Boundaries of coarse proximity spaces and boundaries of compactifications
aa r X i v : . [ m a t h . GN ] S e p BOUNDARIES OF COARSE PROXIMITY SPACES ANDBOUNDARIES OF COMPACTIFICATIONS
PAWEL GRZEGRZOLKA AND JEREMY SIEGERT
Abstract.
In this paper, we introduce the boundary U X of a coarse proximityspace ( X, B , b ) . This boundary is a subset of the boundary of a certain Smirnovcompactification. We show that U X is compact and Hausdorff and that everycompactification of a locally compact Hausdorff space induces a coarse proximitystructure whose corresponding boundary is the boundary of the compactifica-tion. We then show that many boundaries of well-known compactifications arisenaturally as boundaries of coarse proximity spaces. In particular, we give 4natural coarse proximity structures whose boundaries are the Gromov, visual,Higson, and Freudenthal boundaries. Contents
1. Introduction 12. Proximities and the Smirnov Compactification 33. Coarse Proximities 74. Discrete Extensions and Boundaries of Coarse Proximity Spaces 95. Coarse Proximities from Compactifications 146. The Gromov Boundary 167. The Visual Boundary 198. The Higson Corona 219. The Freudenthal Boundary 22References 261.
Introduction
The field of coarse geometry (occasionally called coarse topology) can be pur-sued using two different, but by no means mutually exclusive perspectives. Thefirst perspective is one we may call “geometry going to infinity” in which one typ-ically takes notions and concepts such as uniformly bounded families and scales
Date : September 22, 2020.2010
Mathematics Subject Classification.
Key words and phrases. coarse geometry, coarse topology, large-scale geometry, coarse proxim-ity, proximity, compactifications, boundaries, Smirnov compactification, Higson corona, Freuden-thal boundary, Gromov boundary, Visual Boundary, δ -hyperbolic spaces, Cat(0) spaces. as primitives and pursues large scale properties of spaces (metric or otherwise)by considering properties of uniformly bounded families at ever growing “scales.”Examples of structures well-suited to this perspective are metric spaces, coarsespaces (see [15]), and large scale spaces (see [4]). Coarse properties that exem-plify the utility of this perspective are asymptotic dimension originally defined byGromov in [6] (and expanded to more general structures in [15] and [4]), PropertyA defined by Yu in [17], and amenability (see for example [15]). The first two ofthese properties are particularly well-known for their relationship to the CoarseBaum-Connes and Novikov conjectures (see [17]). The other perspective may becalled “geometry at infinity.” This perspective takes unboundedness and asymp-totic disjointness as primitives. The pursuit of large-scale properties of spaces isthen conducted by considering how unbounded sets interact with each other “atinfinity.” Typically, this is done by assigning a topological space to an unboundedspace (metric, coarse, or large scale) and considering how the large-scale proper-ties of the base space are reflected in this topological space. The chief example incoarse geometry of such an assignment is the Higson corona (see [15]) assigned toa proper metric space. Another such assignment is the Gromov boundary assignedto hyperbolic metric spaces (see [3]). Techniques for studying these spaces includethe use of asymptotic neighbourhoods (see [2]), coarse neighbourhoods (see [5]),asymptotic resemblance spaces (see [10]), and coarse proximity spaces (see [7]).In this paper, we will be exclusively concerned with the “geometry at infinity”perspective. Our primary contribution will be providing a geometrically intuitive“common language” with which one can speak of coarse invariants such as theHigson corona of proper metric spaces and the Gromov boundary of hyperbolicmetric spaces. Specifically, we will show how to assign a certain boundary spaceto each coarse proximity space (see section 4). These boundaries are compactHausdorff spaces that arise as the boundaries of certain small-scale proximity rela-tions naturally induced by coarse proximities. We will show how basic topologicalnotions of the boundary of a coarse proximity space are captured naturally bythe underlying coarse proximity structure (see for example Proposition 4.7 andProposition 4.8). Our secondary contribution is showing that every compactifi-cation of a locally compact Hausdorff space induces a coarse proximity structurewhose corresponding boundary is the boundary of the compactification (see Theo-rem 5.2). We use this result to describe coarse proximity structures that give riseto well known boundaries in later sections. As a corollary, we will show that everycompact Hausdorff space arises as the boundary of some coarse proximity space(one may compare this result to the results of [12]).The structure of this paper is as follows. In section 2, we review the Smirnovcompactification of a separated proximity space and associated concepts. Like-wise, in section 3, we review the basic definitions and concepts surrounding coarseproximity spaces, as found in [7] and [8]. Then in section 4, we introduce thediscrete extension of a coarse proximity and use it to define the boundary of a OUNDARIES OF COARSE PROXIMITY SPACES 3 coarse proximity space. After exploring a few basic properties of boundaries ofcoarse proximity spaces, we show that the assignment of the boundary to a coarseproximity space makes up a functor from coarse proximity spaces to compactHausdorff spaces. In section 5, we show how every compactification of a locallycompact Hausdorff space induces a coarse proximity structure whose boundary ishomeomorphic to the boundary of the compactification. Finally, in sections 6, 7,8, and 9 we describe the coarse proximity structures on proper hyperbolic met-ric, complete proper Cat(0), proper metric, and locally compact Hausdorff spaceswhose boundaries are the the Gromov boundary, the visual boundary, the Higsoncorona, and the Freudenthal boundary, respectively.Throughout the paper, we will use the following notation and conventions: • If X is a topological space and A is a subset of X , the closure of A in X willbe denoted by cl X ( A ). When there is no risk of ambiguity, the subscript X will be dropped, and the closure of A in X will be denoted by cl ( A ) . • A compactification of a topological space X will be denoted by X , withthe exception of the Smirnov compactification, which will be denoted by X . • If X is a topological space and X is its compactification, then by the traceof A ⊆ X in B ⊆ X we will mean the intersection of the closure of A in X with B . We will denote this trace by tr X,B ( A ). In other words, tr X,B ( A ) = cl X ( A ) ∩ B. • By a proper metric space we mean a metric space (
X, d ) whose closed andbounded sets are compact.The authors are grateful to the reviewers whose helpful comments and sugges-tions which greatly improved the quality of this paper.2.
Proximities and the Smirnov Compactification
In this section, we recall necessary definitions and theorems related to proximi-ties from [13]. In particular, we discuss clusters and the Smirnov Compactification.For an introduction to the subject, see [13].
Definition 2.1.
Let X be a set. A binary relation δ (or δ X if the base space isnot clear from the context) on the power set of X is called a proximity on X ifit satisfies the following axioms for all A, B, C ⊆ X :(1) AδB = ⇒ BδA, (2)
AδB = ⇒ A, B = ∅ , (3) A ∩ B = ∅ = ⇒ AδB, (4) Aδ ( B ∪ C ) ⇐⇒ AδB or AδC, (5) A ¯ δB = ⇒ ∃ E ⊆ X, A ¯ δE and ( X \ E )¯ δB, PAWEL GRZEGRZOLKA AND JEREMY SIEGERT where A ¯ δB means ” AδB ” does not hold. Axiom (5) is called the strong axiom .A pair (
X, δ ) where X is a set and δ is a proximity on X is called a proximityspace . Definition 2.2.
A proximity space (
X, δ ) is called separated if { x } δ { y } ⇐⇒ x = y, for all x, y ∈ X. Example 2.3.
Let (
X, d ) be a metric space. Then the relation δ d defined by Aδ d B ⇐⇒ d ( A, B ) = 0 , is a separated proximity called the metric proximity , where d ( A, B ) = inf { d ( a, b ) | a ∈ A, b ∈ B } . Example 2.4.
Let (
X, δ X ) be a proximity space and let Y ⊆ X. Then the relation δ Y defined by Aδ Y B ⇐⇒ Aδ X B, where A and B are subsets of Y , is a proximity relation on Y, called the subspaceproximity . Definition 2.5.
Let (
X, δ X ) and ( Y, δ Y ) be two proximity spaces. A function f : X → Y is called a proximity map if Aδ X B = ⇒ f ( A ) δ Y f ( B ) . A bijective proximity map whose inverse is also a proximity map is called a prox-imity isomorphism . A proximity isomorphism onto a subspace of a proximityspace in called a proximity embedding . Theorem 2.6.
Let ( X, δ ) be a proximity space. Then δ induces a topology on X defined by A is closed ⇐⇒ ( Aδ { x } = ⇒ x ∈ A ) This topology is always completely regular, and is Hausdorff if and only if δ isseparated.Proof. See [13]. (cid:3)
It is easy to show that the metric proximity induces the metric topology, andthat the subspace proximity induces the subspace topology. From now on, we willalways assume that a given proximity space is equipped with the induced topology,as in Theorem 2.6.
Example 2.7.
Let X be a compact Hausdorff space. Then the relation δ definedby AδB ⇐⇒ cl ( A ) ∩ cl ( B ) = ∅ , is the unique separated proximity relation on X inducing the original topology on X . OUNDARIES OF COARSE PROXIMITY SPACES 5
Proposition 2.8.
Let ( X, δ ) be a proximity space and let A, B ⊆ X. Then(1) cl ( A ) = { x ∈ X | xδA } (2) AδB ⇐⇒ ( cl ( A )) δ ( cl ( B )) . Proof.
See [13]. (cid:3)
We now introduce clusters, which are the building blocks of the Smirnov Com-pactification.
Definition 2.9.
Let (
X, δ ) be a proximity space. Let σ be a collection of subsetsof X . Then σ is called a cluster if the following hold:(1) if A, B ∈ σ, then AδB, (2) if
AδB for all B ∈ σ, then A ∈ σ, (3) if ( A ∪ B ) ∈ σ, then either A ∈ σ or B ∈ σ . Definition 2.10.
A cluster in a proximity space (
X, δ ) is called a point cluster if { x } ∈ σ for some x ∈ X . If ( X, δ ) is separated, we denote the point clustercontaining x by σ x , as each cluster in a separated proximity space contains at mostone singleton because distinct points are not close in separated proximity spaces.A simple result about clusters that will be useful to us in the future is thefollowing: Proposition 2.11.
Let ( X, δ ) be a separated proximity space and A, B ⊆ X suchthat AδB . Then there is a cluster σ in X that contains both A and B .Proof. See [13]. (cid:3)
We now focus on the construction of the Smirnov Compactification.
Definition 2.12.
Let (
X, δ ) be a proximity space. Define X to be the set of allclusters in X. Let
A ⊆ X . We say that a subset A ⊆ X absorbs A if A ∈ σ forall σ ∈ A . Theorem 2.13.
Let ( X, δ ) be a separated proximity space and X the correspondingset of all clusters in X. For two subsets A , B ⊆ X , define: A δ ∗ B ⇐⇒
AδB for all
A, B ⊆ X that absorb A and B , respectively. The relation δ ∗ , called theSmirnov proximity , is a separated proximity on X that induces a compact Haus-dorff topology on X . The mapping f : X → X defined by x → σ x is a dense prox-imity embedding. The space ( X , δ ∗ ) is called the Smirnov compactification of ( X, δ ) . It is the unique (up to proximity isomorphism) compact Hausdorff spaceinto which X embeds as a dense subspace through a proximity embedding.Proof. See section 7 in [13]. (cid:3)
PAWEL GRZEGRZOLKA AND JEREMY SIEGERT
Given a proximity space (
X, δ ), we will always denote its Smirnov compactifica-tion by X and the corresponding proximity by δ ∗ . Also, it is customary to identify X with its image in X . We are going to follow this practice. Consequently, given A ⊆ X, one can think of A as a subset of X . It is then easy to show that cl X ( A ) = { σ ∈ X | A ∈ σ } . In other words, A ∈ σ ⇐⇒ σ ∈ cl X ( A ) . The following proposition is used to show that every proximity map extendsuniquely to a proximity map between the Smirnov compactifications.
Proposition 2.14.
Let ( X, δ X ) and ( Y, δ Y ) be proximity spaces. Let f : X → Y be a proximity map. Then to each cluster σ in X, there corresponds a cluster σ in Y defined by σ = { A ⊆ Y | Aδ Y f ( B ) for all B ∈ σ } . Proof.
See [13]. (cid:3)
Notice that in the setting of the above proposition, it is immediate that theimage of any set in σ is in σ i.e., f ( σ ) ⊆ σ . Theorem 2.15.
Let ( X, δ X ) and ( Y, δ Y ) be separated proximity spaces, and let ( X , δ ∗ X ) and ( Y , δ ∗ Y ) be the respective Smirnov compactifications. Let f : X → Y be a proximity map. Then f extends to a unique proximity map f ∗ : X → Y , whichis defined using Proposition 2.14.Proof. The proof of this theorem (with the unnecessary and unused assumption ofthe surjectivity of f ) can be found in [13]. (cid:3) Definition 2.16.
Given a separated proximity space (
X, δ ) and its correspondingSmirnov compactification ( X , δ ∗ ), the Smirnov boundary of X is the subspace X \ X (with the subspace proximity inherited from ( X , δ ∗ )).Now we investigate a few basic properties of the Smirnov boundary. Proposition 2.17.
Let ( X, δ ) be a separated proximity space. Let σ be an elementof X \ X . Then K / ∈ σ for any compact subset K of X .Proof. Let σ ∈ X \ X be given. Assume towards a contradiction that K ⊆ X isa compact set such that K ∈ σ . Identify K with the corresponding set of pointclusters. If B ⊆ X absorbs K and k ∈ K, then { k } δB (since { k } is an element of σ k and B absorbs σ k ). Consequently, K ⊆ cl X ( B ) . Also, given any A ∈ σ , we musthave that AδK by the definition of a cluster. Since K ⊆ cl X ( B ) for any absorbingset B of K , we must have that Aδ ( cl X ( B )) . Consequently, by Proposition 2.8, wehave that
AδB for any A ∈ σ and any set B absorbing K. This yields that { σ } δ ∗ K in X . However, because K is closed in X we must have that σ ∈ K , which is tosay that σ is a point cluster based at some point of K . This contradicts σ beingan element of X \ X . (cid:3) OUNDARIES OF COARSE PROXIMITY SPACES 7
Proposition 2.18.
Let ( X, δ ) be a separated proximity space and σ ∈ X \ X . Thenfor every A ∈ σ and every compact K ⊆ X, we have that ( A \ K ) ∈ σ .Proof. This follows immediately from axiom (3) in the definition of a cluster andProposition 2.17. (cid:3) Coarse Proximities
In this section, we recall the needed definitions and theorems from [7] and [8].To provide the reader with the geometric intuition behind the definitions, mostdefinitions are followed with an example in a metric setting.
Definition 3.1.
Let X be a set. A bornology B (or B X if the base space is notclear from the context) is a family of subsets of X satisfying:(1) { x } ∈ B for all x ∈ X, (2) A ∈ B and B ⊆ A implies B ∈ B , (3) If A, B ∈ B , then A ∪ B ∈ B . Elements of B are called bounded and subsets of X not in B are called un-bounded . Example 3.2.
Let (
X, d ) be a metric space. Then the collection of all metricallybounded sets is a bornology. We denote this bornology by B d . Example 3.3.
Let X be a topological space. Then the collection of all precompactsets (i.e., sets whose closure is compact) is a bornology. We denote this bornologyby B C . Definition 3.4.
Let X be a set equipped with a bornology B . A coarse prox-imity on X (or ( X, B ) if the bornology is not clear from the context) is a relation b (or b X if the base space is not clear from the context) on the power set of X satisfying the following axioms for all A, B, C ⊆ X :(1) A b B = ⇒ B b A, (2) A b B = ⇒ A / ∈ B and
B / ∈ B , (3) A ∩ B / ∈ B = ⇒ A b B, (4) ( A ∪ B ) b C ⇐⇒ A b C or B b C, (5) A ¯ b B = ⇒ ∃ E ⊆ X such that A ¯ b E and ( X \ E )¯ b B, where A ¯ b B means “ A b B is not true.” If A b B , then we say that A is coarselyclose to (or coarsely near ) B. Axiom (5) is called the strong axiom . A triple( X, B , b ) where X is a set, B is a bornology on X , and b is a coarse proximityrelation on X, is called a coarse proximity space .Even though the strong axiom was already defined for proximity spaces, themeaning behind the strong axiom will always be clear from the context. Example 3.5.
Let (
X, d ) be a metric space with the bornology B d . For any
A, B ⊆ X , define A b d B if and only if there exists ǫ < ∞ such that for all bounded sets PAWEL GRZEGRZOLKA AND JEREMY SIEGERT D , there exists a ∈ ( A \ D ) and b ∈ ( B \ D ) such that d ( a, b ) < ǫ. Then b d is acoarse proximity, called the metric coarse proximity . The triple ( X, B d , b d ) iscalled the metric coarse proximity space. To define maps between coarse proximity spaces, we need the concept of theweak asymptotic resemblance.
Theorem 3.6.
Let ( X, B , b ) be a coarse proximity space. Define φ to be therelation on the power set of X by AφB if and only if the following hold:(1) for every unbounded B ′ ⊆ B we have A b B ′ , (2) for every unbounded A ′ ⊆ A we have A ′ b B. Then φ is an equivalence relation satisfying AφB and
CφD = ⇒ ( A ∪ C ) φ ( B ∪ D ) for any A, B, C, D ⊆ X. We call this equivalence relation the weak asymptoticresemblance induced by the coarse proximity b .Proof. See [7]. (cid:3)
We will later show that the relation b captures “closeness at infinity” (see Propo-sition 4.7), and the relation φ captures “equality at infinity” (see Proposition 4.8). Example 3.7.
Let ( X, B d , b d ) be a metric coarse proximity space. Then the weakasymptotic resemblance induced by b d is denoted by φ d . It can be shown (see [7])that for nonempty A, B ⊆ X , Aφ d B ⇐⇒ A and B have finite Hausdorff distance . Definition 3.8.
Let ( X, B X , b X ) and ( Y, B Y , b Y ) be coarse proximity spaces. Let f : X → Y be a function. Then f is a coarse proximity map provided that thefollowing are satisfied for all A, B ⊆ X :(1) B ∈ B X = ⇒ f ( B ) ∈ B Y , (2) A b X B = ⇒ f ( A ) b Y f ( B ) . Notice that item 2 from the above definition implies that item 1 can be strength-ened to B ∈ B X ⇐⇒ f ( B ) ∈ B Y . Definition 3.9.
Let X be a set and ( Y, B , b ) a coarse proximity space. Twofunctions f, g : X → Y are close if for all A ⊆ Xf ( A ) φg ( A ) , where φ is the weak asymptotic resemblance relation induced by the coarse prox-imity structure b . OUNDARIES OF COARSE PROXIMITY SPACES 9
Definition 3.10.
Let ( X, B X , b X ) and ( Y, B Y , b Y ) be coarse proximity spaces.We call a coarse proximity map f : X → Y a coarse proximity isomorphism ifthere exists a coarse proximity map g : Y → X such that g ◦ f is close to id X and f ◦ g is close to id Y . We say that ( X, B X , b X ) and ( Y, B Y , b Y ) are coarse proximityisomorpic (or just isomorpic ) if there exists a coarse proximity isomorphism f : X → Y. The collection of coarse proximity spaces and closeness classes of coarse proxim-ity maps makes up the category
CrsProx of coarse proximity spaces. For details,see [7].4.
Discrete Extensions and Boundaries of Coarse Proximity Spaces
In this section, we introduce discrete extensions of coarse proximities. We usethem to define the boundaries of coarse proximity spaces. We also prove a fewproperties of such boundaries. In particular, we show that every such boundary iscompact and Hausdorff. Finally, we show the existence of a nontrivial functor fromthe category of coarse proximity spaces (with closeness classes of coarse proximitymaps) to the category of compact separated proximity spaces (with proximitymaps). To learn more about the category of coarse proximity spaces, the reader isreferred to section 7 in [7].The following definition has been inspired by the definition of the Higson prox-imity of a proper metric space (see Definition 8.1).
Proposition 4.1.
Let ( X, B , b ) be a coarse proximity space. Define a binaryrelation δ dis (or δ dis, b if the coarse proximity is not clear from the context) on thepower set of X by Aδ dis B ⇐⇒ A ∩ B = ∅ or A b B. Then δ dis is a separated proximity on X. We will call this proximity the discreteextension of b and the space ( X, δ dis ) the discrete extension of ( X, B , b ) .Proof. All the axioms besides the strong axiom are immediate. To prove the strongaxiom, let A ¯ δ dis B. Then A ∩ B = ∅ and A ¯ b B. Consequently, there exists E ⊆ X such that A ¯ b E and ( X \ E )¯ b B. Since A ∩ E is bounded, the set E ′ = E \ A isdisjoint from A , and we still have A ¯ b E ′ and ( X \ E ′ )¯ b B. Since B is contained in E ′ up to a bounded set, B \ E ′ is bounded. Consequently, E ′′ = E ′ ∪ ( B \ E ′ ) is stilldisjoint from A (we added a subset of B , which does not intersect A ), and we stillhave that A ¯ b E ′′ and ( X \ E ′′ )¯ b B. By construction, E ′′ fully contains B, i.e., X \ E ′′ does not intersect B. In conclusion, we found E ′′ such that A ∩ E ′′ = ∅ , A ¯ b E ′′ , ( X \ E ′′ ) ∩ B = ∅ , and ( X \ E ′′ )¯ b B. This means that A ¯ δ dis E ′′ and ( X \ E ′′ )¯ δ dis B, which completes the proof of the strong axiom. (cid:3) Now we are ready to define boundaries of coarse proximity spaces.
Definition 4.2.
Let ( X, B , b ) be a coarse proximity space,. Let X be the Smirnovcompactification induced by δ dis . Define U X ⊆ X (or U b X if the coarse proximityinducing the discrete extension is not clear from the context) to be the set ofclusters in ( X, δ dis ) that do not contain any bounded sets. In particular, U X ⊆ X \ X. The space U X (equipped with the subspace proximity inherited from X ) iscalled the boundary of the coarse proximity space ( X, B , b ) . Given a coarse proximity space ( X, B , b ), we will always assume that ( X , δ ∗ dis )(or just X ) denotes the Smirnov compactification induced by δ dis . To show thatthe boundary of a coarse proximity space is compact and Hausdorff, we need thefollowing lemmas. Lemma 4.3.
Let ( X, B , b ) be a coarse proximity space. Let σ be an arbitraryelement of U X. Then for any
A, B ∈ σ, we have A b B. Proof.
Since A and B are in σ, we know that Aδ dis B. Then A ∩ B = ∅ or A b B. If A ∩ B is unbounded, then clearly A b B. Suppose that A ∩ B is bounded. Byaxiom 3 of a cluster, we know that A \ ( A ∩ B ) is in σ or A ∩ B is in σ. However, A ∩ B is bounded, so it cannot belong to σ. Thus, it has to be that A \ ( A ∩ B ) isin σ. But then ( A \ ( A ∩ B )) δ dis B (since A \ ( A ∩ B ) is in σ and B is in σ ). Sincethese sets are disjoint, this means that ( A \ ( A ∩ B )) b B. Since A ∩ B is bounded,this in turn implies that A b B. (cid:3) Lemma 4.4.
Let ( X, B , b ) be a coarse proximity space. Then every element of U X is closed under the weak asymptotic resemblance φ induced by b . In otherwords, if A ∈ σ ∈ U X, then for every B such that AφB, we have B ∈ σ. Proof.
Let σ be an element of U X, A an element of σ, and B a subset of X such that AφB.
We need to show that B ∈ σ. Notice that since
AφB and A is unbounded,we know that B is unbounded. We are going to utilize axiom 2 of a cluster. Let C be an arbitrary element of σ. We want to show that B b C, and consequently Bδ dis C. We know that Aδ dis C (because A and C belong to the same cluster σ ).By the previous lemma, this means that A b C. Since A and B are φ related, theyare coarsely close to the same subsets of X (see Corollary 6.18 in [7]). Thus, ithas to be true that B b C, and consequently, Bδ dis C. Thus, we have shown that foran arbitrary element C of σ, we have Bδ dis C. By axiom 2 of a cluster, this showsthat B is in σ. (cid:3) Proposition 4.5.
For every coarse proximity space ( X, B , b ) , the space U X iscompact and Hausdorff.Proof. First note that if X is bounded, then U X is empty. Thus, assume that X is unbounded. Because X is a compact Hausdorff space and U X ⊆ X it will sufficeto show that U X is closed in X . By Proposition 2.8, it suffices then to show thatif σ ∈ X \ U X, then { σ } is not close to U X in X . Suppose σ ∈ X \ U X. Thismeans that there is some bounded set B ⊆ X such that B ∈ σ . Let φ be the OUNDARIES OF COARSE PROXIMITY SPACES 11 weak asymptotic resemblance induced by b . Since B is bounded, ( X \ B ) φX. ByLemma 4.4, every element of U X is closed under the φ relation. Because X is anelement of each cluster of X we have that the set X \ B belongs to each elementof U X . Because B ¯ δ dis ( X \ B ), B absorbs { σ } , and X \ B absorbs U X we havethat { σ } is not close to U X in the Smirnov compactification. Therefore, U X isclosed in X , and is consequently compact and Hausdorff. (cid:3) For the reminder of this section, for any A ⊆ X , tr ( A ) will denote tr X , U X ( A ) , as defined in the introduction.What follows are some basic facts regarding the boundary of a coarse proximityspace. Proposition 4.6.
Let ( X, B , b ) be a coarse proximity space, and A ⊆ X. Then(1) tr ( A ) = ∅ if A is bounded,(2) tr ( A ) = ∅ if A is unbounded.Proof. To see (1), let A be bounded. Then A ∈ σ for any σ ∈ cl X ( A ) (it is because cl X ( A ) = { σ ∈ X | A ∈ σ } ). Consequently, tr ( A ) = cl X ( A ) ∩ U X = ∅ . To see(2), notice that for each D ∈ B , we have that A \ D is unbounded. The collection A = { cl X ( A \ D ) | D ∈ B} is the collection of nonempty closed sets in X thattrivially has the finite intersection property, and thus T A 6 = ∅ . Let σ be a pointin this intersection. Clearly σ ∈ cl X ( A ) = cl X ( A \ ∅ ) . Also, σ must be an elementof U X. To see that, for contradiction assume that there exists C ∈ B such that C ∈ σ. Since σ ∈ cl X ( A \ D ) for each D ∈ B , we must have that A \ D ∈ σ for each D ∈ B . Consequently, Cδ dis ( A \ D ) for each D ∈ B . In particular, Cδ dis ( A \ C ) . However, this is a contradiction, since C is bounded and A \ C is unbounded (andthus C ¯ b ( A \ C )) and C ∩ ( A \ C ) = ∅ . Therefore, tr ( A ) = ∅ . (cid:3) The following proposition explains the intuitive notion of coarse proximitiescapturing “closeness at infinity.” In particular, it shows that two subsets of acoarse proximity space are coarsely close if and only if they are “close at theboundary.”
Proposition 4.7.
Let ( X, B , b ) be a coarse proximity space. Let A and B besubsets of X. Then A b B if and only if tr ( A ) ∩ tr ( B ) = ∅ .Proof. The proposition is clearly true if at least one of the subsets is bounded, solet us assume that both A and B are unbounded. To prove the forward direction,assume A b B. Consequently, we have that ( A \ D ) b ( B \ D ) for all D , D ∈ B .This implies that ( A \ D ) δ dis ( B \ D ) for all D , D ∈ B . Because X is compact wehave that cl X ( A \ D ) ∩ cl X B = ∅ for all D ∈ B . We then have that the collection C = { cl X ( B ) } ∪ { cl X ( A \ D ) | D ∈ B} is a collection of closed sets in X thathas the finite intersection property. To see this, notice that for any bounded sets D , . . . , D n ∈ B , we have n \ i =1 cl X ( A \ D i ) ∩ cl X ( B ) ⊇ cl X A \ n [ i =1 D i ∩ cl X ( B ) = ∅ . The compactness of X then tells us that T C 6 = ∅ . Let σ ∈ T C . The proof ofProposition 4.6 gives us that σ ∈ U X. Moreover, it is clear that σ ∈ cl X ( A ) and σ ∈ cl X ( B ) . Thus, σ ∈ tr ( A ) and σ ∈ tr ( B ) , which shows that tr ( A ) ∩ tr ( B ) = ∅ . To see the converse, assume that tr ( A ) ∩ tr ( B ) = ∅ . Let σ ∈ tr ( A ) ∩ tr ( B ) . Then
A, B ∈ σ and σ ∈ U X, and thus Lemma 4.3 gives us that A b B. (cid:3) The following proposition shows that two subsets of a coarse proximity spaceare φ related if and only if they are “the same at the boundary.” Proposition 4.8.
Let ( X, B , b ) be a coarse proximity space and φ the corre-sponding weak asymptotic resemblance. If A, B ⊆ X, then AφB if and only if tr ( A ) = tr ( B ) . Proof. If A and B are bounded, then clearly AφB and tr ( A ) = tr ( B ) = ∅ . Ifonly one of the sets is bounded, then clearly A ¯ φB and tr ( A ) = tr ( B ) . So assumethat A and B are unbounded. To prove the forward direction, assume AφB.
Let σ ∈ tr ( A ) . This means that A ∈ σ and σ ∈ U X. Then Lemma 4.4 implies that B ∈ σ, i.e., σ ∈ cl X ( B ) . Since σ ∈ U X as well, this shows that σ ∈ tr ( B ) . Thus, tr ( A ) ⊆ tr ( B ) and by symmetry of the argument it follows that tr ( A ) = tr ( B ) . To prove the converse, assume tr ( A ) = tr ( B ) . Let C ⊆ A be an unboundedsubset. For contradiction, assume that C ¯ b B. Then C ∩ B ∈ B . We then havethat C = ( C \ ( C ∩ B )) is an unbounded subset of A such that C ¯ b B . Because C ⊆ A we trivially have that tr ( C ) ⊆ tr ( A ) = tr ( B ). Proposition 4.6 tells usthat tr ( C ) = ∅ . Thus, tr ( C ) ∩ tr ( B ) = ∅ . Proposition 4.7 implies then that C b B, a contradiction. Thus, C b B . One can similarly show that if C ⊆ B is anunbounded set, then C b A . Therefore, AφB . (cid:3) For the remainder of this section, we will focus on the construction of the afore-mentioned functor (from the category of coarse proximity spaces to the categoryof compact Hausdorff spaces).
Proposition 4.9.
Let f : ( X, B X , b X ) → ( Y, B Y , b Y ) be a coarse proximity map.Then f : ( X, δ dis, b X ) → ( Y, δ dis, b Y ) is a proximity map. Moreover, if σ is in U X, then the associated cluster σ in Y, as described in Proposition 2.14, is in U Y. Proof.
That f is a proximity map is clear from the fact that coarse proximity mapspreserve b and all sets functions preserve nontrivial intersections. Now let σ be acluster in X that does not contain any unbounded sets. By Proposition 2.14, theassociated cluster σ in Y is given by σ = { A ⊆ Y | ∀ C ∈ σ , Aδ dis, b Y f ( C ) } . OUNDARIES OF COARSE PROXIMITY SPACES 13
For contradiction, assume that B ⊆ Y is a bounded set such that B ∈ σ . Then forall C ∈ σ , we have that Bδ dis, b Y f ( C ) . Since B is bounded and f ( C ) is unbounded,this shows that for all C ∈ σ , we have that B ∩ f ( C ) = ∅ . Consequently, f − ( B )is a bounded set that intersects all C ∈ σ . Thus, f − ( B ) ∈ σ , a contradiction to σ ∈ U X. Thus, σ ∈ U Y. (cid:3) Corollary 4.10.
Let f : ( X, B X , b X ) → ( Y, B B , b B ) be a coarse proximity map.Then the unique extension f ∗ : X → Y between Smirnov compactifications maps U X to U Y .Proof. Follows immediately from Theorem 2.15 and Proposition 4.9. (cid:3)
Definition 4.11.
The map f ∗ in Corollary 4.10 restricted in domain and codomainto U X and U Y will be denoted by U f . Proposition 4.12.
Let f, g : ( X, B X , b X ) → ( Y, B Y , b Y ) be close coarse proximitymaps. Then U f = U g .Proof. Let σ be an element of U X. Let σ and σ ′ be the clusters in U Y corre-sponding to the images of σ under f and g, respectively. Let A ∈ σ . Then for all C ∈ σ , we have that Aδ dis, b Y f ( C ) . Since f ( C ) φ b X g ( C ) (where φ b X is the weakasymptotic resemblance induced by b X ), this shows that Aδ dis, b Y g ( C ) . Since C was an arbitrary element of σ , this implies that A ∈ σ . Thus, σ ⊆ σ ′ . Theopposite inclusion follows similarly. (cid:3)
Proposition 4.13.
Let f : ( X, B X , b X ) → ( Y, B Y , b Y ) and g : ( Y, B Y , b Y ) → ( Z, B Z , b Z ) be coarse proximity maps. Then the following are true:(1) U id X = id U X , (2) U ( g ◦ f ) = U g ◦ U f .Proof. Recall that given any proximity space and any clusters σ and σ ′ in thatproximity space, then σ ⊆ σ ′ implies σ = σ ′ . Keeping that in mind, (1) is imme-diate. To see (2) , let σ be an element of U X, σ the associated cluster in U Y (through f ), and σ the associated cluster in U Z (through g ), i.e., σ = { B ⊆ Y | ∀ A ∈ σ , Bδ dis, b Y f ( A ) } ,σ = { C ⊆ Z | ∀ B ∈ σ , Cδ dis, b Z g ( B ) } . Let σ ′ be the associated cluster in U Z (through g ◦ f ) , i.e., σ ′ = { C ⊆ Z | ∀ A ∈ σ , Cδ dis, b Z ( g ◦ f )( A ) } . Let C be any element of σ . Then Cδ dis, b Z g ( B ) for any B in σ . Since f ( A ) isin σ for any A in σ , this in particular shows that Cδ dis, b Z g ( f ( A )) for any A in σ . Thus, C is in σ ′ . Consequently, σ ⊆ σ ′ , and thus σ = σ ′ . (cid:3) Theorem 4.14.
The assignment of the compact separated proximity space U X to the coarse proximity space ( X, B , b ) together with the assignment of the prox-imity map U f to a closeness class of coarse proximity maps [ f ] : ( X, B X , b X ) → ( Y, B Y , b Y ) makes up a functor from the category CrsProx of coarse proximityspaces with closeness classes of coarse proximity maps to the category
CSProx ofcompact separated proximity spaces with proximity maps.Proof.
This is immediate from Corollary 4.10, Proposition 4.12, and Proposition4.13. (cid:3)
Corollary 4.15. If f : ( X, B X , b X ) → ( Y, B Y , b Y ) is a is a coarse proximity iso-morphism, then U f is a proximity isomorphism. In particular, if ( X, B X , b X ) and ( Y, B Y , b Y ) are coarse proximity isomorphic, then U X and U Y are homeomorphic.Proof. Immediate from Theorem 4.14. (cid:3) Coarse Proximities from Compactifications
In this section, we will show that every compactification X of a locally compactHausdorff space X induces a coarse proximity structure on X whose boundaryis homeomorphic to X \ X . Keeping with the convention introduced in the pre-vious section, for any A ⊆ X , tr ( A ) will denote tr X,X \ X ( A ) , as defined in theintroduction.We begin by fixing a locally compact space X with compactification X . Recallthat B C denotes the bornology of precompact sets. We can then define a naturalcoarse proximity on ( X, B C ) using the compactification X . Proposition 5.1.
The relation b on subsets of X defined by A b C ⇐⇒ tr ( A ) ∩ tr ( C ) = ∅ is a coarse proximity on ( X, B C ) .Proof. We are only going to verify the strong axiom as the other axioms arestraightforward. Let
A, C ⊆ X be such that A ¯ b C . This means that tr ( A ) and tr ( C ) are disjoint and closed subsets of X \ X. In particular, they are disjoint andclosed subsets of X . Since X is normal and Hausdorff, there exist open sets U and U of X such that tr ( A ) ⊆ U , tr ( C ) ⊆ U and cl X ( U ) and cl X ( U ) are disjoint.Define E = U ∩ X . Notice that E is nonempty, since U is open in X and X isdense in X . We claim that A ¯ b ( X \ E ) and C ¯ b E . The second of these follows from tr ( E ) ⊆ tr ( U ) ⊆ cl X ( U ) and tr ( C ) ∩ cl X ( U ) = ∅ . Likewise, A ¯ b ( X \ E ) follows from tr ( X \ E ) = tr ( X \ U ) ⊆ cl X ( X \ U ) = X \ U and tr ( A ) ∩ ( X \ U ) = ∅ . OUNDARIES OF COARSE PROXIMITY SPACES 15
This establishes the strong axiom for b . (cid:3) Theorem 5.2.
The boundary U X of the coarse proximity space ( X, B C , b ) . where b is defined as in Proposition 5.1, is homeomorphic to X \ X .Proof. Let δ be the subspace proximity on X \ X , as in Example 2.7. Let δ ∗ dis, b beSmirnov proximity on the set of all clusters in X induced by the discrete extensionof b . We will show the results via three claims. Claim 1: x σ = { A ⊆ X | x ∈ tr ( A ) } is a cluster of X for all x ∈ X \ X . Proof of Claim 1.
Notice that the only axiom that is not clear is that if Cδ ∗ dis, b A forall A ∈ x σ, then C ∈ x σ . However, if C / ∈ x σ, then by definition cl X ( C ) ∩ { x } = ∅ .In particular, x is disjoint from cl X ( C ) . Since X is compact Hausdorff, we can findan open set U in X such that x ∈ U and cl X ( U ) is disjoint from cl X ( C ) . Consider U ∩ X. Since U is open and Y is dense in X, U ∩ X is a nonempty subset of X .Also, U ∩ X is in x σ. Finally, since cl X ( U ) is disjoint from cl X ( C ) , we know that C ¯ δ dis, b ( U ∩ X ) , a contradiction to Cδ dis, b A for all A ∈ x σ. (cid:3) Claim 2: ˜ X = { x σ | x ∈ X \ X } , (equipped with the proximity δ ∗ dis, b ) ishomeomorphic to X \ X . Proof of Claim 2.
Define f : ( X \ X ) → ˜ X by f ( x ) = x σ. It is clear that f issurjective. To show that f is injective, let x and y be elements of X \ X suchthat x = y. Consequently, there exist two open sets U and U of X such that x ∈ U , y ∈ U and cl X ( U ) is disjoint from cl X ( U ) . Thus, U ∩ X and U ∩ X are nonempty subsets of X whose closures in X don’t intersect and ( U ∩ X ) ∈ x σ and ( U ∩ X ) ∈ y σ. In particular, since the closures of U ∩ X and U ∩ X in X don’t intersect, they cannot be in the same cluster. Thus, x σ = y σ. To seethat f is a proximity map, let A and B subsets of X \ X such that AδB, i.e., cl X \ X ( A ) ∩ cl X \ X ( B ) = ∅ . Notice that f ( A ) = { x σ | x ∈ A } ,f ( B ) = { x σ | x ∈ B } . Let C and D be subsets of X that absorb f ( A ) and f ( B ) , respectively. We claimthat this implies that cl X \ X ( A ) ⊆ tr ( C ) and cl X \ X ( B ) ⊆ tr ( D ). To see this, notethat C absorbing f ( A ) implies that C ∈ a σ for all a ∈ A, which by definition givesus that a ∈ tr ( C ) for all a ∈ A . Because A ⊆ X \ X , this gives us that A ⊆ tr ( C ).Since tr ( C ) is closed in X \ X, we have that cl X \ X ( A ) ⊆ tr ( C ). A similar argumentshows that cl X \ X ( B ) ⊆ tr ( D ). Because cl X \ X ( A ) ⊆ tr ( C ), cl X \ X ( B ) ⊆ tr ( D ), and cl X \ X ( A ) ∩ cl X \ X ( B ) = ∅ we have that tr ( C ) ∩ tr ( D ) = ∅ . This implies that C b D, and consequently we have that Cδ dis, b D. Since C and D were arbitrary absorbingsets, this shows that f ( A ) δ ∗ f ( B ) . Thus, f is a bijective proximity map from acompact space to a Hausdorff space, which shows that f is a homeomorphism. (cid:3) Claim 3: U X = ˜ X Proof of Claim 3.
Notice that for any x ∈ X \ X, we have that x σ contains onlyunbounded sets (for if x σ contains a bounded set A, then tr ( A ) is empty, and thuscannot contain x ). Thus, ˜ X ⊆ U X. To see the opposite inclusion, we can showthat ˜ X is dense in U X. Showing that ˜ X is dense in U X is sufficient, since ˜ X ishomeomorphic to X \ X, and consequently it is compact (and thus closed) in U X. Being a dense closed subset of U X, ˜ X will have to equal U X. To show that ˜ X isdense in U X, let σ ∈ U X and assume that { σ } ¯ δ ∗ dis, b ˜ X . Then there are absorbingsets C for σ and A for ˜ X such that tr ( C ) ∩ tr ( A ) = ∅ . Because tr ( A ) must equal X \ X (since A is in x σ for all x ∈ X \ X ), this implies that tr ( C ) = ∅ , whichimplies that C is bounded. But this cannot be, since C is in σ and σ by definitioncontains only unbounded sets. Thus, by contradiction it has to be that { σ } δ ∗ dis, b ˜ X. Since σ was an arbitrary element of U X, this shows that ˜ X is dense in U X , whichshows the claim and finishes the proof of the theorem. (cid:3)(cid:3) Given a compactification X of a locally compact Hausdorff X , we will call thecoarse proximity space ( X, B C , b ) the coarse proximity structure induced on X bythe compactification X , where b is defined as in Proposition 5.1. An immediateconsequence of the preceding results of this section is the following: Corollary 5.3.
Every compact Hausdorff space arises as the boundary of a coarseproximity space.Proof. If X is a given compact Hausdorff space, then we may take X × [0 ,
1] to bea compactification of Y = X × [0 , Y induced by X × [0 ,
1] is such that U Y is homeomorphic to X . (cid:3) In the next 4 sections, we are going to show how some boundaries of well-knowncompactifications can be realized as boundaries of coarse proximity spaces. In thenext two sections, we use Theorem 5.2 to show how the Gromov boundary and thevisual boundary arise as boundaries of coarse proximity structures. In the final2 sections, we describe coarse proximities whose boundaries are homeomorphic tothe Higson corona and the Freudenthal boundary without using theorem 5.2.6.
The Gromov Boundary
In this section, we will briefly review the construction of the Gromov boundaryof hyperbolic metric spaces. The results and definitions outlining the construc-tion are as they appear in [3] and [11]. As the Gromov boundary of a hyperbolicmetric space X compactifies X, we may treat it as the boundary of a compactifi-cation. This allows us (by using known characterizations of the Gromov boundary OUNDARIES OF COARSE PROXIMITY SPACES 17 and Theorem 5.2) to describe the natural coarse proximity structure on a hyper-bolic metric space whose corresponding boundary is homeomorphic to the Gromovboundary of that space (see Theorem 6.6).
Definition 6.1.
Let (
X, d ) be a metric space and x, y, p ∈ X . The Gromovproduct of x and y with respect to p is( x, y ) p = 12 ( d ( x, p ) + d ( y, p ) − d ( x, y )) . Definition 6.2.
Let (
X, d ) be a metric space. Then X is said to be δ -hyperbolic for some real number δ < ∞ if for all x, y, z, p ∈ X, ( x, y ) p ≥ min { ( x, z ) p , ( y, z ) p } − δ. We note that this definition of hyperbolicity for a metric space is compatiblewith an alternative characterization of hyperbolicity within geodesic metric spacesdue to Rips (see [15] or [14]).For the remainder of this section, let (
X, d ) be an arbitrary δ -hyperbolic metricspace with a fixed based point p. Definition 6.3.
A sequence ( x n ) in X is said to converge at infinity iflim ( m,n ) →∞ ( x m , x n ) p = ∞ . Definition 6.4.
Two sequences ( x n ) and ( y n ) in X converging at infinity are saidto be equivalent , denoted ( x n ) ∼ ( y n ), iflim n →∞ ( x n , y n ) p = ∞ . The relation ∼ is an equivalence relation on sequences in X that converge atinfinity. We denote the equivalence class of a sequence ( x n ) in X converging atinfinity by [( x n )] and the set of all equivalence classes of such sequences in X by ∂X. We will proceed to define a topology (as in in [3] and [11]) on ∂X and X = X ∪ ∂X that makes both ∂X and X into compact Hausdorff spaces.To do that, identify the points of X with the set of sequences in X that convergeto x . Then, extend the Gromov product on X to X in the following way: for η = [( x n )] ∈ ∂X and ξ = [( y n )] ∈ ∂X, define:( η, ξ ) p = inf lim inf ( m,n ) →∞ ( x m , y n ) p , where the infimum is taken over all representative sequences. If y ∈ X, then define( η, y ) p = inf lim inf n →∞ ( x n , y ) p These products can be written most generally for x, y ∈ X by writing( x, y ) p = inf { lim inf i →∞ ( x i , y i ) p } , where the infimum is taken over all representative sequences for x and y . The topology on X , called the Gromov topology , is given by equipping X with its metric topology and defining a neighbourhood basis at each η ∈ ∂X bydefining the sets, U η,R = { x ∈ X ∪ ∂X | ( η, x ) p > R } . The topology on ∂X and X are such that X and ∂X are both compact Hausdorffspaces (a more detailed presentation of this construction can be found in [3] or[11]). Since we identified the points of X with the set of sequences in X thatconverge to x, we can think of X as a dense subset of X. Consequently, we willcall X the Gromov compactification of X and ∂X the Gromov boundary .By using the definition of the topology on X , one can show that a sequence ( x n )in X converges to some η ∈ ∂X if and only iflim n →∞ ( x n , η ) p = ∞ . In particular, we have the following:
Proposition 6.5.
Given a δ -hyperbolic metric space X with the Gromov boundary ∂X , a sequence ( x n ) in X converges to a point η ∈ ∂X if and only if ( x n ) convergesat infinity and [( x n )] = η. (cid:3) Proof.
See 11.101 in [11]. (cid:3)
Note that X equipped with the Gromov topology is first countable. Thus, if A ⊆ X , then we have that η ∈ tr ( A ) = tr X,∂X ( A ) if and only if there is a sequence( x n ) in A that converges at infinity and [( x n )] = η .The above characterization of the intersection of the closure of A in X with theGromov boundary suggests the definition of a coarse proximity structure. Theorem 6.6.
Let ( X, d ) be a proper δ -hyperbolic metric space with the Gromovboundary ∂X . For any two sets A, B ⊆ X, define A b G B if and only if there aresequences ( x n ) in A and ( y n ) in B that converge at infinity and are equivalentvia the relation ∼ . Then the triple ( X, B d , b G ) is a coarse proximity space whoseboundary U b G X is homeomorphic to the Gromov boundary ∂X .Proof. In this case, the bornology B d of metrically bounded subsets is identical tothe bornology B C of subsets of X whose closures in X are compact. Proposition6.5 tells us that if A ⊆ X is unbounded, then a point η ∈ ∂X is in tr ( A ) ifand only if there is a sequence ( x n ) in A that converges at infinity and whoseequivalence class is η . Said differently, η ∈ tr ( A ) if and only if there is a sequencein A that converges to η in the topology on X described above. Then for subsets A, C ⊆ X we have that A b G C if and only if tr ( A ) ∩ tr ( C ) = ∅ . That relation b G is then precisely the coarse proximity structure on ( X, B C ) induced by thecompactification X . Theorem 5.2 then gives us that U b G X is homeomorphic to ∂X , the Gromov boundary of X . (cid:3) OUNDARIES OF COARSE PROXIMITY SPACES 19
Definition 6.7.
For a proper δ -hyperbolic metric space X , the coarse proximitystructure ( X, B d , b G ) as described in Theorem 6.6 will be called the Gromovcoarse proximity structure on X and b G will be called the Gromov coarseproximity . 7.
The Visual Boundary
Our next example of a boundary of a coarse proximity space will be the visualboundary assigned to a proper, complete Cat(0) metric space. Our very briefintroduction to the visual boundary is as found in [1]. The main theorem ofthis section is Theorem 7.10, which describes the coarse proximity on a proper,complete Cat(0) space which induces a boundary homeomorphic to the visualboundary.
Definition 7.1. A geodesic ray in a metric space ( X, d ) is a map c : [0 , ∞ ) → X such that d ( c ( t ) , c ( t ′ )) = | t − t ′ | for all t ∈ [0 , ∞ ). The geodesic ray c is said to be based at x if c (0) = x . Definition 7.2.
Let (
X, d ) be a metric space and a, b, c ∈ X any three points. A comparison triangle for the triple ( a, b, c ) is another triple of points (¯ a, ¯ b, ¯ c ) suchthat ¯ a, ¯ b, ¯ c ∈ R (with the usual Euclidean metric) and d ( a, b ) = d (¯ a, ¯ b ) , d ( b, c ) = d (¯ b, ¯ c ) , and d ( a, c ) = d (¯ a, ¯ c ). Definition 7.3.
Given a geodesic metric space (
X, d ), a geodesic triangle in X consists of three vertices a, b, c ∈ X and three geodesic segments between the threepossible pairs of distinct vertices. Definition 7.4.
Let (
X, d ) be a geodesic metric space. We say that X is a Cat(0)space if given a geodesic triangle ∆ ⊆ X with comparison triangle ∆ ⊆ R , wehave that for all x, y in ∆ , d ( x, y ) ≤ d (¯ x, ¯ y ). Definition 7.5.
Two geodesic rays c and c ′ in a metric space ( X, d ) are said to be asymptotic if there is a constant K such that d ( c ( t ) , c ′ ( t )) ≤ K for all t ∈ [0 , ∞ ).In what follows, we will assume that ( X, d ) is a Cat(0) space with a completemetric. We will also denote the equivalence class of a geodesic ray γ in X by [ γ ],the set of all such equivalence classes ∂X , and denote X ∪ ∂X by X .Given a point x in X, there is a bijection from X to the inverse limit of closedballs around x , denoted lim ←− B ( x , r ) . The bijection is defined by mapping anequivalence class [ γ ] to the geodesic ray in [ γ ] that begins at x and maps a point x ∈ X to the map c x : [0 , ∞ ) → X whose restriction to [0 , d ( x , x )] is the geodesicsegment joining x to x and whose restriction to [ d ( x , x ) , ∞ ) is the constant mapat x . We will denote the topology for which this bijection is a homeomorphism by T ( x ) and refer to it as the cone topology on X . Remark . If X is a complete Cat(0) space, then the cone topology on X isindependent of the basepoint chosen. That is, if x , x ′ are distinct points in X ,then the cone topologies T ( x ) and T ( x ′ ) on X are homeomorphic. For full details,see [1]. Definition 7.7.
Let X be a complete Cat(0) space, x ∈ X , and let X be giventhe cone topology based at x . Then ∂X equipped with the subspace topologyinherited from X is called the visual boundary of X. A basis for the topology on X is the collection of all open metric balls aroundpoints in X , together with neighbourhoods centered on points in ∂X defined in thefollowing way: given x ∈ X , a geodesic ray η starting at x , and given positive realnumbers r, ǫ > , let ρ r : X = lim ←− B ( x , r ) → B ( x , r ) be the standard projection,and then define the neighbourhood as U ( η, r, ǫ ) = { x ∈ X | d ( x, η (0)) > r, d ( ρ r ( x ) , c ( r )) < ǫ } Remark . If X is a proper and complete Cat(0) space with visual boundary ∂X, then the space X equipped with the cone topology (defined using any basepoint)is a compact, first countable, Hausdorff space. Moreover, in such a case the visualboundary ∂X is also compact.In light of the previous remark, we may view the cone topology on X as acompactification of X with boundary. We can then equip X with the coarseproximity structure induced by the compactification. The next proposition willlet us recharacterize the coarse proximity induced by this compactification usingproperties internal to X . Proposition 7.9.
Let X be a proper complete Cat (0) space, x ∈ X , and let X begiven the cone topology based at x . A sequence ( x n ) in X (with X viewed viewedas a subspace of X ) converges to a point [ η ] ∈ ∂X if and only if the geodesicsjoining x to x n converge (uniformly on compact sets) to the geodesic ray thatbegins at x and belongs to the equivalence class of η. Proof.
See Chapter II.8 in [1]. (cid:3)
Theorem 7.10.
Let ( X, d ) be a proper and complete Cat (0) space. Define a rela-tion b on ( X, B ) by defining for all A, B ⊆ X : A b V B ⇐⇒ ∃ ( x n ) ⊆ A, ( y n ) ∈ B, [ η ] ∈ ∂X, lim x n = lim y n = [ η ] ⇐⇒ ∃ ( x n ) ⊆ A, ( y n ) ∈ B, [ η ] ∈ ∂X, [lim[ x , x n ]] = [lim[ x , y n ]] = [ η ] where [ a, b ] denotes a geodesic from a to b. The triple ( X, B C , b V ) is a coarseproximity space whose boundary U b V X is homeomorphic to the visual boundary ∂X . OUNDARIES OF COARSE PROXIMITY SPACES 21
Proof.
As the compactification X is first countable, we have that a point [ η ] ∈ ∂X is in cl X ( A ) for some A ⊆ X if and only if there is a sequence ( x n ) in A thatconverges to [ η ] in the sense of Proposition 7.9. Then, for subsets A, B ⊆ X wehave that A b V B if and only if tr X,∂X ( A ) ∩ tr X,∂X ( B ) = ∅ . Then, ( X, B C , b V )is precisely the coarse proximity structure induced by the compactification X .Consequently the boundary U b V X is homeomorphic to ∂X , the visual boundaryof X . (cid:3) Definition 7.11.
For a proper and complete Cat(0) space (
X, d ), the coarse prox-imity structure ( X, B C , b V ) will be the visual coarse proximity on the properand complete Cat(0) space X , and b G will be called the visual coarse proximity .8. The Higson Corona
In this section, we show how the Higson corona of a proper metric space is aboundary of a particular coarse proximity space (whose underlying base space isthat proper metric space).Recall that (
X, d ) is called k -discrete for some k >
0, if, for any distinct x, y ∈ X, we have d ( x, y ) ≥ k. Theorem 8.1.
Let ( X, d ) be a proper metric space, and let δ H be a binary relationon the power set of X defined by: Aδ H B ⇐⇒ d ( A, B ) = 0 or A b d B, where d ( A, B ) = inf { d ( x, y ) | x ∈ A, y ∈ B } . Then δ H is a separated proximitythat is compatible with the topology on X. We will call this proximity the
Higsonproximity associated to the metric space ( X, d ) . Proof.
The proof is straightforward. (cid:3)
In other words, two subsets A and B of a proper metric space X are close viathe Higson proximity if the distance between them is 0 or if they are metricallycoarsely close. If ( X, d ) is a k -discrete proper metric space for some k > , then thecondition d ( A, B ) = 0 can be replaced with A ∩ B = ∅ . Thus, when the k -discreteproper metric space is equipped with the metric coarse proximity b d , the Higsonproximity is the discrete extension of b d , i.e., δ H = δ dis, b d . Definition 8.2.
Let (
X, d ) be a proper metric space. The
Higson compactifica-tion of X , denoted hX , is the Smirnov compactification of ( X, δ H ). The Higsoncorona (i.e.,
Higson boundary ) is the corresponding Smirnov boundary hX \ X and is customarily denoted by νX .The equivalence of the above definition with the standard way of constructingthe Higson compactification (see [15] or [16] for the standard construction) hasbeen given in [2]. Proposition 8.3.
Let ( X, d ) be a proper k -discrete metric space for some k > .Then a cluster σ ∈ hX is an element of the Higson corona νX if and only if σ does not contain any bounded sets.Proof. Bounded sets in X are necessarily compact (actually finite) by the proper-ness and the k -discreteness of X . Consequently, the forward direction follows fromProposition 2.17. Conversely, if σ ∈ hX does not contain any bounded sets, thenclearly σ = σ x for any x ∈ X. Thus, σ ∈ hX \ X = νX. (cid:3) Theorem 8.4. If ( X, d ) is a proper metric space, then U b d X is homeomorphic tothe Higson corona νX. Proof.
If (
X, d ) is a proper metric space, then it is well-known that X is coarselyequivalent to a 1-discrete proper subset X ′ of itself. Because δ H = δ dis, b d , Proposition 8.3 tells us that U b d X ′ is the Higson corona νX ′ of X ′ . We also knowthat • coarsely equivalent spaces have homeomorphic Higson coronas (see Corol-lary 2.42 in [15]), • U b d X is homeomorphic to U b d X ′ (coarsely equivalent metric spaces arecoarse proximity isomorphic by Corollary 7 . ≈ denotes a homeomorphism, we have that U b d X ≈ U b d X ′ = νX ′ ≈ νX which shows that U b d X is homeomorphic to the Higson corona νX. (cid:3) The Freudenthal Boundary
In this section, we explicitly construct (without using Theorem 5.2) a coarseproximity structure on a locally compact Hausdorff space such that the boundaryof that coarse proximity space is the Freudenthal boundary.
Definition 9.1.
Let X be a topological space and A and B two subsets of X. Wesay that A and B are separated by a compact set if there exists a compact set K ⊆ X and open sets U , U ⊆ X such that(1) X \ K = U ∪ U ,(2) U ∩ U = ∅ , (3) A ⊆ U and B ⊆ U . The following proximity characterization of the Freudenthal compactification ofa locally compact Hausdorff space comes from [9].
OUNDARIES OF COARSE PROXIMITY SPACES 23
Definition 9.2. If X is a locally compact Hausdorff space, then the Freudenthalcompactification of X is the Smirnov compactification of the separated proximity δ F on X , called the Freudenthal proximity , defined by A ¯ δ F B ⇐⇒ A and B are separated by a compact set.We will denote the Freudenthal compactification of X by F X .Recall that Proposition 2.17 tells us the elements of
F X \ X do not contain anycompact sets. Thus, if a bornology on X consists of all precompact sets (denoted B C ) , then elements of F X \ X do not contain any bounded sets (since clustersare closed under taking supersets). Proposition 9.3.
Let X be a locally compact Hausdorff space. Define the relation b F on the power set of X by: A ¯ b F B if and only if there are compact sets D, K ⊆ X such that ( A \ D ) and ( B \ D ) are separated by K. Then ( X, B C , b F ) is a coarse proximity space.Proof. The only axiom that is not clear is the strong axiom. Let
A, B ⊆ X besuch that A ¯ b F B . Then by definition there are compact sets D and K such thatthere are disjoint open sets U, V ⊆ X that are disjoint from K , cover X \ K , andcontain A \ D and B \ D , respectively. Then V is a subset of X such that A ¯ b F V and ( X \ V )¯ b F B . Thus, the triple ( X, B C , b F ) is a coarse proximity space. (cid:3) Notice that with the notation from Definition 9.2 and Proposition 9.3, we havethat A b F B ⇐⇒ ( A \ D ) δ F ( B \ D ) for all bounded sets D. Definition 9.4.
For a locally compact Hausdorff space X, the coarse proximitystructure ( X, B C , b F ) will be called the Freudenthal coarse proximity struc-ture on X , and b F will be called the Freudenthal coarse proximity .Our goal is to show that
F X \ X is homeomorphic to U X. The two followinglemmas are used to prove Proposition 9.7, which says that clusters in
F X \ X areexactly those in U b F X. Lemma 9.5.
Let X be a locally compact Hausdorff space. If A, B ⊆ X are setssuch that A, B ∈ σ for some σ ∈ F X \ X, then A b F B. Proof.
Since
A, B ∈ σ, Proposition 2.17 implies that both A and B are unbounded.For contradiction, assume that A ¯ b F B. Then there is a compact set D ⊆ X suchthat ( A \ D )¯ δ F ( B \ D ) . However, by Proposition 2.18, ( A \ D ) and ( B \ D ) areelements of σ, which in particular means that ( A \ D ) δ F ( B \ D ) , a contradiction. (cid:3) Lemma 9.6.
Let X be a locally compact Hausdorff space. Let A and B be susbetsof X. Then ( Aδ F B and A ¯ b F B ) = ⇒ cl X ( A ) ∩ cl X ( B ) is compact. Proof.
Since A ¯ b F B, there exist compact D and compact K such that X \ K = U ∪ U for some open sets U and U that are disjoint and A \ D ⊆ U and B \ D ⊆ U . If we show that cl X ( A ) ∩ cl X ( B ) ⊆ K ∪ D, then cl X ( A ) ∩ cl X ( B )is a closed subset of a compact space, and consequently is compact. To see that cl X ( A ) ∩ cl X ( B ) ⊆ K ∪ D, let x be an element of cl X ( A ) ∩ cl X ( B ) . For contradiction,assume that x / ∈ K ∪ D. Since x / ∈ K, without loss of generality we can assume that x ∈ U. Since X is regular, let V and V be open sets such that x ∈ V , K ∪ D ⊆ V , and V ∩ V = ∅ . Then x belongs to an open set V ∩ U that is disjoint from B (since B \ D does not intersects U and D does not intersect V ). This contradictsthe fact that x is in the closure of B. (cid:3) Proposition 9.7.
Let X be a locally compact Hausdorff space. Let σ be a collectionof susbets of X . Then σ ∈ F X \ X ⇐⇒ σ ∈ U b F X. Proof. (= ⇒ ) Let σ ∈ F X \ X be given. By Proposition 2.17, σ does not containany bounded sets. Thus, it is enough to show that σ satisfies all the axioms of acluster under δ dis, b F . This will show that σ ∈ U b F X. To see the first axiom of acluster, notice that by Lemma 9.5, we have that if
A, B ∈ σ then A b F B, whichshows that Aδ dis, b F B . To prove the second axiom of a cluster, notice that Aδ dis, b F B ∀ B ∈ σ ⇐⇒ ∀ B ∈ σ, A ∩ B = ∅ or A b F B ⇐⇒ ∀ B ∈ σ, A ∩ B = ∅ or ( A \ D ) δ F ( B \ D ) ∀ D ∈ B = ⇒ ∀ B ∈ σ, A ∩ B = ∅ or Aδ F B = ⇒ ∀ B ∈ σ, Aδ F B = ⇒ A ∈ σ. The last axiom of a cluster is true, since σ is also a cluster in F X \ X by assumption.( ⇐ =) Let σ ∈ U b F X be given. By definition, σ does not contain any compactsets. Thus, if σ is a cluster in F X, then it clearly is a cluster in
F X \ X. To seethat σ is a cluster in F X, we need to show the three axioms of a cluster:(1) if
A, B ∈ σ then Aδ F B ,(2) if C ⊆ X is such that Cδ F A for all A ∈ σ , then C ∈ σ (3) if A ∪ B ∈ σ , then A ∈ σ or B ∈ σ. Axiom 3 is obvious, since σ is also a cluster in U b F X by assumption. To seeaxiom 1, assume A, B ∈ σ . Note that if A ¯ δ F B, then A and B can be separated by acompact set. In particular, they are disjoint and A ¯ b F B . But this is a contradictionto Aδ dis, b F B. Thus, Aδ F B .Finally, we need to prove the last axiom, which will finish the proof. Let C ⊆ X be such that Cδ F A for all A ∈ σ . For contradiction, assume that C / ∈ σ. Thenthere has to exist B ∈ σ such that C ¯ δ dis, b F B (otherwise, since σ is a cluster in U b F X it would imply that C ∈ σ by the second axiom of a cluster). In particular,this implies that C ¯ b F B. Since we have that Cδ F B and C ¯ b F B, Lemma 9.6 tells
OUNDARIES OF COARSE PROXIMITY SPACES 25 us that W = cl X ( C ) ∩ cl X ( B ) is compact. Since X is also locally compact, we canfind a finite open cover U , ..., U n of W such that each element in that cover hascompact closure. Define W ′ = n [ i =1 cl X ( U i ) . It is clear that W ⊆ W ′ and W ′ is compact. Since B ∈ σ and W ′ is compact, B \ W ′ ∈ σ. Consequently, by the assumption about C we have Cδ F ( B \ W ′ ) . Proposition 2.11 implies then that there exists a cluster σ in F X that containsboth C and B \ W ′ . To show that σ is not a point cluster, we prove a series ofclaims: Claim 1 : cl X ( C ) ∩ cl X ( B \ W ′ ) = ∅ . Notice that cl X ( C ) ∩ cl X ( B \ W ′ ) ⊆ cl X ( C ) ∩ cl X ( B ) . Thus, showing that cl X ( B \ W ′ )is disjoint from cl X ( C ) ∩ cl X ( B ) , proves Claim 1. Thus, to prove Claim 1 it isenough to prove: Claim 2 : cl X ( B \ W ′ ) is disjoint from cl X ( C ) ∩ cl X ( B ) . To see that claim 2 is true, notice that cl X ( C ) ∩ cl X ( B ) = W ⊆ int( W ′ ) ⊆ X \ cl X ( B \ W ′ ) , where int( W ′ ) denotes the interior of W ′ in X . This proves Claim 2 and conse-quently Claim 1. Claim 3 : σ is not a point cluster, i.e., σ ∈ F X \ X. To see that Claim 3 is true, for contradiction assume that there exists x ∈ X suchthat { x } ∈ σ . Under the assumptions of Claim 3, we will show that
Claim 4: neither cl X ( C ) nor cl X ( B \ W ′ ) contains x .If x ∈ cl X ( C ) , then by Claim 1 we have that cl X ( C ) ∩ cl X ( B \ W ′ ) = ∅ . Thus, byregularity we can find an open set V that contains x and whose closure is disjointfrom cl X ( B \ W ′ ) . Then the boundary of V is a separating compact set for x and cl X ( B \ W ′ ) . In other words, { x } ¯ δ F cl X ( B \ W ′ ) . But this is a contradiction, sinceboth x and cl X ( B \ W ′ ) are in σ and thus should be δ F -close. Consequently, x / ∈ cl X ( C ) . Similarly one can show that x / ∈ cl X ( B \ W ′ ) . Thus, x is in neither cl X ( C ) nor cl X ( B \ W ′ ) . This proves Claim 4.
Claim 3 proof cont.:
Notice that Claim 4 implies that x ∈ X \ ( cl X ( C ) ∪ ( cl X ( B \ W ′ ))) . But since ( cl X ( C ) ∪ ( cl X ( B \ W ′ ))) is closed, by the same argument thatuses regularity we can show that this implies that { x } ¯ δ F ( cl X ( C ) ∪ ( cl X ( B \ W ′ ))) , a contradiction (since cl X ( C ) is in σ , ( cl X ( C ) ∪ ( cl X ( B \ W ′ ))) is in σ , and con-sequently ( cl X ( C ) ∪ ( cl X ( B \ W ′ ))) should be δ F -close to x ). Thus, it has to bethat σ is not a point cluster, i.e., σ ∈ F X \ X. This shows Claim 3.Since both C and B \ W ′ belong to a cluster σ ∈ F X \ X, by Proposition 2.18we have that both C \ K and ( B \ W ′ ) \ K belong to σ for any compact set K. This implies that ( C \ K ) δ F (( B \ W ′ ) \ K ) for any compact K, which implies that( C \ K ) δ F ( B \ K ) for any compact set K. Consequently, C b F B, which contradicts the original assumption about C . Thus, C ∈ σ, finishing the proof that σ is anelement of F X \ X. (cid:3) Theorem 9.8.
Let X be a locally compact Hausdorff space. Then U b F X is home-omorphic to F X \ X .Proof. In light of Proposition 9.7 it will be enough to show that the identity map id : U X → F X \ X is continuous. As both of these spaces are compact andHausdorff, it is enough to show that the identity map is a proximity map. Let A , B ⊆ U b F X be such that A δ ∗ dis, b F B where δ ∗ dis, b F is the proximity relation of U X . We wish to show that A δ ∗ F B where δ ∗ F is the proximity on F X . Let
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