (X1, ρ1), (X2, ρ2) are coarsely equivalent if thereexist large subsets Y1, Y2 of X1, X2 such that (Y1, ρ1), (Y2, ρ2) are asymorphic.We show that, in this case, X̌1 and X̌2 are homeomorphic. Let f : Y1 → Y2 bean asymorphism. Since f and f−1 are ≺-mappings, applying [6, Proposition 1],we conclude that, for any p, q ∈ Y #1 , p ∼ q if and only if fβ(p) ∼ fβ(q). Thenthe mapping f̌ : Y̌1 → Y̌2 defined by f̌([x]) = [fβ(x)], where [x] and [fβ(x)] areequivalence classes containing x and fβ(x), is a homeomorphism. To see that X̌iand Y̌i, i ∈ {1, 2} are homeomorphic, we pick mi ∈ ω such that Xi = B(Yi,mi)and, for each x ∈ Xi, choose hi(x) ∈ Yi such that ρi(x, hi(x)) ≤ mi. Thus themapping ȟi : X̌i → Y̌i is a homeomorphism.Theorem 1. For a metric space X , the following statements hold:(i) every non-empty open subset of X̌ contains a copy of ω∗;(ii) every non-empty Gδ-subset of X̌ has non-empty interior;(iii) if X is ultrametric then X̌ is zero-dimensional F -space;(iv) if X is countable then X̌ is of weight c.Proof: We need some notations. Let x0 be a fixed point of X . Given an un-bounded subset P of X and a function f : ω → ω, we putΨP,f =⋃i∈ωB(P \B(x0, f(i)), i),ΨP,f = {q ∈ X# : ΨP,f ∈ q},Ψ̌P,f = {q̌ ∈ X̌ : q ∈ ΨP,f},where q̌ = {r ∈ X# : r ∼ q}. By [5, Theorem 2.1], for every p ∈ X#,p̌ =⋂{ΨP,f : P ∈ p, f : ω → ω},and the family Ψp = {Ψ̌P,f : P ∈ p, f : ω → ω} is a base of neighbourhoods of p̌in X̌.(i) Let P be an unbounded subset of X . We choose an injective sequence(tn)n∈ω in P such that, for each n ∈ ω,B({t1, . . . , tn}, n) ∩B(tn+1, n+ 1) = ∅,and put T = {tn : n ∈ ω}. Let q, r be distinct ultrafilters from T ∗, Q ∈ q, R ∈ rand Q∩R = ∅. By the choice of T , for each n ∈ ω, B(Q,n)∩B(R, n) is bounded,Coronas of ultrametric spaces 305so we can choose h : ω → ω and h′ : ω → ω such that ΨQ,h ∩ΨR,h′ = ∅. Hence,the mapping from T ∗ to X̌ , defined by p 7→ p̌ is injective. It follows that, forevery f : ω → ω, Ψ̌P,f contains a copy of ω∗.(ii) Let p ∈ X#, {Pn : n ∈ ω} be a decreasing family of members of p,{fn : n ∈ ω} be a family of functions fn : ω → ω. It suffices to show that⋂n∈ω Ψ̌Pn,fn has non-empty interior. We choose a sequence (an)n∈ω in X suchthat an ∈ Pn \B(x0, n), where x0 is taken from definition of ΨP,f , put A = {an :n ∈ ω}, define a function f : ω → ω byf(i) = max{i, f0(i), . . . , fi(i)},and note thatA \B(x0, f(i)) ⊆ Pn \B(x0, fn(i))for all i ≥ n. Since the subsetΨA,f \⋂i≥nB(A \B(x0, f(i)), i)is bounded, we get Ψ̌A,f ⊆ Ψ̌Pn,fn .(iii) To show that X̌ is zero-dimensional, we fix p ∈ X#, P ∈ p and f : ω → ω.By the definition, ΨP,f is closed. We put Φ = ΨP,f . Since X is ultrametric,B(B(x, n), i) = B(x, n) for all x ∈ X and n ≥ i. It follows that B(Φ, i) \ Φ isbounded for each i ∈ ω. Therefore we can define a function h : ω → ω such thatΨΦ,h ⊆ Φ, so Ψ̌Φ,h ⊆ Ψ̌P,f and Ψ̌P,f is open.To prove that X̌ is an F -space, in view of [4, Lemma 1.2.2(b)], it suffices toverify that any two disjoint open Fδ subsets Y, Z of X̌ have disjoint closures. Wemay suppose thatY =⋃n∈ωΨ̌Yn,fn , Z =⋃n∈ωΨ̌Zn,fn .Since Ψ̌Yn,fn ∩ Ψ̌Zm,fm is bounded for all m,n ∈ ω, we can choose inductively thesequences of functions (f ′n)n∈ω, (h′n)n∈ω such thatΨ̌Yn,fn = Ψ̌Yn,f ′n , Ψ̌Zm,hm = Ψ̌Zm,h′m , ΨYn,f ′n ∩ΨZm,h′m = ∅for all m,n ∈ ω.For every n ∈ ω, we putΨn =⋂i≥nB(Yn \B(x0, f′n(i)), i),Ψ′n =⋂i≥nB(Zn \B(x0, h′n(i)), i),and note thatΨ̌n = Ψ̌Yn,fn , Ψ̌′n = Ψ̌Zn,hn .306 I.V. ProtasovNow suppose that p̌ ∈ clX̌⋃n∈ω Ψ̌n and pick p′ ∈ p̌ such that p′ ∈ clX#⋃Ψn,so⋃n∈ω Ψn ∈ p′. We put P =⋃n∈ω Ψn and define a function f : ω → ω byf(i) = max{f0(i), . . . , fi(i)}.Since X is ultrametric, B(Ψn, i) = Ψn for every i ≤ n. Hence, ΨP,f ⊆⋃n∈ω Ψnand p̌ ⊆⋃n∈ω Ψn.Analogously, for every q̌ ∈ clX̌⋃n∈ω Ψ̌′n, we have q̌ ⊆⋃n∈ω Ψ′n. Since(⋃n∈ω Ψn) ∩ (⋃n∈ω Ψ′n) = ∅, we conclude that clX̌ Y ∩ clX̌ Z = ∅.(iv) By (i), w(X̌) ≥ c. The family{ΨP,f : P is an unbounded subset of X, f : ω → ω}is a base of topology of X̌ , so w(X̌) ≤ c. Theorem 2. Let X be an ultrametric space such that X = B(A, n) for somecountable subset A of X and n ∈ ω. Then, under CH, X̌ is homeomorphic to ω∗.Proof: Since X and A are coarsely equivalent, X̌ and Ǎ are homeomorphic,so we may suppose that X is countable. By Theorem 1, X̌ is a compact zero-dimensional F -space of weight c in which every non-empty Gδ-subset has aninfinite interior. Thus, we can apply a characterization [4, Corollary 1.2.4] of ω∗under CH. Corollary 1. Under CH, the Higson’s corona νX of a proper ultrametric spaceX is homeomorphic to ω∗.Proof: To apply Theorem 2, we note that νX is homeomorphic to X̌ and X =B(A, 1) for some countable subset A of X . Let G be an infinite discrete group. A subset A ⊆ G is called almost invariantif gA\A is finite for every g ∈ G. We denote by A the family of all infinite almostinvariant subsets of G and by εG the set of all maximal filters in A endowed withthe topology defined by the family {{ϕ ∈ εG : A ∈ ϕ} : A ∈ A} as a base for theopen sets. Then εG is the remainder of the Freudental-Hopf compactification ofG and every element of εG is called an end of G (for this approach to definitionof ends see [3]). If G is countable and locally finite, by [7, Theorem 3.1.1] and[5, Proposition 2], there is an ultrametric on G such that Ǧ is homeomorphicto εG. Recall that G is locally finite if every finite subset of G generates a finitesubgroup.Corollary 2. Under CH, the space of ends of a countable locally finite group Gis homeomorphic to ω∗.Acknowledgment. The author is thankful to the referee who helped to improvethe presentation of the paper.Coronas of ultrametric spaces 307References[1] Dranishnikov A., Asymptotic topology , Russian Math. Surveys 55 (2000), 1085–1129.[2] Engelking R., General Topology , PWN, Warzsawa, 1985.[3] Houghton C.H., Ends of groups and associated first cohomology groups, J. London Math.Soc. 6 (1972), 81–92.[4] van Mill J., Introduction to βω, in Handbook of Set-Theoretical Topology, Chapter 11,Elsevier Science Publishers B.V., Amsterdam, 1984.[5] Protasov I.V., Normal ball structures, Math. Stud. 20 (2003), 3–16.[6] Protasov I.V., Coronas of balleans, Topology Appl. 149 (2005), 149–160.[7] Protasov I., Zarichnyi M., General asymptology, Math. Stud. Monogr. Ser. 12, VNTL, Lviv,2007.Department of Cybernetics, Kyiv University, Volodimirska 64, Kyiv 01033,UkraineE-mail: I.V.Protasov@gmail.com(Received November 17, 2010, revised January 12, 2011)

Coronas of ultrametric spaces

Czech digital mathematics library

A N Dranishnikov

Crossref