Metric compactifications and coarse structures

METRIC COMP A CTIFICTIONS AND CO ARSE STRU CTURES KOT ARO MINE AND A TSUSHI Y AMASHIT A Abstract. Let TB b e the cate gory of totally bounded, lo cally compact metric spaces with the C 0 coarse structures. W e show that if X and Y are i n TB then X and Y ar e coarsely equiv alent i f and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor giv es an equiv alence of categories TB → K , where K is the category of compact metrizable spaces. W e use this fact to sho w that the cont inuously con trolled coarse structure on a lo cally compact space X i nduced by some metrizable compactification ˜ X i s determined only by the top ology of the remainder ˜ X \ X . 1. Introduction When studying “lar ge-scale ” or “a symptotic” str uctures of metric spa ces, one is often led to co nsider a kind of “b ounda ry at infinity” o f them, for example the bo undary sphere ∂ ∞ H n = S n − 1 of the Poincar ´ e ba ll H n . This b oundary sphere reflects the geo metry of H n in the sense that the isometries of H n are in one- to-one corres p o ndenc e with the M¨ obius transformations of S n − 1 . In many situations w e can asso c ia te a b ounda ry at infinit y to a metr ic space, and in the optimal case, the large- s cale structure in question is re c ov e red fro m the b oundary . Results in this directio n a r e pursued by several author s, including Paulin [8], Bonk- Schramm [1], Buyalo-Schroe de r [3] and Jordi [7 ]. As an example, let X and Y be Gromov hyperb olic g eo desic spaces and ∂ ∞ X and ∂ ∞ Y their bo undaries at infinit y . W e can define a visua l metric on each of these b oundar ie s , which is an a nalogue of the angle metric o n ∂ ∞ H n = S n − 1 (see [2, Chapter I I I.H]). Then, under so me niceness condition (for example, it is s atisfied by Cayley gra phs of Gr omov hyperb olic gr oups and their b oundaries), the metric space s X and Y are q uasi-isometr ic if and only if ∂ ∞ X and ∂ ∞ Y are quasi-M¨ obius equiv alent [3, 7]. In the present pap er, w e prov e another suc h co rresp ondence in more topo lo gical settings. Let X b e a lo cally compact, totally b ounded metric space. Then, our main res ult states that a la rge-sca le structure called the C 0 c o arse stru ctur e on X int ro duced by W right [13] (see § 2) is completely recov ered from the top o lo gy o f the bo undary ˜ X \ X , where ˜ X stands for the completion of X (Theor em 4.5). Before int ro ducing o ur results in more details, we informally r eview the notion of coarse structure (see § 2 for formal definitions). “La rge-sca le” prop erties o f spaces, such as quasi-is ometry inv a riant pr op erties o f finitely gener a ted groups, can b e describ ed by coa rse structures. A coar se structure o n a set X is given by a collection of c ontr ol le d subsets of X × X satisfying several a xioms. When E ⊂ X × X is a fixed controlled subset, one think of x and y as “clos e unifor mly” for all ( x, y ) ∈ E . Thus 2010 Mathematics Subje ct Classific ation. 51F99, 53C23, 54C20, 18B30. Key wor ds and phr ases. coarse geomet ry , Hi gson corona, cont inuously con trolled coarse struc- ture, uniform con tinuit y , boundary at infinity . 1 2 K. MINE AND A. Y AMAS HIT A a typical coars e s tructure on a metric space X is the b ounde d c o arse stru ct ur e , where E ⊂ X × X is controlled if and only if there exists C > 0 such that d ( x, y ) ≤ C for all ( x, y ) ∈ E . In this str ucture, the phr ase “close uniformly” ab ov e has its usua l meaning. The C 0 coarse str ucture on a lo cally compact metric spac e, mentioned ab ov e, is another kind o f co arse structure. Roughly , the phrase “ close uniformly” in the C 0 structure a ctually means “b ecoming closer and clo ser as p oints appr oach to infinit y”. Given a suita ble c oarse s tr ucture on a lo cally compact Hausdorff space, we can define the Higson c omp actific ation hX of X (see § 2), a co mpactification of X defined in terms o f the r ing o f “s lowly os cillating” functions ca lled the Higson functions. The remainder ν X = hX \ X is called the Higson c or ona , and ν X can b e regar ded as a boundar y o f X . The corona ν X is a coar se inv ariant, in the sense that “coar sely equiv alent” coar s e spaces hav e homeo morphic Higson coronas [9, C o rollar y 2 .42]. Then, it is now natura l to as k whether the conv erse holds: if ν X a nd ν Y ar e homeomorphic, then a re X and Y coarsely equiv alent? As we mentioned earlier, an analog ous statement is true for Gromov h yp erb olic gr oups. The pap er of Cuchillo-Ib´ a˜ nez, Dydak, Koy ama and Mor´ on [4] giv es an a ffirmative answer to this question abo ut Higs on coronas in so me special case. They considered Z -sets (which a re “ thin” closed subsets in some sense) in the Hilb ert cub e and their complements, where each Z - set can b e reg arded a s the Higso n co rona o f the complement equipp ed with the C 0 structure. Their result then states that the category of Z -s e ts in the Hilber t cub e (and the co n tinuous ma ps b etw een them) is isomorphic to the ca tegory of the C 0 coarse space s formed by their c omplements. In the pres ent pap er , we extend the ar gument in [4] to gener al lo ca lly compact metric spaces equipp ed with the C 0 structure. F ormally stated, our ma in r e s ult claims a n equiv a lence of categor ies TB → K , wher e TB is the categor y of totally bo unded lo ca lly c o mpact metric spa ces and C 0 coarse maps modulo clos eness, and K is the catego ry of compact metrizable spa ces and contin uo us ma ps (Theorem 4.5). This equiv alence is realized b y the Higso n corona functor, whic h in this case reduce s to the op eratio n of taking the complement in the completion. As a c onsequence of the eq uiv alence TB ≃ K , it follows that the C 0 coarse structure o n M \ Z , wher e Z is a nowhere dense clo sed set in a compact metric s pa ce M , is determined (up to coarse equiv alence) o nly from the top olo gical type of Z , regar dless of the space M or how Z is embedded in M (Corollar y 4.6). A compa ctification ˜ X of a (lo ca lly compact Hausdorff ) space X in general in- duces a natura l coarse structure on X , ca lled the c ontinuously c ontro l le d c o arse structur e (se e § 2 ). Since this structur e ca n b e rega rded as a C 0 coarse str ucture with the Higso n compactificatio n ˜ X (see Cor ollary 4.3 and Remark 4.4), we hav e that the contin uous ly co nt rolle d structure on X is determined, up to co arse equiv- alence, by the top ologica l type of the r emainder ˜ X \ X (Cor ollary 4.7). 2. Preliminaries on coarse strcutures and Higson coronas W e refer the reader to Ro e’s mono graph [9] as a basic reference for this se ction. A c o arse st ructur e on a set X is defined as a collection E of subsets of X × X , called c ontr ol le d sets, sa tisfying the following five c o nditions: (i) the diagonal ∆ X = { ( x, x ) | x ∈ X } belo ngs to E , (ii) if E ∈ E and E ′ ⊂ E then E ′ ∈ E , (iii) if E ∈ E then its inv erse E − 1 = { ( x, y ) ∈ X × X | ( y , x ) ∈ E } b elongs to E , (iv) if E , F ∈ E then the comp osition E ◦ F = { ( x, z ) ∈ X × X | there exists y ∈ X such tha t ( x, y ) ∈ METRIC COMP ACT IFICA TIONS AND COARSE STRUC TURES 3 E and ( y , z ) ∈ F } b elongs to E , and (v) if E , F ∈ E then the union E ∪ F b elongs to E . The pair ( X , E ) (o r briefly X ) is then called a c o arse sp ac e . A subset B ⊂ X is called b ounde d in the co arse space X if B × B is controlled. Let X and Y b e coarse spaces. W e can define a class of ma ps fr om X to Y that resp ect coa rse s tructures, namely the coar se maps, as follows. A map f : X → Y is called pr op er if the inv e rse image f − 1 ( B ) is b ounded for every b ounded set B of Y . The map f is called b ornolo gous if ( f × f )( E ) ⊂ Y × Y is controlled for every controlled set E ⊂ X × X . Then, we say that f : X → Y is a c o arse map if it is b oth prop er and b o r nologo us . A coa rse map f : X → Y is calle d a c o arse e quivalenc e if there exists a coarse map g : Y → X such that bo th g ◦ f a nd f ◦ g are c lo se to their resp ective identities. Her e maps h, k : S → Z fro m a s e t S to a coarse space Z ar e ca lled close if the set { ( h ( s ) , k ( s )) | s ∈ S } is controlled. Coa rse spaces X and Y are then called c o arsely e quivalent . A coar se str uctur e on a parac o mpact Hausdorff space X is called pr op er (in w hich case w e say that X is a pr op er c o arse sp ac e ) if (1) there is a controlled neighbor ho o d of the diagonal ∆ X and (2) every bo unded s ubset has compact clo sure. F o r a proper coarse space X , the co n verse statemen t of (2) is also true if X is c o arsely c onne cte d , that is, each singleton { ( x, y ) } is controlled (see [9, Prop ositio n 2.2 3]). Notice a lso that a pro p e r co arse space is nece s sarily lo cally co mpact. As mentioned in the introduction, a standar d ex ample of a coar se structure is the b oun de d c o arse struct ur e o n a metric spac e ( X , d ), where E ⊂ X × X is defined to b e controlled if there exists C > 0 such that d ( x, y ) ≤ C for every ( x, y ) ∈ E . In this structure, the b o unded sets are exactly the b ounded sets in the metric s ense. The b ounded coa rse structure o n X is pr op er if and o nly if X is pr op er as a metric space, that is, every closed b ounded subset of X is compac t. It is not difficult to show that tw o g eo desic metric spaces with the b ounded co arse structur e s a re coarsely equiv alent if and o nly if they a re quasi- isometric. F or a lo cally compact metric space ( X, d ), we can define a coar se structure other than the b ounded s tructure, called the C 0 c o arse structu r e which is introduce d by W right [13]. In the C 0 coarse structur e , a subset E of X × X is defined to be controlled if for every ε > 0 we can find a compact set K ⊂ X such that d ( x, y ) < ε for every ( x, y ) ∈ E \ K × K . The following is prov ed for completeness. Prop ositio n 2. 1. L et ( X, d ) b e a lo c al ly c omp act metric sp ac e. Then, the ab ove definition of the C 0 c o arse structu r e inde e d gives a c o arse stru ct ur e on X , wher e a subset is b ounde d if and only if it has c omp act closur e. In c ase X is sep ar able, t his structur e is pr op er. Pr o of. Let ( X , d ) b e a lo cally compact metric s pace. It is easy to verify the con- ditions (i), (ii), (iii) and (v). T o see (iv), take any controlled sets E , F and ε > 0. W e prov e that E ◦ F is a lso controlled. Since E ∪ F is controlled, we can cho o se a compact set K 0 of X such that d ( x, y ) < ε/ 2 whenever ( x, y ) ∈ ( E ∪ F ) \ K 0 × K 0 . Since X is lo c a lly compa c t, there is an ε ′ > 0 with ε ′ ≤ ε/ 2 such that the closed ε ′ -neighborho o d N ( K 0 , ε ′ ) o f K 0 is compact. Then, we can choose a compact set K of X containing N ( K 0 , ε ′ ) such that d ( x, y ) < ε ′ whenever ( x, y ) ∈ ( E ∪ F ) \ K × K . W e claim that d ( x, y ) < ε holds for every ( x, y ) ∈ ( E ◦ F ) \ K × K . Given ( x, y ) ∈ ( E ◦ F ) \ K × K , we can find a z ∈ X such that ( x, z ) ∈ E and ( z , y ) ∈ F . Since ( x, y ) / ∈ K × K , either x / ∈ K or y / ∈ K ho lds. W e first consider the case when x / ∈ K . Then, we see from ( x, z ) ∈ E \ K × K that d ( x, z ) < ε ′ . Since N ( K 0 , ε ′ ) ⊂ K , we hav e z / ∈ K 0 , a nd in par ticular, ( z , y ) ∈ F \ K 0 × K 0 . This in 4 K. MINE AND A. Y AMAS HIT A turn implies that d ( z , y ) < ε / 2, a nd hence d ( x, y ) ≤ d ( x, z ) + d ( z , y ) < ε ′ + ε/ 2 ≤ ε . Since the case when y / ∈ K can b e treated in a s imila r wa y , the co ndition (iv) is verified. It is c le a r fro m the definition o f the C 0 coarse structure tha t every s ubset of X with co mpact clos ure is b ounded. T o show the conv erse, let B ⊂ X be a b ounded set w ith res pec t to the C 0 structure, a nd supp ose that B do es not hav e compact closure. Then, in par ticular, there are tw o distinct po int s p, q ∈ B , and we set the distance ε = d ( p, q ) > 0. Since B is b ounded, the s q uare B × B is controlled, and hence there exists a co mpact se t K ⊂ X such that d ( x, y ) < ε/ 2 whenever ( x, y ) ∈ B × B \ K × K . Since the clo sure of B is no t compact, B is not contained in K . Fix a p oint r ∈ B \ K a nd obser ve that ( p, r ) , ( q , r ) ∈ B × B \ K × K . This implies that ε = d ( p, q ) ≤ d ( p, r ) + d ( q , r ) < ε/ 2 + ε/ 2 = ε , which is a contradiction. W e further as sume that X is separa ble . T o prov e that the C 0 structure is prop er, it remains only to show that there is a co nt rolled neighbor ho o d of the diago nal ∆ X . Since X is lo cally compact and separable metrizable, we can take a co un table lo ca lly finite op en cov er { U n | n ∈ N } such tha t each U n has compact closur e. Then, we can define a contin uo us function f : X → (0 , ∞ ) by f ( x ) = X i ∈ N min { 2 − i , d ( x, X \ U i ) } . Then, it is easy to see that the function f v anishes at infinity , that is, for a ll ε > 0 there is a c o mpact set K ⊂ X such that 0 < f ( x ) < ε for every x / ∈ K . This implies that the set E = { ( x, y ) ∈ X × X | d ( x, y ) < min { f ( x ) , f ( y ) }} is a controlled neighborho o d of ∆ X .  Let X = ( X , E ) b e a coa r se spa ce. A b ounded (not necess a rily contin uous) function f : X → R is a Higson funct ion on X if for every c ontrolled set E ∈ E and ε > 0 there is a bo unded set B ⊂ X such that | f ( x ) − f ( y ) | < ε whenever ( x, y ) ∈ E \ B × B . The Higso n functions on X fo r m a unital Banach alge br a which is denoted by B h ( X ). A coarse spa ce is usually eq uipp ed with a top o logy , and it makes sense to s pea k of contin uous functions on the co arse space . Let X b e a lo ca lly compact Hausdorff coarse space, and let C h ( X ) b e the Banach a lgebra of c ontinuous Higson functions on X . Let e : X → R C h ( X ) be an embedding into a pro duct of lines defined by e ( x ) = ( f ( x )) f ∈ C h ( X ) . Then, the compactifica tion hX = c l R C h ( X ) e ( X ) o f X is homeomor phic to the max ima l ideal space o f C h ( X ). W e call hX the Higson c omp actific ation of X , a nd its b oundary ν X = hX \ X is then calle d the Higson c or ona of X . The next le mma co nnects Higs o n functions and coar se maps. The pr o of is straightforward and left to the r eader. Lemma 2. 2. L et X and Y b e lo c al ly c omp act Hausdorff c o arse sp ac es satisfying the c ondition ( ⋆ ) and f : X → Y a c o arse map. Then for every Higson function ϕ on Y , the c omp osition ϕ ◦ f is a H igson funct ion on X . Conse quently, f induc es a ring homomorphism f ∗ : B h ( Y ) → B h ( X ) . If mor e over f is c ontinu ous, f induc es f ∗ : C h ( Y ) → C h ( X ) .  Remark 2.3. In the definition of Higso n functions, we used the notion o f b ounded sets which is purely coa rse one. In many cases a coar se space has a top olog y , and METRIC COMP ACT IFICA TIONS AND COARSE STRUC TURES 5 it is natur al to assume tha t the b ounded sets hav e so me r elation with the top ology . F or a lo cally co mpact Hausdorff coar s e s pace X , w e consider the followin g condition: ( ⋆ ) A subset of X is b ounded if and only if it has compact clo s ure. Hereafter we will cons ider the Higs on cor ona of X only when this condition is s atisfied. The co ndition ( ⋆ ) is satisfied by the following co arse structure s : the bo unded s tr uctures on prop er metric spa ces, the contin uously co ntrolled structures (defined b elow), the C 0 structures o n lo cally co mpa ct metric spa ces (Pr o p o sition 2.1), and all co arsely connected prop er coar s e s pa ces. F or a set X and subsets E ⊂ X × X a nd K ⊂ X , we define E [ K ] to b e the se t of x ∈ X s uch that ( x, y ) ∈ E for some y ∈ K . This set is the “image” of K under E , wher e E is considered to b e a multiv a lued function from the se c ond co or dinate to the fi rs t co o rdinate. Now assume that X has a top ology . Then E ⊂ X × X is called pr op er if each of E [ K ] and E − 1 [ K ] ha s compact clo s ure for every compact subset K of X . Let X b e a lo ca lly compa c t Hausdorff space w ith a (Hausdorff ) compa ctification ˜ X . Denote the b oundar y ˜ X \ X by ∂ X . Then, s ince X is lo ca lly compact, X is op en in ˜ X and hence ∂ X is compa ct. A subset E ⊂ X × X is then defined to b e c ontinuou s ly c ontr ol le d by ˜ X if one of (hence all of ) the following three equiv alent conditions is sa tis fie d: (a) the closure of E in ˜ X × ˜ X intersects the co mplement of X × X only in the diag o nal ∆ ∂ X = { ( ω , ω ) | ω ∈ ∂ X } , (b) E is prop er (in the sense defined in the prev io us parag r aph), and for every net  ( x λ , y λ )  in E , if ( x λ ) conv erges to ω ∈ ∂ X , then ( y λ ) also conv erges to ω , (c) E is pr op er, and for every po int ω ∈ ∂ X a nd every neighborho o d V of ω in ˜ X , ther e is a neighborho o d U ⊂ V of ω in ˜ X such that E ∩ ( U × ( X \ V )) = ∅ . Then, the c ollection of all contin uously controlled subsets is shown to b e a c o arse structur e called the c ontinuou s ly c ontr ol le d c o arse st r u ctur e induced by ˜ X (see [9, Section 2.2]). Remark 2.4. F or a contin uo us ly co nt rolle d structure, it is eas y to see that the condition ( ⋆ ) is alwa ys satisfied, while it may happ en that there is no controlled neighborho o d of the dia gonal, even if the spa c e is pa racompac t. This means that such a structure need not b e prop er. (In [9, Theorem 2.27 ], it is asser ted that every contin uo us ly controlled structure o n a paraco mpact s pa ce is prop er, but the pro of given there is actually incor rect, as p ointed o ut by Be rndt Grave: s ee [10].) As an example, let X = [0 , ∞ ) and consider the Stone- ˇ Cech compactifica tion β X of X . Let U b e any neig h b orho o d of ∆ X in X × X . F or ea ch n ∈ N , let a n = n and take b n so that 0 < b n − a n < 2 − 1 and ( a n , b n ) ∈ U are satisfied. Then A = { a n | n ∈ N } a nd B = { b n | n ∈ N } a re disjoint clo sed s ubsets in X , and henc e ther e exis ts a co nt inuous map f : X → [0 , 1 ] with f ( A ) = { 0 } and f ( B ) = { 1 } . This f a dmits a contin uous extension ˜ f : β X → [0 , 1] a nd we hav e cl β X A ⊂ ˜ f − 1 (0) a nd cl β X B ⊂ ˜ f − 1 (1). In particular , cl β X A and cl β X B ar e disjoint. Since A is noncompact, there exists a p oint ω ∈ (cl β X A ) \ X and a net ( a n λ ) in A co nv er gent to ω . Then the ne t ( b n λ ) has a subnet ( b n ′ µ ) conv ergent to some p oint ω ′ ∈ cl β X B . The cor resp onding subnet ( a n ′ µ ) converges to ω . Then ( a n ′ µ , b n ′ µ ) ∈ U and ( a n ′ µ , b n ′ µ ) → ( ω , ω ′ ) / ∈ ∆ β X \ X , showing that U is not cont rolle d. In the rest of this sectio n, we discuss how a noncontin uous co a rse map b etw een prop er co arse spac es induces a contin uous map betw een their Hig s on cor onas. The results will b e a pplied to prov e our main theor em (Theorem 4.5). 6 K. MINE AND A. Y AMAS HIT A F or a prop er coar se space X satis fying ( ⋆ ), let B 0 ( X ) denote the s e t of b ounded, real-v alued functions that v anish at infinity , in the sense that for all ε > 0 ther e exists a compact set K such that we hav e | f ( x ) | < ε for a ll x ∈ X \ K . L e t C 0 ( X ) denote the subalgebr a of all contin uo us functions in B 0 ( X ). The Ba nach alg ebra C ( ν X ) o f rea l-v alued contin uous functions o f the Hig son corona is then is o morphic to C h ( X ) /C 0 ( X ). There is a na tural isomor phism C h ( X ) /C 0 ( X ) ∼ = B h ( X ) /B 0 ( X ) by [9, Lemma 2.40 ], and hence C ( ν X ) ∼ = B h ( X ) /B 0 ( X ). Now let X and Y be tw o pr o p e r coarse s paces satisfying ( ⋆ ) and f : X → Y a (not nece s sarily contin uous) c o arse map. By Lemma 2.2 there is a n induced map f ∗ : B h ( Y ) → B h ( X ), a nd by the pr op erness of f , we have f ∗ ( B 0 ( Y )) ⊂ B 0 ( X ). Therefore, we hav e a map f ∗ : C ( ν Y ) ∼ = B h ( Y ) /B 0 ( Y ) → B h ( X ) /B 0 ( X ) ∼ = C ( ν X ). Then, ν f : ν X → ν Y is defined a s the contin uous ma p co rresp onding to the last f ∗ by Gel’fand-Naimark dua lit y . This makes the op era tion ν a functor, ca lle d the Higson c or ona functor , fro m the c ategory o f pro pe r coa rse spaces to the categor y of compact Hausdor ff spaces. Of co urse, we can exp ect the ma p ν f to be a “contin uous extensio n” of f in some sense. In fact, we hav e the fo llowing: Prop ositio n 2.5 . L et f : X → Y b e a c o arse map b etwe en pr op er c o arse s p ac es satisfying the c ondition ( ⋆ ) . Then the map ν f : ν X → ν Y is char acterize d by the pr op erty that f ∪ ν f : hX → hY is c ontinuous at e ach p oint of ν X . Pr o of. W e first show that ν f : ν X → ν Y satisfies this prop erty . Since ν f is contin- uous, we need only to show that for ea ch net ( x λ ) conv erging to a p o int ω ∈ ν X , the net ( f ( x λ )) converges to ν f ( ω ). If this is no t the case, there exists a s ubnet ( x λ µ ) of ( x λ ) such that ( f ( x λ µ )) is conv ergent to ω ′ ∈ ν Y \ { ν f ( ω ) } . Then, there exists a contin uous function ˜ ϕ : hY → R with ˜ ϕ ( ν f ( ω )) = 0 and ˜ ϕ ( ω ′ ) = 1, which restricts to a Higson function ϕ = ˜ ϕ | Y ∈ C h ( Y ) ⊂ B h ( Y ) . Then, since f is coa rse, we have ϕ ◦ f ∈ B h ( X ) by Lemma 2 .2. Using Tietze’s theorem, we can ta ke a contin uo us ex tension ψ : hX → R o f ˜ ϕ ◦ ( ν f ) : ν X → R . The definition of ν f yields that ϕ ◦ f − ( ψ | X ) ∈ B 0 ( X ). This implies, by the contin uit y of ψ , lim ϕ ◦ f ( x λ µ ) = lim ψ ( x λ µ ) = ψ ( ω ) = ˜ ϕ ◦ ( ν f )( ω ) = 0 . On the other hand, by the contin uit y of ˜ ϕ , lim ϕ ◦ f ( xλ µ ) = ˜ ϕ ( ω ′ ) = 1 , which is a contradiction. The ma p ν f is uniquely determined by the prop erty we ha ve now demonstrated, since every p oint of ν X is a limit of some net in X . This co mpletes the pro of.  Remark 2.6 . The a bove prop ositio n mea ns that ν f is characterized by the fact that f ∪ ν f : ( hX , ν X ) → ( hY , ν Y ) is eventu al ly c ontinu ous in the sense of [5, Definition 1.14 ] and [11, Definition 2.4], or is ultimately c ont inuous in the se ns e of [6, Section 2]. This obser v ation is a lready ma de in the sp ecial cas e that b oth X and Y a re contin uously controlled by some metrizable compactifications ˜ X and ˜ Y , resp ectively [6]. In fact, the Higs on compactifications hX a nd hY a re equiv ale n t to ˜ X and ˜ Y in this sp ecial cas e [9, Pro p o s ition 2 .48]. In so me s itua tion, it is also tr ue that f must b e coa rse whenever f admits an extension as in the la s t prop osition. F or a precise sta tement w e need the following notion: a map f : X → Y b etw een c oarse spaces is called pr e-b ornolo gous if f ( B ) ⊂ METRIC COMP ACT IFICA TIONS AND COARSE STRUC TURES 7 Y is b ounded for every b ounded set B ⊂ X . Notice that every b orno logous map betw een coa rse spaces is pr e-b ornolog ous. Prop ositio n 2. 7. Le t X and Y b e pr op er c o arse sp ac es satisfying ( ⋆ ) and f : X → Y a (not ne c essarily c ontinuous) pr e-b ornolo gous map. Supp ose that Y has t he c ontinuou s ly c ontr ol le d c o arse stu ctur e induc e d by some c omp actific ation ˜ Y of Y . Then, f is c o arse if and only if t her e exists ˜ f : ν X → ν Y (which is ne c essarily e qual to ν f ) such t hat f ∪ ˜ f : hX → hY is c ontinu ou s at e ach p oint of ν X . Pr o of. The “ only if ” pa rt is Pro p o sition 2 .5. W e prove the “if ” part. Supp ose that there is a map ˜ f : ν X → ν Y a s ab ov e. T o see that f is pro per , it is enoug h to s how that f − 1 ( K ) has c o mpact closure in X whenever K ⊂ Y is co mpact, s inc e b oth X and Y satisfy the conditio n ( ⋆ ). Let K be a compact subs et o f Y . If f − 1 ( K ) do es not hav e co mpact closur e in X , then there exists a p oint ω ∈ ν X ∩ cl hX f − 1 ( K ). Then we hav e ˜ f ( ω ) ∈ ν Y , but the contin uity of f ∪ ˜ f at ω implies ˜ f ( ω ) ∈ cl hY K = K ⊂ Y . This is a contradiciton, which means that f − 1 ( K ) has compact closur e in X . T o prove that f is b ornolo gous, let E b e a co ntrolled subset of X × X a nd consider the imag e F = ( f × f )( E ) ⊂ Y × Y . It is straightforward to show that F is pr op er as a subset of Y × Y , using the fact that E is prop er (see [9, Pro po sition 2.23]) and that f is a prop er, pre-b orno lo gous map. Let  ( f ( x λ ) , f ( x ′ λ ))  be a net in F with ( x λ , x ′ λ ) ∈ E and f ( x λ ) → ω ∈ ˜ Y \ Y . It remains to sho w that f ( x ′ λ ) → ω . Suppo se that this is not the case. Then, there exis t subnets ( x λ µ ) and ( x ′ λ µ ) (with the same index set) s uch that f ( x ′ λ µ ) → ω ′ for some ω ′ 6 = ω . W e w r ite x λ µ = x µ , x ′ λ µ = x ′ µ to simplify notation. Cho ose a contin uo us function ˜ ϕ : ˜ Y → [0 , 1] such that ˜ ϕ ( ω ) = 0 and ˜ ϕ ( ω ′ ) = 1 , and let ϕ denote the r estriction ˜ ϕ | Y : Y → [0 , 1 ] ⊂ R . By [9, Prop ositio n 2.45 (b)], there exists a contin uous map π : hY → ˜ Y tha t r estricts to the identit y on Y . Then, the comp os ition F = ˜ ϕ ◦ π ◦ ( f ∪ ˜ f ) : hX → R gives an extension o f ϕ ◦ f ov er hX which is c o ntin uous at each po int in ν X . By Tietze’s theorem, there exists a contin uous extension G : hX → R of ˜ ϕ ◦ π ◦ ˜ f = F | ν X . Then, we hav e G | X ∈ C h ( X ) and ( G − F ) | X ∈ B 0 ( X ), which in turn implies ϕ ◦ f = F | X = G | X − ( G − F ) | X ∈ C h ( X ) + B 0 ( X ) = B h ( X ) . This ca uses a contradiction, since it ca n als o b e shown that ϕ ◦ f / ∈ B h ( X ), as follows. Given a compact set K ⊂ X , we can take µ s o lar ge that | ϕ ◦ f ( x µ ) | < 1 / 3, | ϕ ◦ f ( x ′ µ ) − 1 | < 1 / 3 , and x µ / ∈ E [ K ]. Then x ′ µ / ∈ K and it follows tha t ( x µ , x ′ µ ) ∈ E \ K × K and | ϕ ◦ f ( x µ ) − ϕ ◦ f ( x ′ µ ) | ≥ 1 / 3. This shows that ϕ ◦ f / ∈ B h ( X ).  3. C 0 and continuousl y controlled coarse structures In this se c tion, al l lo c al ly c omp act metric sp ac es ar e assume d to have the C 0 c o arse structure s. Contr ol le d sets, c o arse maps and Higson functions wil l b e with r esp e ct to the C 0 structur e. F or such structures, we first make c le ar how the notions of Hig son functions and coa rse maps ar e rela ted to uniform contin uit y (Pro po sition 3.1, Cor ollary 3.4). Then, w e prov e that the contin uously controlled coar se structur e induced by the Higson compa c tifica tion is the original C 0 structure (Theorem 3.5). Prop ositio n 3.1. L et ( X , d ) b e a lo c al ly c omp act metric sp ac e. Then t he c ontinu - ous Higson functions on X ar e exactly the b ounde d uniformly c ontinu ous fun ctions on X . 8 K. MINE AND A. Y AMAS HIT A Pr o of. First assume tha t f : X → R is b ounded and uniformly contin uous. T ake any controlled set E in the C 0 structure a nd ε > 0. Then, we can choose a δ > 0 such that d ( x, y ) < δ implies | f ( x ) − f ( y ) | < ε , and then we c a n choose a compac t set K such that ( x, y ) ∈ E \ K × K implies d ( x, y ) < δ . Then, | f ( x ) − f ( y ) | < ε holds for every po in t ( x, y ) ∈ E \ K × K . This pr ov e s that f is a Higson function. T o show the con verse, supp ose that f is co nt inuous but not unifor mly contin uo us. The latter condition means that there ar e ε > 0 and seq uences ( x n ) n ∈ N , ( x ′ n ) n ∈ N in X s uch tha t d ( x n , x ′ n ) < 1 /n and | f ( x n ) − f ( x ′ n ) | ≥ ε . Then, the set { x n | n ∈ N } is no t contained in any co mpact set. Indeed, if it were contained in a co mpa ct set, then the closure of { x n , x ′ n | n ∈ N } would be compact, where f must b e uniformly contin uo us , contrary to the choice of ( x n ) and ( x ′ n ). T o s how that f is not a Higson function, we first notice that the set E = { ( x n , x ′ n ) | n ∈ N } is controlled, and take any co mpa ct s ubset K of X . As s e en a b ove, the set { x n | n ∈ N } is not contained in K . Thus, we can find an N such that x N / ∈ K . This means ( x N , x ′ N ) ∈ E \ K × K , but we hav e a lso that | f ( x N ) − f ( x ′ N ) | ≥ ε . The r efore, f is not a Higson function.  In what follows, we g ive a characterization of co a rse ma ps be t ween lo c a lly com- pact metric spaces without ass uming contin uit y . W e reca ll from the last section that f : X → Y b etw een coa rse spaces is pr e-b ornolo gous if for every b ounded B ⊂ X the image f ( B ) is b ounded. Since lo ca lly compa ct metric spa ces sa tisfy the condition ( ⋆ ) in Remark 2.3 by Prop ositio n 2.1, we obtain the following: Lemma 3.2. L et X and Y b e lo c al ly c omp act metric sp ac es and f : X → Y a (not ne c essarily c ontinuous) map. Then, f is pr op er if and only if f − 1 ( K ) has c omp act closur e for every c omp act set K of Y . S imilarly, f is pr e- b ornolo gous if and only if f ( K ) has c omp act closur e for every c omp act set K of X .  Prop ositio n 3.3. L et X and Y b e lo c al ly c omp act metric sp ac es and f : X → Y a (not ne c essarily c ontinuous) pr op er, pr e-b ornolo gous map. The fol lowing ar e e quivalent: (a) f is a c o arse map. (b) F or every ε > 0 , ther e exist a c omp act set K ⊂ X and a δ > 0 such that d ( f ( x ) , f ( x ′ )) < ε whenever ( x, x ′ ) / ∈ K × K and d ( x, x ′ ) < δ . Pr o of. (b) ⇒ (a): Assume (b) and let f : X → Y b e a pr op er, pre-b ornolo gous map. It is eno ugh to show that f is b o r nologo us . T ake an y co n trolled set E ⊂ X × X and put F = ( f × f )( E ). T o show tha t F is controlled, take a n y ε > 0 . By (b), we can take a compact set K ⊂ X and a δ > 0 such that d ( x, x ′ ) < δ and ( x, x ′ ) / ∈ K × K imply d ( f ( x ) , f ( x ′ )) < ε . Since E is controlled, there is a compact set K ′ ⊃ K such that d ( x, x ′ ) < δ whenever ( x, x ′ ) ∈ E \ K ′ × K ′ . Then, by Lemma 3.2, L = cl Y f ( K ′ ) is compact, since f is pre- b o r nologo us . Let ( y , y ′ ) ∈ F \ L × L . Then, ( y , y ′ ) = ( f ( x ) , f ( x ′ )) for s ome ( x, x ′ ) ∈ E \ K ′ × K ′ . It fo llows that d ( x, x ′ ) < δ , and hence d ( y , y ′ ) = d ( f ( x ) , f ( x ′ )) < ε , since ( x, x ′ ) / ∈ K × K . (a) ⇒ (b): Assume that f : X → Y is prop er and pre-b ornolo gous, and tha t (b) is not the case. W e then prov e that f is not b o r nologo us to obtain a contradiction. There exists r > 0 such tha t for each n ∈ N a nd each compact set K ⊂ X , we ca n take x K,n and x ′ K,n , not b oth of which are in K , with d ( x K,n , x ′ K,n ) < 1 /n a nd d ( f ( x K,n ) , f ( x ′ K,n )) ≥ r . W e may exchange x K,n and x K ′ ,n if necessa r y to assume that x K,n / ∈ K . Fix a lo ca lly finite c ov e r ( U λ ) of X by ope n sets U λ with compact METRIC COMP ACT IFICA TIONS AND COARSE STRUC TURES 9 closure D λ = c l X U λ . Let K 1 = ∅ and inductively , define K n +1 as the union o f all D λ that intersects K n ∪ { x K n ,n , x ′ K n ,n } . Since ( D λ ) is lo ca lly finite, we s ee by induction that K n is compact for ea ch n . L e t us define x n = x K n ,n and x ′ n = x ′ K n ,n . Notice that K n ⊂ K n +1 , x n / ∈ K n and x n , x ′ n ∈ K n +1 . W e show that the set E = { ( x n , x ′ n ) | n ∈ N } ⊂ X × X is controlled. T o see this, let ε > 0 . T ake N ∈ N so large that 1 / N < ε holds, a nd let K = K N . If ( x n , x ′ n ) / ∈ K × K , then it follows that n ≥ N , and hence d ( x n , x ′ n ) < 1 / n ≤ 1 / N < ε . This shows that E is controlled. Next, we cla im that, the set { x n | n ∈ N } is no t co nt ained in any compac t set. Indeed, if this set is contained in a co mpa ct set, then so me subsequence ( x n k ) conv erges to a p o int x ∞ ∈ X , and D λ is a neighborho od of x ∞ for some λ . Then, for a la rge k , b oth x n k and x n k +1 are in D λ . Since x n k ∈ D λ , we have D λ ⊂ K n k +1 . Then, x n k +1 ∈ D λ ⊂ K n k +1 ⊂ K n k +1 (using n k + 1 ≤ n k +1 ), which is contrary to x n k +1 / ∈ K n k +1 . Thus, { x n | n ∈ N } is not contained in an y compa ct set. Finally , we show that ( f × f )( E ) = { ( f ( x n ) , f ( x ′ n )) | n ∈ N } is not controlled to prove that f is not bo rnologo us (and hence not coarse ). T o this end, take any compact s et K ⊂ Y . Then, by Lemma 3 .2, f − 1 ( K ) has compact closur e, a nd hence ther e is some n s uch that x n / ∈ f − 1 ( K ) b y the las t par agra ph, which implies ( f ( x n ) , f ( x ′ n )) / ∈ K × K . How ever, we hav e d ( f ( x n ) , f ( x ′ n )) = d ( f ( x K n ,n ) , f ( x ′ K n ,n )) ≥ r. Notice that r > 0 is irr elev ant to o ur choice of K . This mea ns ( f × f )( E ) is no t controlled.  Since contin uous maps b etw een coar se spaces satisfying ( ⋆ ) are pr e - bo rnolog o us, and are uniformly contin uo us on every compact set, we obtain the following cor ol- lary: Corollary 3.4. A c ontinuous map b etwe en lo c al ly c omp act m etric sp ac es is c o arse with re sp e ct to the C 0 c o arse structu r es if and only if it is pr op er and uniformly c ontinuou s .  Let us consider the Higson compactification h 0 X with resp ect to the C 0 struc- ture. Then, in turn, h 0 X induces a contin uously controlled structure o n X . As a generaliza tion of [4, Pr op osition 6], we asser t that this is the same as the or iginal C 0 structure: Theorem 3.5. The C 0 c o arse structu r e on a lo c al ly c omp act metric sp ac e X is e qual t o the c ontinu ously c ont ro l le d structu r e induc e d by the Higson c omp actific ation h 0 X . T o s how this theor em, the next lemma will b e useful: Lemma 3.6. L et X b e a lo c al ly c omp act metric sp ac e and E a su bset of X × X with E = E − 1 . Then, E is c ont r ol le d if and only if d ( x n , x ′ n ) → 0 holds for every se quenc e  ( x n , x ′ n )  n ∈ N in E s u ch that ( x n ) has no c onver gent su bse quenc e. Pr o of. The “o nly if ” par t is clear . T o show the “if ” part, we use the c o nstruction in the pro of o f Pr op osition 3 .3 (a) ⇒ (b), as follows. First choo se a lo cally finite cov ering ( U λ ) λ ∈ Λ of X by o pen se ts U λ with compact closure D λ = cl X U λ . Assume that E = E − 1 ⊂ X × X is not co ntrolled. Then, there ex ists ε > 0 s uch that for each compact set K ⊂ X , w e hav e d ( x K , x ′ K ) ≥ ε for some ( x K , x ′ K ) ∈ E \ K × K . 10 K. MINE AND A. Y AMAS HIT A Here we can cho o se ( x K , x ′ K ) so tha t x K / ∈ K , since o therwise we ca n ex change x K and x ′ K using E = E − 1 . Let K 1 = ∅ , and indutively , define K n +1 to b e the union o f all D λ that inter- sects K n ∪ { x K n } . Since ( D λ ) is lo ca lly finite, it follows by induction that K n is compact for each n . P ut x n = x K n and x ′ n = x ′ K n . Then, clear ly , ( x n , x ′ n ) ∈ E . Moreov er, ( x n ) do es no t have a co nv er gent subsequence. T o see this, a ssume that a subsequence ( x n k ) co nv er ges to a p oint x ∞ ∈ X . Then, there ex ists a λ such that D λ is a compac t neighbo rho o d of x ∞ . T ake a lar ge k such that b oth of x n k and x n k +1 belo ng to D λ . Then x n k +1 ∈ D λ ⊂ K n k +1 ⊂ K n k +1 , which co ntradicts the choice of x n k +1 .  The nex t lemma, also needed to prov e Theorem 3 .5, is v alid for general metr ic spaces: Lemma 3. 7. L et ( x n ) and ( x ′ n ) b e se quenc es in a metric sp ac e X and assume that d ( x n , x ′ n ) ≥ r for every n ∈ N . Then, ther e exist subse quenc es ( x n k ) and ( x ′ n k ) such that d ( A, A ′ ) ≥ r / 3 , wher e A = { x n k | k ∈ N } and A ′ = { x ′ n k | k ∈ N } . Pr o of. F or n ∈ N define the subsets I n , J n of N as follows: I n = { i ∈ N | d ( x n , x ′ i ) < r / 3 } , J n = { i ∈ N | d ( x i , x ′ n ) < r / 3 } . Then, for i , j ∈ I n , we have ( ♣ ) d ( x i , x ′ j ) ≥ r / 3 . Indeed, d ( x ′ i , x ′ j ) ≤ d ( x ′ i , x n ) + d ( x n , x ′ j ) < 2 r / 3, and hence d ( x i , x ′ j ) ≥ d ( x i , x ′ i ) − d ( x ′ i , x ′ j ) ≥ r − 2 r / 3 = r / 3, a s desir ed. Similar ly , the inequality ( ♣ ) also holds for i, j ∈ J n . Thus if I n (or J n ) is infinite for s o me n , the enumeration I n = { n k | k ∈ N } (or J n = { n k | k ∈ N } ) with n 1 < n 2 < · · · gives the desire d subseq uences ( x n k ) and ( x ′ n k ). W e are left with the ca se where I n and J n are finite fo r all n . W e inductively construct a s equence ( n k ) which will give the desir ed subse - quences. Let n 1 = 1, and supp ose that we have constructed n 1 < · · · < n k − 1 satisfy- ing d ( x n i , x ′ n j ) ≥ r / 3 for every i, j < k . Notice that the set S = S i 0 s uch that d ( x n , x ′ n ) ≥ r for every n . By Lemma 3.7, we can further pa ss to subsequences to obta in d ( A, A ′ ) ≥ r/ 3, where A = { x n | n ∈ N } and A ′ = { x ′ n | n ∈ N } . Now define ϕ : X → R by ϕ ( x ) = d ( x, A ) d ( x, A ) + d ( x, A ′ ) . Notice that ϕ ( A ) = { 0 } and ϕ ( A ′ ) = { 1 } . The function ϕ is uniformly contin uous and bo unded, a nd hence is a Higson function by P rop osition 3.1. Thus, ϕ admits a contin uous extension ˜ ϕ : h 0 X → R . METRIC COMP ACT IFICA TIONS AND COARSE STRUC TURES 11 On the o ther hand, we can take a subnet ( x n λ ) o f ( x n ) such that x n λ → ω for some ω ∈ h 0 X \ X . Since E is co n tinuously controlled by h 0 X , we have x ′ n λ → ω . How ever, we then obta in 0 = lim ϕ ( x n λ ) = ˜ ϕ ( ω ) = lim ϕ ( x ′ n λ ) = 1 , which is a contradiction.  4. C 0 coarse structures on tot all y bounded sp aces The S mirnov c omp actific ation uX of a metr ic space X is defined as the maxima l ideal space of the unital Ba na ch a lgebra C u ( X ) of rea l- v alued b ounded uniformly contin uo us functions . Th us, a bo unded co nt inuous function ϕ : X → R is ex tend- able contin uo usly ov er uX if and only if it is unifor mly contin uo us , and any co m- pactification with this pr op erty is equiv alent to uX . Her e, tw o c o mpactifications γ X and δ X of a s pa ce X a re called e quivalent if there exists a homeomo rphism h : γ X → δ X such that h | X = id. Prop o s ition 3.1 immediately implies the follow- ing: Prop ositio n 4 .1. F or any lo c al ly c omp act metric sp ac e X , the Smirnov c omp act- ific ation uX of X is e quivalent to t he Higson c omp actific ation of X with r esp e ct to the C 0 c o arse structu r e.  On the o ther hand, there is a useful characteriza tion of the Smirnov compa ctifi- cation of a g e ne r al metric spac e : Theorem 4.2. [12, Theor em 2 .5] Le t γ X b e a ( H ausdorff ) c omp actific ation of a metric sp ac e X = ( X , d ) . Then, γ X is e quivalent to the Smirnov c omp actific ation uX if and only if cl γ X A ∩ cl γ X B = ∅ for al l subsets A, B ⊂ X with d ( A, B ) > 0 .  Corollary 4.3. F or any c omp act metric sp ac e X = ( X , d ) and its dense subsp ac e Y , the sp ac e X c oincides with the Smirnov c omp actific ation uY . If mor e over Y is lo c al ly c omp act (or e quivalently, op en in X ), then the C 0 structur e on Y c oincides with the c ontinuously c ontr ol le d structu re induc e d fr om X , and X is the Higson c omp actific ation for this structu r e. Pr o of. The first half of the statement is immediate from Theor em 4.2 . If Y is lo cally compact, we can cons ider the C 0 structure on Y with resp ect to the metric d induced from X , as well as the co nt inuously controlled structure on Y induced by X . Then, by Prop osition 4 .1, X = u Y is the Higson compa ctification of Y for the C 0 structure. Finally , it follows fro m Theorem 3.5 that the contin uo us ly controlled structure on Y induced by X = uY is equal to the C 0 structure.  T o state our main result, we define tw o ca tegories . Let K be the categ ory of compact metrizable space s and contin uous maps. W e define another ca teg ory TB a s follows: the ob jects of TB are totally b ounded lo cally compac t metric spaces with the C 0 coarse structur e s . The set Hom TB ( X, Y ) of morphisms betw een ob jects X and Y consists o f the equiv a le nce classes o f coar se maps by the equiv alence relation ∼ , wher e f ∼ g if f and g a r e close (that is , { ( f ( x ) , g ( x )) | x ∈ X } is a controlled set). Such a ca teg ory can be defined, s inc e the closeness re la tion is compatible with co mpo s ition from left and right. Remark 4. 4. The ca tegory TB is related to c o ntin ously controlled structures. Indeed, as seen from Co r ollary 4.3, the ca tegory TB is equiv alent to the following 12 K. MINE AND A. Y AMAS HIT A category CC : the ob jects o f CC are the lo ca lly compact spaces with the co n- tin uously controlled str uctures induced b y metriza ble co mpactifications, and the morphisms b e tween them are the coars e maps mo dulo closeness. On the other ha nd, Cuchillo-Ib´ a˜ nez, Dydak, K oy ama and Mor´ on [4] co nsidered the catego ry Z o f Z -sets in the Hilbe r t cub e Q a nd contin uous maps, and they hav e shown that Z is isomo rphic to the ca tegory C 0 ( Z ) of the complements of Z -sets in Q with the C 0 coarse structures and coars e maps mo dulo closene s s (here Q is assumed to have a fixed metric). Since every compact metrizable space is homeomorphic to some Z -set in Q , the ca teg ory K is equiv ale n t to Z . It follows that the categor ies K , Z and C 0 ( Z ) a re equiv alent to each other. The next Theor em 4 .5 implies that they are equiv alent to TB , and hence to CC . Let us consider the Higson cor ona functor ν intro duced b e fore Pro p o s ition 2.5. This functor sends close coa rse maps to the s ame cont inuous map (see [9, Prop osi- tion 2.41]), and thus coa rsely equiv alent pro pe r coa rse space s hav e homeomo rphic Higson coronas . Natura lly , we can ask the co nverse, namely whether X and Y a re coarsely equiv a lent if ν X and ν Y are homeomo rphic. This q uestion has a nega- tive answer in g eneral (see [9, Exa mple 2.44, P r op osition 2.45 (c)]), but the next theorem states that we have a n affirma tive answer for o b jects of TB . If X is a n ob ject of TB , then the completion ˜ X o f X is compac t since X is totally b ounded. By Corolla r y 4.3, ˜ X is the Higson c ompactification of X and ˜ X \ X is the Higson co rona. In par ticular, ν X is co mpa ct and metrizable. Thus, we can define a functor ν : TB → K . Theorem 4.5. The functor ν : TB → K is an e quivalenc e of c ate gories. Pr o of. It is enough to show that ν is full and faithful, and that every ob ject in K is isomorphic to ν X for some o b ject X in TB . W e shall fir st show that ν is full, namely that ν gives a surjective map from Hom TB ( X, Y ) to the set Hom K ( ν X, ν Y ) of c o nt inuous maps fro m ν X to ν Y , fo r each X and Y in TB . L e t h : ν X → ν Y be a contin uous map. Recall that the completion ˜ X of X gives the Higson compa ctification hX = X ∪ ν X of X , a nd the same holds for hY . Th us we use the notation ˜ X and ˜ Y rather than hX and hY , and their metr ic s extended fro m X and Y are denoted by d when neces sary . W e construct (a represe ntative of ) a mor phis m f : X → Y in TB s uc h tha t ν f = h . The basic idea her e is as follows: for x ∈ X , we take a p oint a ∈ ν X close to x and define f ( x ) to b e a p oint of Y close to h ( a ), to the sa me extent as x is close to a . W e explain this constructio n in detail. Le t us de fine U n as the o pen 1 /n -neighbo rho o d of ν X in ˜ X for n ∈ N , and let U 0 = ˜ X . Using the compactness of ν Y , for each n ∈ N , take finitely many p oints y n, 1 , y n, 2 ,. . . , y n,k ( n ) in Y such that ν Y ⊂ S k ( n ) i =1 B ( y n,i , 1 /i ). F or conv enience, let k (0) = 1 a nd let y 0 , 0 be an a rbitrarily fixed p oint in Y . T o define f : X → Y , let x ∈ X and take the large s t n ≥ 0 such that x ∈ U n . I f n = 0 , then we define f ( x ) = y 0 , 0 . If n ≥ 1 , cho ose x ′ ∈ ν X such that d ( x, x ′ ) = d ( x, ν X ). Then we can cho ose i ∈ { 1 , 2 , . . . , k ( n ) } such that h ( x ′ ) ∈ B ( y n,i , 1 /n ). W e finally define f ( x ) = y n,i ∈ Y . W e c la im that f : X → Y is a co arse map a nd ν f = h . First, notice that f is pr e- bo rnologo us, since C 0 coarse s tructures s a tisfy the co ndition ( ⋆ ) in Remark 2.3 a nd f ( X \ U n ) is co n tained in the finite set { y m,i | m < n, 1 ≤ i ≤ k ( m ) } for each n ∈ N . By Theorem 3.5, the C 0 coarse structure on Y is the continously c ontrolled structure METRIC COMP ACT IFICA TIONS AND COARSE STRUC TURES 13 induced by ˜ Y . Also , we easily see that f ∪ h : X ∪ ν X = ˜ X → ˜ Y is c ontin uous a t each p oint in ν X . Then, it follows by Prop ositio n 2 .7 (and Pro po sition 2.5) that f is coars e a nd ν f = h . The fullness of ν is now prov ed. Next we show that ν : TB → K is faithful, namely that ν maps each Hom TB ( X, Y ) injectiv ely to Hom K ( ν X, ν Y ). T o see this, let f , g : X → Y be coar se maps such that ν f = ν g . W e hav e to s how that f a nd g are close, in other words, E = { ( f ( x ) , g ( x )) | x ∈ X } ⊂ Y × Y is controlled. By Theor em 3 .5, it is enough to show that E is contin uously controlled by ˜ Y . T o this end, take any ( η , η ′ ) ∈ E \ Y × Y , where E deno tes the clos ur e of E in ˜ Y × ˜ Y . Then, ther e exis ts a net ( x λ ) in X such that ( f ( x λ ) , g ( x λ )) → ( η , η ′ ). Since f is pro pe r, we can take a subnet ( x λ µ ) of ( x λ ) such that x λ µ → ω fo r some ω ∈ ν X = ˜ X \ X . Then by Pr op osition 2.5, we hav e η = lim f ( x λ µ ) = ν f ( ω ) = ν g ( ω ) = lim g ( x λ µ ) = η ′ ∈ ν Y = ˜ Y \ Y , which shows that E is contin uously co nt rolled by ˜ Y . Finally , we have to show that every ob ject in K is isomorphic to ν X for some ob ject X in TB . T o see this, let K b e any compact metriza ble space, and fix any admissible metric d on K × [0 , 1 ]. Let X = K × (0 , 1]. Then, X = ( X , d ) is an ob ject of TB and K × [0 , 1] is its Higson compactification by Corolla ry 4.3. It follows that ν X = K × { 0 } and hence K is homeomor phic to ν X . The pro of is completed.  The following is an immediate c o nsequence of Theorem 4.5 (and Coro llary 4 .3): Corollary 4.6. S upp ose that M 1 and M 2 ar e c omp act metric sp ac es and t hat Z 1 ⊂ M 1 and Z 2 ⊂ M 2 ar e close d nowher e dense subsp ac es. Then, M 1 \ Z 1 and M 2 \ Z 2 ar e c o arsely e quivalent as C 0 c o arse sp ac es if and only if Z 1 and Z 2 ar e home omorphic.  Moreov er, Theor em 4 .5 and the ab ov e cor ollary tra nslate to the lang uage o f the category CC intro duced in Remark 4.4, in view of Corolla ry 4.3: Corollary 4.7. T he Higson c or ona functor ν : CC → K is an e quivalenc e of c ate- gories. In p articular, t wo m etrizable c omp actific ations ˜ X 1 and ˜ X 2 of a lo c al ly c om- p act sp ac e X determine c o arsely e quivalent c ontinuously c ontr ol le d c o arse str u ctur es if and only if their r emainders ar e home omorphic, ˜ X 1 \ X ≈ ˜ X 2 \ X .  Corollary 4.8. E very obje ct in CC is c o arsely e quivalent t o an obje ct in CC that is c ont ra ctible, whose Higson c omp actific ation is also c ontr actible. Pr o of. F or any ob ject X in CC , which ha s the contin uously controlled structur e induced by a metrizable compa ctification ˜ X , consider the r emainder Z = ˜ X \ X . Let ˜ Y b e the cone over Z , which is co mpact metr izable and is a compactifica tio n o f the op en cone Y = ˜ Y \ Z . W e can then equip Y with the contin uously controlled structure induced by ˜ Y . By Co rollary 4.7, the coa rse space Y is a n o b ject of CC coarsely equiv alent to X . Clea r ly Y and ˜ Y a re contractible, and ˜ Y is the Higso n compactification of Y b y Coro llary 4.3.  Example 4. 9. Applying Coro llary 4.7, we can construct three pr op er coarse struc- tures E i ( i = 1 , 2 , 3) on the same top ologic al spa ce X with E 1 ⊂ E 2 ⊂ E 3 for whic h E 1 and E 3 are c oarsely equiv a lent, but E 2 fails to b e eq uiv alent to E 1 (or E 3 ). Indeed, it suffices to take three metrizable co mpactifications γ i X of the sa me lo ca lly compact space X that a dmit maps γ 1 X → γ 2 X → γ 3 X ex tending the identit y , with the remainders Z i = γ i X \ X sa tisfying Z 1 ≈ Z 3 but Z 1 6≈ Z 2 . Then, the contin uously 14 K. MINE AND A. Y AMAS HIT A controlled strutcures induced by γ i X ( i = 1 , 2 , 3) give an example. It is eas y to construct an explicit ex a mple where X = [0 , 1] × [0 , 1 ), Z 2 is a circle a nd Z 1 , Z 3 are arcs. W e conclude this pap er with results c oncerning embeddings of C 0 coarse spaces, stating that there is a “universal” C 0 coarse s pace in which all ob ject in TB can be embedded. W e say that a map f : X → Y betw een coar se spaces is a c o arse emb e dding if the map f : X → f ( X ) is a coarse equiv alence. Here f ( X ) is assumed to hav e the induced coar se structure { F ⊂ f ( X ) × f ( X ) | F is controlled in Y } . The pro of of the next le mma is stra ightforw ard: Lemma 4.10. L et X b e a lo c al ly c omp act metric sp ac e with the C 0 c o arse structu re and Y ⊂ X b e a close d set. Then, the induc e d c o arse str u ctur e on Y c oincides with the C 0 structur e for t he lo c al ly c omp act met r ic sp ac e Y .  First, we consider coars e embeddings that are top o logical embedding s at the same time: Prop ositio n 4. 11. Ther e exists a sep ar able lo c al ly c omp act metric s p ac e X such that for every obje ct Y in TB admits a map f : Y → X that is simultane ously a top olo gic al and c o arse emb e dding. Pr o of. W e can take X = Q × [0 , 1), where Q = [0 , 1 ] N is the Hilb ert cub e. W e define a metric on X as the restriction o f a n y compatible metr ic on Q × [0 , 1]. Let Y b e an y ob ject of TB and ˜ Y be its completion. W e fix a c ontin uous function ϕ : ˜ Y → [0 , 1 ] such that ϕ − 1 (1) = ˜ Y \ Y a nd a top ologica l embedding j : ˜ Y → Q . Then, the map i : ˜ Y → Q × [0 , 1] defined by i ( y ) = ( j ( y ) , ϕ ( y )) gives a top olo g ical embedding such that i − 1 ( X ) = i − 1 ( Q × [0 , 1)) = Y . Let us show tha t f = i | Y : Y → X is the r equired map. The maps f : Y → f ( Y ) and f − 1 : f ( Y ) → Y are prop er since they are ho meomophisms, and are uniformly contin uous since they a re restrictions of contin uous maps, namely i and i − 1 , defined on c ompact metric spaces. W e conclude from Cor ollary 3.4 that f : Y → X is a co arse embedding.  If w e admit coars e embeddings that are not top o logical em b eddings (and not even contin uous maps), we hav e the following r e sult b y using Theorem 4 .5: Theorem 4.1 2. F or every nonc omp act lo c al ly c omp act sep ar able metrizable sp ac e X , t her e exists a c omp atible total ly b ounde d metric d on X su ch that every obje ct in TB c an b e c o arsely emb e dde d into ( X , d ) with r esp e ct to the C 0 structur e. Corollar y 4.8 turns every o b ject in TB into a contractible s pace, whic h is “co n- tin uous” in na tur e. The nex t co rollar y of Theorem 4.12 is a result in the opp os ite direction, s aying that every o b ject in TB can b e expr essed as a discrete metric space. Her e, a discr ete metric spa c e mea ns a metric spac e whose top olo gy is dis- crete. Corollary 4.13. Ther e ex ist s a c oun t able discr ete met ric sp ac e X such t hat every obje ct in TB c an b e c o arsely emb e dde d into X with r esp e ct t o the C 0 structur es. Mor e over, every obje ct in TB is c o arsely e qu ivalent to some c ountable discr ete metric sp ac e with the C 0 structur e. Pr o of. The first part r eadily follows fro m Theorem 4.12. The seco nd part follows from the firs t part using Lemma 4.10.  METRIC COMP ACT IFICA TIONS AND COARSE STRUC TURES 15 T o pr ov e Theo rem 4.12 (and Corollar y 4.1 3), we need some technical le mma s: Lemma 4.14. L et X b e a lo c al ly c omp act sep ar able metric sp ac e with the C 0 c o arse structur e and A, B ⊂ X with the induc e d s tructur es, wher e cl X A = B . Then, the inclusion A → B is a c o arse e quivalenc e. Pr o of. Let i : A → B b e the inclusion, whic h is clear ly a coa rse map. By Pro po sition 2.1, there exists a controlled neighbo rho o d E 0 of the diag onal ∆ X in X × X . W e define h : B → A by choo sing a p oint h ( b ) ∈ A with ( b, h ( b )) ∈ E 0 for ea ch b ∈ B . It is easy to chec k that h : B → A is a ls o a coar se ma p. Then, i ◦ h is clos e to the identit y id B since the s et { ( b, h ( b )) | b ∈ B } is contained in E 0 and hence is controlled. Similarly , the other comp osition h ◦ i is close to the identit y id A . W e conclude that i : A → B is a c o arse equiv a lence.  Remark 4. 15. Clearly , this lemma is true for a coar se space X equipp ed with a top ology for which there is a controlled neighbo rho o d of the diagona l ∆ X in X × X , in particular for a ll prop er coarse spaces. F urthermor e, if X is such a co arse space , subsets A and B of X are coars ely equiv alent with res pec t to the induced str uctur es whenever they have the same closure, cl X A = c l X B . Lemma 4.16. L et X , Y b e sp ac es in TB and ν X , ν Y b e their Higson c or onas with r esp e ct to t he C 0 structur es. L et j : ν X → ν Y b e a t op olo gic al emb e dding. Th en, ther e exists a c o arse emb e dding f : X → Y s u ch that ν f = j . Pr o of. Let ˜ Y = Y ∪ ν Y be the Higs on compactification which coincides with the completion. As shown in the pr o of of Theorem 4.5, ther e ex is ts a co arse map f : X → Y such that ν f = j . Let f ( X ) be the closur e of f ( X ) in Y . By Pro po sition 2.5, it is easy to see that cl ˜ Y f ( X ) = cl ˜ Y f ( X ) = f ( X ) ∪ j ( ν X ). Hence by Corollary 4.3, we hav e j ( ν X ) = ν f ( X ). Let f 0 : X → f ( X ) and j 0 : ν X → j ( ν X ) be the maps which ar e equal to f and j r esp ectively , with their ranges restric ted. Then we hav e ν f 0 = j 0 by Pro p o sition 2.5 . Notice that f ( X ) is closed in Y , hence its C 0 structure coincides with the structure induced from Y by Lemma 4.10. Since j 0 is a homeomor phism, f 0 : X → f ( X ) is a coarse equiv alence by Theorem 4 .5. The coarse equiv alence f 0 factors as X → f ( X ) → f ( X ), and the second map is a coa rse equiv alence by Le mma 4 .14. Then, it ea sily follows that the first map X → f ( X ) is a coar se equiv alence, as desired.  Pr o of of The or em 4.12 . Recall that every compact metrizable s pa ce can b e embed- ded int o Q = [0 , 1 ] N . In v iew of Lemma 4.16, to pr ov e this theor e m it is enough to notice that there exists a metrizable compactifica tio n γ X of X with the remainder homeomorphic to Q . Then the restr iction to X o f any co mpatible metric on γ X satisfies our requir ement (then, γ X is the Higso n compactifica tion with resp ect to the C 0 structure by Coro llary 4.3 ). F o r co mpleteness, we explain how to construct γ X . Since X is no ncompact and metrizable, there exists a sequence ( x n ) n ∈ N of dis - tinct p oints in X without co nv er gent subsequences. Fix a countable dense subset { y n | n ∈ N } in Q . The map { x n | n ∈ N } → Q = [0 , 1] N that s e nds each x n to y n can b e extended to a contin uous map h : X → Q by T ie tze ’s theorem. Let K b e the pro duct ( X ∪ {∞} ) × Q , wher e X ∪ { ∞} denotes the one- po int compactification. The ma p i : X → K defined by i ( x ) = ( x, h ( x )) is a top ologica l embedding, and the clo s ure of its imag e in K is i ( X ) ∪ ( {∞} × Q ), which is clea rly a metriza ble compactification of X with the r emainder homeomorphic to Q .  16 K. MINE AND A. Y AMAS HIT A ackno wledgements The s econd author deeply ackno wledges the ho spitality of Ob er wolfach ma the- matical research institute in Germany , where he has undertaken a part of this work. 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Mine) Institute of Ma thema tics, Un iversity of Tsukuba, Tsukuba, 305-8571 , Jap an E-mail addr ess : pen@math.tsu kuba.ac.jp (A. Y amashita) G radua te School of Informa tion Sciences, Tohoku University, Sendai 980-8579 , Jap an E-mail addr ess : yamyam@rock. sannet.ne.jp