Dugundji's Canonical Covers, asymptotic and covering dimension
DDugundji’s Canonical Covers, asymptotic and covering dimension
Jesús P. Moreno-Damas
Abstract
Given a nowheredense closed subset X of a metrizable compact space (cid:101) X , wecharacterize the dimension of X in terms of the multiplicity of the canonicals coversof the complementary of X , specially in some particular cases, like when (cid:101) X is theHilbert cube or the finite dimensional cube and X , a Z-set of (cid:101) X . In this process, wesolve some questions in the literature. This paper reinforces the relation between the canonical covers, used by Dugundji in [9]to prove his famous Extension Theorem, and the way to describe the dimension of certainsubsets of compact spaces in terms of the corresponding complement. Such a relation wastreated in [5] in the special case when a compact metrizable space is embedded as a Z-setin the Hilbert cube. Concretely, in [5], the authors proved that a compact Z-subset of theHilbert cube has finite topological dimension if and only if there exists a canonical coverof finite order in its complement (for us, the order of a cover α is less the supremum ofthe number elements of α with non empty intersection, that is, less the multiplicity of α .) The above kind of relations are in the core of so called Higson-Roe functor with allowsus to compare coarse properties of a coarse structure with topological properties of thecorresponding Higson-Roe corona (see [19] and [4] for a specific example). In particular,asymptotic dimensional properties with covering dimensional properties. Several works goin this way, for example [2], [3], [7] and [8].When the coarse structure is the topological one attached to a metrizable compactifica-tion, Grave proved in [11] and [12] that the asymptotic dimension of that coarse structureexceeds just by the covering dimension of the corona. That result allowed the authorsin [4] to prove that the gap between the C -asymptotic dimension of the complement of aZ-set in the Hilbert and the topological dimension of the corresponding Z-set is precisely . As proved in [5], the particularity of the canonical covers in the complementary is that1 a r X i v : . [ m a t h . GN ] J un very one has information about the dimension of the Z-set. In fact, what was reallyproved in that article is that, if a compact Z-set in the Hilbert cube has dimension n , thenthe order of any canonical cover of the complement is greater than or equal to n and thereexists a canonical cover which order is less than or equal to n + 1 .As pointed out in that article, the definition of canonical cover is in the core of the C coarse geometry (see [19], [22] and [23] for definitions). It is natural to ask if, for everycanonical cover, the gap between its order and the dimension of the Z-set is at least and,as it was did in [5] page 3713, if one can always find a canonical cover whose gap is just .In this paper, we use canonical covers to strength and improve relations pointed out in[5]. In particular, we are able to work in a more general framework and to answer positivelythe questions mentioned above. We also describe accurately the relations between thecanonical covers and the C coarse structure. Finally, we use our results here to give aneasier proof of a Grave’s result in [11] and [12], relating the topological dimension of thecorona with the corresponding asymptotic dimension.In section 2 we introduce the basic needed definitions and notation and state an easycriterium to detect coarse equivalences.Section 3, which is the core of the paper, contains the main technical tools and resultsin this paper. Firstly, we relate the canonical covers with the coarse structures, gettingthat a canonical cover is precisely an open and locally finite cover which is uniform for thetopological coarse structure attached to a compactification. In our case, that compactifi-cation is metrizable and that means that the cover is uniform for the C coarse structure.That states the relation pointed out in [5].Taking it into account, we are able to relate the multiplicity of canonical covers withthe asymptotic dimension of the C coarse spaces, using the Grave’s result mentionedabove, to obtain:(A) If X is a nowheredense closed subset of a metrizable compact space (cid:101) X , then dim X ≤ n if and only if for every canonical cover α of (cid:101) X \ X , there exists a canonical cover β which has α as a refinement and whose order is less than or equal to n + 1 .To improve this equivalence when (cid:101) X is the Hilbert cube and X is a Z-set of (cid:101) X , wedefine a more general category, the cylindrical subsets . This allows us to extends resultswhich work for the Z-sets of the Hilbert cube, to other cases, like the Z-sets of the finitedimensional cube [0 , n . We get:(B) If X is a cylindrical, nowheredense and closed subset of a metrizable compact space (cid:101) X (in particular, if X is a Z-set of (cid:101) X and (cid:101) X is the Hilbert Cube or the finite dimensionalcube), then dim X ≤ n if and only if there exists a canonical cover of (cid:101) X \ X whoseorder is less than or equal to n + 1 .Actually, (B) is satisfied not only for canonical covers, but also for open and C -uniform covers. Consequently, the order of every open and uniform cover is at least the2 -asymptotic dimension. That means that, when X is a cylindrical subset, from theasymptotic dimensional point of view, the open and uniform covers are big enough. Thatsugests a natural question: In the general case, when is a uniform cover big enough fromthe asymptotic dimensional point of view? We finish the section by answering that.In section 4 we recover Grave’s result in an easier way. To do it, we give some resultsin Topological Dimension Theory (one of them, developed by us in [16]). We say that compactification pack is a vector ( X, (cid:98) X, (cid:101) X ) such that (cid:101) X is a compact Hausdorffspace, X is a nowheredense closed subset of (cid:101) X and (cid:98) X = (cid:101) X \ X . Observe that (cid:101) X is acompactification of (cid:98) X and X is its corona.For us, a cover α of a set Z is a collection of subsets of Z whose union is Z .If Z is a topological space and α is a family of subsets of Z , we say that α is open ifevery U ∈ α is open. If Z is a metric space, we say that mesh α = sup { diam U : U ∈ α } .If and A ⊂ Z and α and β are families of subsets of Z , we denote α ( A ) = (cid:83) U ∈ αU ∩ A (cid:54) = ∅ U .By β ≺ α we mean that β is a refinement of α . The multiplicity of α is mult α = sup { A : A ⊂ α, (cid:84) U ∈ A U (cid:54) = ∅ } (where, for every set B , B means the cardinal of B ).The concept of canonical cover was used by Dugundji in [9]. In our language, acanonical cover α of a compactification pack ( X, (cid:98) X, (cid:101) X ) is an open and locally finite coverof (cid:98) X such that for every x ∈ X and every neighborhood V x of x in (cid:101) X there exists aneighborhood W x of x in (cid:101) X with α ( W x ) ⊂ V x .Let us give some definitions of coarse geometry. For more information, see [19]. Let E, F ⊂ Z × Z , let x ∈ Z and let K ⊂ Z . The product of E and F , denoted by E ◦ F ,is the set { ( x, z ) : ∃ y ∈ Z such that ( x, y ) ∈ E, ( y, z ) ∈ F } , the inverse of E , denotedby E − , is the set E − = { ( y, x ) : ( x, y ) ∈ E } , the diagonal, denoted by ∆ , is the set { ( z, z ) : z ∈ Z } . If x ∈ Z , the E -ball of x , denoted by E x is the set E x = { y : ( y, x ) ∈ E } and, if K ⊂ Z , E ( K ) is the set { y : ∃ x ∈ K such that ( y, x ) ∈ E } . We say that E issymmetric if E = E − .A coarse structure E over a set Z is a family of subsets of Z × Z which contains thediagonal and is closed under the formation of products, finite unions, inverses and subsets.The elements of E are called controlled sets. B ⊂ Z is said to be bounded if there exists x ∈ Z and E ∈ E with B = E x (equivalently, B is bounded if B × B ∈ E ).A map f : ( Z, E ) → ( Z (cid:48) , E (cid:48) ) between coarse spaces is called coarse if f × f ( E ) iscontrolled for every controlled set E of Z and f − ( B ) is bounded for every bounded subset B of Z (cid:48) . We say that f is a coarse equivalence if f is coarse and there exists a coarse map g : ( Z (cid:48) , E (cid:48) ) → ( Z, E ) such that { ( g ◦ f ( x ) , x ) : x ∈ Z } ∈ E and { ( f ◦ g ( y ) , y ) : y ∈ Z (cid:48) } ∈ E (cid:48) .In this case, g is called a coarse inverse of f .A subset A ⊂ Z is coarse dense if Z = E ( A ) for some E ∈ E .3iven E ⊂ Z × Z , we denote K ( E ) = { E x : x ∈ Z } . If α is a family of subsets of Z ,we say that D ( α ) = (cid:83) U ∈ α U × U (observe that ( x, y ) ∈ D ( α ) if and only if x, y ∈ U forsome U ∈ α ).A family of subsets α of Z is called uniform if D ( α ) ∈ E . Note that, if E ∈ E , then K ( E ) is uniform. Dydak and Hoffland showed in [10] that the coarse structures can bedescribed in terms of the uniform covers.Intuitively, the uniform families of subsets of ( Z, E ) behave like the controlled sets. Forexample, suppose that f : ( Z, E ) → ( Z (cid:48) , E (cid:48) ) is a coarse map between are coarse spaces, E is a controlled set of ( Z, E ) , α is a uniform family of subsets of ( Z, E ) and β is a family ofsubsets of Z such that β ≺ α . Then, E ( α ) and β are uniform families of subsets of ( Z, E ) and f ( α ) = { f ( U ) : U ∈ α } is a uniform family of subsets of ( Z (cid:48) , E (cid:48) ) . For every B ⊂ Z , B is bounded if and only if there exists a uniform family of subsets γ of ( Z, E ) such that B ∈ γ (equivalently, if { B } is uniform).If Z is a topological space and E ⊂ Z × Z , we say that E is proper if E ( K ) and E − ( K ) are relatively compact for every relatively compact subset K ⊂ Z . If E is a coarsestructure over Z , we say that ( Z, E ) is a proper coarse space if Z is Hausdorff, locallycompact and paracompact, E contains a neighborhood of the diagonal in Z × Z and thebounded subsets of ( Z, E ) are precisely the relatively compact subsets of Z .By Q we denote the Hilbert cube [0 , N . Let (cid:101) X = [0 , n with n ∈ N or (cid:101) X = Q andsuppose that d is a metric in (cid:101) X . X is a Z-set of (cid:101) X if it is closed and for every ε > thereexists a continuous function f : (cid:101) X → (cid:101) X such that d (cid:48) ( f, Id ) < ε —where d (cid:48) is the suprememetric— and f ( (cid:101) X ) ∩ X = ∅ (the definition of Z-set given in [6], chapter I-3, pag. 2, isequivalent in this context).For us, an increasing function between two ordered sets f : ( X, < ) → ( X (cid:48) , < (cid:48) ) is afunction such that x < y implies f ( x ) ≤ (cid:48) f ( y ) .The following identities will be useful along this article. Let Z, Z (cid:48) be sets, suppose x ∈ Z , x (cid:48) ∈ Z (cid:48) , A, B ⊂ Z and E, F ⊂ Z × Z . Consider a family of subsets α of Z and amap f : Z → Z (cid:48) . Then: D ( α )( A ) = α ( A ) (1) ( E ◦ F ) x = E ( F x ) (2) E ( A ) = (cid:91) a ∈ A E a (3) E ( A ) ∩ B (cid:54) = ∅ if and only if A ∩ E − ( B ) (cid:54) = ∅ . (4) E ( B ) ⊂ A if and only if E ∩ ( Z \ A ) × B = ∅ (5) ( f × f ( E )) x (cid:48) = (cid:91) a ∈ f − ( x (cid:48) ) f ( E a ) = f ( E ( f − ( x (cid:48) ))) (6)The following proposition provides a criterion to detect coarse equivalences:4 roposition 1. If f : ( X, E ) → ( Y, F ) is a map between coarse spaces, thena) f is a coarse equivalence.b) There exists g : Y → X such that f × f ( E ) ∈ F for every E ∈ E , g × g ( F ) ∈ E forevery F ∈ F , { ( x, g ◦ f ( x )) : x ∈ X } ∈ E and { ( y, f ◦ g ( y )) : y ∈ Y } ∈ F .c) f × f ( E ) ∈ F for every E ∈ E , ( f × f ) − ( F ) ∈ E for every F ∈ F and there exists g : Y → X such that { ( y, f ◦ g ( y )) : y ∈ Y } ∈ F .d) f × f ( E ) ∈ F for every E ∈ E , ( f × f ) − ( F ) ∈ E for every F ∈ F and f ( X ) iscoarse dense in ( Y, F ) .are equivalent. Moreover, if g is the map of b) or c), then g is a coarse inverse of f .Proof. Throughout the proof, every time we use a map called g : Y → X , we will denoteby G and H the sets G = { ( x, g ◦ f ( x )) : x ∈ X } H = { ( y, f ◦ g ( y )) : y ∈ Y } It is obvious that a) implies b).To see that b) implies a), it is suffices to show that f − ( U ) is bounded for everybounded subset U of X and g − ( V ) is bounded for every bounded subset V of Y . But itis easily deduced from: f − ( V ) × f − ( V ) ⊂ G ◦ ( g × g ( V × V )) ◦ G − ∈ E g − ( U ) × g − ( U ) ⊂ H ◦ ( f × f ( U × U )) ◦ H − ∈ F Moreover, g is a coarse inverse of f .To see that b) implies c) it is sufficient to show that ( f × f ) − ( F ) ∈ E for every F ∈ F .But it follows from: ( f × f ) − ( F ) ⊂ G ◦ ( g × g ( F )) ◦ G − ∈ E To see that c) implies b), it is sufficient to show that g × g ( F ) ∈ E for every F ∈ F and that G ∈ E . But it follows from: g × g ( F ) ⊂ ( f × f ) − ( H − ◦ F ◦ H ) ∈ E G ∈ ( f × f ) − ( H ) ∈ E Moreover, since g satisfies b), g is a coarse inverse of f .To see that c) implies d), it suffices to show that f ( X ) is coarse dense in ( Y, F ) . Andit happens, since Y = H ( f ( X )) .To see that d) implies c) we only need to define a map g : Y → X such that H ∈ F .To show that, take M ∈ F such that Y = M ( f ( X )) . By (3), Y = (cid:83) y ∈ f ( X ) M y , so there5xists a partition of Y { P y : y ∈ f ( X ) } such that P y ⊂ M y for every y ∈ f ( X ) . Choose x y ∈ f − ( y ) for every y ∈ f ( X ) . Let g : Y → X be the map such that, for every y ∈ f ( X ) and every y (cid:48) ∈ P y , g ( y (cid:48) ) = x y .Observe that ( y (cid:48) , f ◦ g ( y (cid:48) )) = ( y (cid:48) , f ( x y )) = ( y (cid:48) , y ) ∈ P y × { y } ⊂ M y × { y } ⊂ M . Then, H ⊂ M ∈ F and H ∈ F . Corollary 2. If f : ( X, E ) → ( Y, F ) is a bijective coarse equivalence, then f − is a coarseinverse of f .Proof. Since f is a coarse equivalence, it satisfies property d) of Proposition 1. Then, f and f − satisfice property c) and hence, f − is a coarse inverse of f . Corollary 3. If f : ( X, E ) → ( Y, F ) is a coarse equivalence and α is a uniform cover of ( Y, F ) , then f − ( α ) is a uniform cover of X .Proof. Since D ( f − ( α )) = ( f × f ) − ( D ( α )) is a controlled set of ( X, E ) , it follows that f − ( α ) is uniform. Firstly, we relate the canonical covers with the C coarse structures.Recall the following definition, see [19]: Definition 4.
Let ( X, (cid:98) X, (cid:101) X ) be a compactification pack. The topological coarse structure E over (cid:98) X attached to the compactification (cid:101) X is the collection of all E ⊂ (cid:98) X × (cid:98) X satisfyingany of the following equivalent properties:a) Cl (cid:101) X × (cid:101) X E meets (cid:101) X × (cid:101) X \ (cid:98) X × (cid:98) X only in the diagonal of X × X .b) E is proper and for every net ( x λ , y λ ) ⊂ E , if { x λ } converges to a point x of X , then y λ converges also to x .c) E is proper and for every point x ∈ X and every neighborhood V x of x in (cid:101) X thereexists a neighborhood W x of x in (cid:101) X such that E ( W x ) ⊂ V x . Remarks 5. • In this coarse structure, the bounded subsets of E are precisely the relatively compactsubsets of (cid:98) X . Moreover, if (cid:101) X is metrizable, then E is proper. • The definition above is Proposition 2.27 and Definition 2.28 of [19] (pags. 26-27)together with the author’s correction in [20]. The property c) above is not the one ofthat proposition, but it is equivalent (see (5)).6 roposition 6.
Let ( X, (cid:98) X, (cid:101) X ) be a compactification pack. Suppose that E is the topologicalcoarse structure over (cid:98) X attached to the compactification (cid:101) X and let E ⊂ (cid:98) X × (cid:98) X . Then, E ∈ E if and only if, for every x ∈ X and every neighborhood V x of x in (cid:101) X , there exist aneighborhood W x of x in (cid:101) X such that ( E ∪ E − )( W x ) ⊂ V x .Proof. If E ∈ E , then also E ∪ E − ∈ E , thus E ∪ E − satisfies property c) of definition 4and we get the necessity.Let us see the sufficiency. To prove that E satisfies property c) of definition 4, it sufficesto show that E is proper, that is, E ( K ) ∪ E − ( K ) is relatively compact in (cid:98) X for everyrelatively compact subset K of (cid:98) X .Take a relatively compact subset K of (cid:98) X . Fix x ∈ X . Since (cid:101) X \ K is a neighborhood of x in (cid:101) X , there exists an open neighborhood W x of x in (cid:101) X such that ( E ∪ E − )( W x ) ⊂ (cid:101) X \ K ,i. e. K ∩ ( E ∪ E − )( W x ) = ∅ . By (4), it is equivalent to ( E ∪ E − )( K ) ∩ W x = ∅ . Since (cid:83) x ∈ X W x is an open neighborhood of X in (cid:101) X with ( E ∪ E − )( K ) ∩ (cid:0)(cid:83) x ∈ X W x (cid:1) = ∅ , wehave that ( E ∪ E − )( K ) = E ( K ) ∪ E − ( K ) is relatively compact in (cid:98) X . Corollary 7.
Let ( X, (cid:98) X, (cid:101) X ) be a compactification pack and suppose that E is the topologicalcoarse structure over (cid:98) X attached to the compactification (cid:101) X . If α is a family of subsets of (cid:98) X , then α is uniform if and only if for every x ∈ X and every neighborhood V x of x in (cid:101) X ,there exists a neighborhood W x of x in (cid:101) X such that α ( W x ) ⊂ V x .Particulary, α is a canonical cover of ( X, (cid:98) X, (cid:101) X ) if and only if it is an open and locallyfinite cover of (cid:98) X which is controlled for E .Proof. α is uniform if and only if D ( α ) ∈ E . By Proposition 6, (1) and the symmetry of D ( α ) , that is equivalent to say that for every x ∈ X and every neighborhood of x in (cid:101) X there exists a neighborhood W x of x ∈ (cid:101) X such that D ( α )( W x ) ⊂ V x , i. e. α ( W x ) ⊂ V x .Recall the following definition of Wright in [22] or [23] (see also of example 2.6 of [19],pag. 22): Definition 8.
Let ( (cid:98) X, d) be a metric space. The C coarse structure is the collection ofall subsets E ⊂ (cid:98) X × (cid:98) X such that for every ε > there exists a compact subset K of (cid:98) X such that d( x, y ) < ε whenever ( x, y ) ∈ E \ K × K . We have the following using the same argument as in [4]:
Proposition 9.
Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack and let d be a metricon (cid:101) X . Then, the topological coarse structure over (cid:98) X attached to the compactification (cid:101) X isthe C coarse structure over (cid:98) X attached to d .Proof. The proof of Proposition 6 of [4] is valid in this context. But, using Proposition 6,we get the following shorter proof:Denote by E and E the topological coarse structure attached to (cid:101) X and the C coarsestructure attached to d respectively. 7et E ∈ E . Observe that E ∪ E − ∈ E . Fix x ∈ X and a neighborhood V x of x in (cid:101) X .Choose an ε > such that B( x, ε ) ⊂ V x and a compact subset K of (cid:98) X such that d( x, y ) < ε whenever ( x, y ) ∈ ( E ∪ E − ) \ K × K . Take δ = min (cid:110) ε , d( K,X )2 (cid:111) and W x = B( x, δ ) . Picka point y ∈ ( E ∪ E − )( W x ) and take z ∈ W x such that ( y, z ) ∈ E ∪ E − . Since z (cid:54)∈ K , itfollows that y, z ∈ ( E ∪ E − ) \ K × K and d( y, z ) < ε . Hence d( y, x ) ≤ d( y, z ) + d( z, x ) < ε + ε = ε and, consequently, ( E ∪ E − )( W x ) ⊂ B( x, ε ) ⊂ V x . Therefore, E ∈ E .Take now E ∈ E and fix ε > . For every x ∈ X , consider a neighborhood W x of x contained in B (cid:0) x, ε (cid:1) such that ( E ∪ E − )( W x ) ⊂ B (cid:0) x, ε (cid:1) . Let K = (cid:101) X \ (cid:83) x ∈ X W x andsuppose ( y, z ) ∈ E \ K × K . Observe that neither y ∈ K nor z ∈ K . If z (cid:54)∈ K , then thereexists x ∈ X with z ∈ W x and y ∈ E ( W x ) . If y (cid:54)∈ K , then there exists x ∈ X such that y ∈ W x and z ∈ E − ( W x ) . In both cases, y, z ∈ W x ∪ ( E ∪ E − )( W x ) ⊂ B (cid:0) x, ε (cid:1) and hence d( y, z ) ≤ d( y, x ) + d( x, z ) ≤ ε + ε = ε . Therefore, E ∈ E .From now on, we will use the following notation: Definition 10.
Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack. The C coarse struc-ture over (cid:98) X attached to ( X, (cid:98) X, (cid:101) X ) , denoted by E ( X, (cid:98) X, (cid:101) X ) or by E when no confusioncan arise, is the topological coarse structure attached to the compactification (cid:101) X , i. e. the C coarse structure attached to any metric of (cid:101) X restricted to (cid:98) X . We use this notation because working with a metric in (cid:101) X simplifies the calculus andis easier to see geometrically (see figure 1).Figure 1. ( X, (cid:98) X, (cid:101) X ) and uniform cover of E ( X, (cid:98) X, (cid:101) X ) Proposition 11.
Let ( (cid:98) X, d) be a locally compact metric space and let E be the C coarsestructure. Consider a family of subsets α of (cid:98) X . Then, α is uniform for E if and only if: • α is proper, i. e. for every relatively compact subset K of (cid:98) X , α ( K ) is relativelycompact in (cid:98) X .and • For every ε > there exists a compact subset K of (cid:98) X such that diam U < ε for every U ∈ α with U ∩ K = ∅ . roof. Suppose that α is uniform. Since D ( α ) is controlled and hence proper, (1) showsthat α ( K ) = D ( α )( K ) is relatively compact for every relatively compact K of (cid:98) X .Let ε > and consider a compact subset K of (cid:98) X such that d( x, y ) < ε for every ( x, y ) ∈ D ( α ) \ K × K . Take U ∈ α with U ∩ K = ∅ and x, y ∈ U . Since x, y (cid:54)∈ K , ( x, y ) ∈ D ( α ) \ K × K and hence d( x, y ) < ε . Thus, diam U ≤ ε .For the reciprocal, fix ε > and consider a compact subset K of (cid:98) X with diam U < ε forevery U ∈ α such that U ∩ K = ∅ . Since α is proper, α ( K ) is relatively compact. Consider K (cid:48) = α ( K ) and pick ( x, y ) ∈ D ( α ) \ K (cid:48) × K (cid:48) . Observe that neither x ∈ K nor y ∈ K .Suppose, without loss of generality, that x (cid:54)∈ K (cid:48) . Let U ∈ α be such that x, y ∈ U . Since x ∈ U and x (cid:54)∈ α ( K ) = (cid:83) U ∩ K (cid:54) = ∅ U ∈ α U , we have that U ∩ K = ∅ . Then, d( x, y ) ≤ diam U < ε .Hence, D ( α ) ∈ E . Lemma 12.
Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack and consider ( (cid:98) X, E ) .Fix a sequence of open subsets { W n } ∞ n =0 of (cid:101) X with W = (cid:101) X , W ⊃ W ⊃ W ⊃ W ⊃ W ⊃ . . . and X = (cid:84) ∞ n =0 W n .Let { β n } ∞ n =0 be a sequence of families of open subsets of (cid:101) X with β = { (cid:101) X } , X ⊂ (cid:83) U ∈ β n U for every n and lim m,n →∞ mesh { U ∩ W m : U ∈ β n } = 0 .Consider α ( { β n } , { W n } ) = { U ∩ ( W n \ W n +2 ) : U ∈ β n , n ≥ } . Then:a) α ( { β n } , { W n } ) is an open and uniform family of subsets of (cid:98) X .b) If each β n if finite, α ( { β n } , { W n } ) is locally finite.c) If γ is a uniform family of subsets of (cid:98) X , then there exists a subsequence { n k } ∞ k =0 with n = 0 such that γ ≺ α ( { β k } , { W n k } ) . α (cid:0) { β n } , { W n } (cid:1) Proof.
Observe that if K is a compact subset of (cid:98) X , then { K ∩ W n } ∞ n =0 is a sequence ofnested compact subsets such that (cid:84) ∞ n =0 K ∩ W n = ∅ and, hence there exists N ∈ N with K ∩ W N = ∅ . Equivalently, if U is an open subset of (cid:101) X containing X , then there exist N ∈ N such that W N ⊂ U .For short, let us denote α ( { β n } , { W n } ) by α . Obviously, α is open.Let us see that α is uniform, i. e. D ( α ) ∈ E . Fix ε > and let N ∈ N besuch that for every m, n ≥ N , mesh { U ∩ W m : U ∈ β n } < ε . Put K = (cid:101) X \ W N +2 ,pick ( x, y ) ∈ D ( α ) \ K × K and take V ∈ α with x, y ∈ V . Let n > and U ∈ β n be such that V = U ∩ W n \ W n +2 . Since ( x, y ) (cid:54)∈ K × K , neither x ∈ K nor y ∈ K .Suppose, without loss of generality, that x (cid:54)∈ K , or equivalently, that x ∈ W N +2 . Since x ∈ V ⊂ (cid:101) X \ W n +2 , we have that W N +2 \ W n +2 ⊃ { x } (cid:54) = ∅ and, consequently, n > N .Thus, d( x, y ) ≤ diam U ∩ W n < ε and D ( α ) ∈ E .Fix x ∈ (cid:98) X and take n ≥ such that x ∈ W n \ W n +1 . Consider the neighborhood of xW n \ W n +2 . It is easy to check that { V ∈ α : V ∩ W n \ W n +2 (cid:54) = ∅ } ⊂ { U ∩ ( W k \ W k +2 ) : U ∈ β k , k = n − , n, n + 1 } . Then, if each β n is finite, α is locally finite.Let γ be a uniform family of subsets of ( (cid:98) X, E ) . Let us construct a subsequence { n k } ∞ k =0 with n = 0 such that γ ≺ α (cid:0) { β k } , { W n k } (cid:1) . We will use the characterization of uniformfamilies of subsets given in Lemma 11.Let L n = sup { diam V : V ∈ γ, V ⊂ W n } for every n ∈ N ∪ { } . Let us see that L n → .Fix ε > and suppose that K is a compact subset of (cid:98) X such that diam V < ε for every V ∈ α with V ∩ K = ∅ . Choose N ∈ N with K ∩ W N = ∅ . Fix n ≥ N and V ∈ γ with V ⊂ W n . Since V ∩ K = ∅ , we have that diam V < ε . Consequently, L n ≤ ε for every n ≥ N .Put n = 0 . Assuming that n , · · · , n k − are defined, let us define n k :Choose m > such that W m ⊂ (cid:83) U ∈ β k U . Consider the cover of (cid:101) X β k ∪ { (cid:101) X \ W m } andlet L be its Lebesgue number. Take m (cid:48) such that L n < L for every n ≥ m (cid:48) .Since γ is a uniform cover of E , it is proper. Since (cid:101) X \ W n k − is relatively compactin (cid:98) X , it follows that γ ( (cid:101) X \ W n k − ) is relatively compact in (cid:98) X . Choose m (cid:48)(cid:48) with W m (cid:48)(cid:48) ∩ γ ( (cid:101) X \ W n k − ) = ∅ .Let n k = max { n k − + 1 , m + 1 , m (cid:48) , m (cid:48)(cid:48) } . By definition of m (cid:48)(cid:48) : γ ( (cid:101) X \ W n k − ) ⊂ (cid:101) X \ W n k (7)Let us see that: { V ∈ γ : V ⊂ W n k } ≺ β k (8)Fix V ∈ γ with V ⊂ W n k . Since n k ≥ m (cid:48) , diam V ≤ L n k < L . Then, there exists U ∈ β k ∪ { (cid:101) X \ W m } such that V ⊂ U . Necessarily U ∈ β k , because V ∩ ( X \ W m ) = ∅ .10et us see now that γ ≺ α (cid:0) { β k } , { W n k } (cid:1) . Fix V ∈ γ . Since V is bounded and hencerelatively compact, there exists N with W n ∩ V = ∅ . Then, V ∩ ( (cid:101) X \ W n ) (cid:54) = ∅ for every n ≥ N . Let k = min { k (cid:48) : V ∩ ( (cid:101) X \ W n k (cid:48) ) (cid:54) = ∅ } . By (7), V ⊂ γ ( (cid:101) X \ W n k ) ⊂ (cid:101) X \ W n k +1 (9)If k = 0 , then (9) implies that V ⊂ (cid:101) X \ W n ⊂ (cid:101) X ∩ ( W n \ W n ) ∈ α (cid:0) { β k } , { W n k } (cid:1) .Suppose k ≥ . By definition of k , V ∩ ( (cid:101) X \ W n k − ) = ∅ , that is, V ⊂ W n k − . By (9), V ⊂ W n k − \ W n k +1 .Since V ⊂ W n k − , by applying (8) to k − , we get that there exists U ∈ β k − such that V ⊂ U . Therefore, V ⊂ U ∩ ( W n k − \ W n k +1 ) ∈ α (cid:0) { β k } , { W n k } (cid:1) . Proposition 13.
Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack and consider ( (cid:98) X, E ) .For every uniform family γ of subsets of (cid:98) X , there exists a canonical cover α with γ ≺ α .Proof. Set W = (cid:101) X and β = { (cid:101) X } . Let k = sup x ∈ (cid:101) X d( x, X ) . For every n ∈ N , put W n = B (cid:0) X, k n (cid:1) . By the compactness of X , for every n ∈ N we may choose be a finitesubfamily β n of { B( x, n ) : x ∈ X } such that X ⊂ (cid:83) U ∈ β n U .By Lemma 12, there exists a subsequence { n j } with n = 0 such that γ ∪ {{ x } : x ∈ (cid:98) X } ≺ α (cid:0) { β j } , { W n j } (cid:1) . Moreover, α is open, locally finite and uniform for E .Since γ ∪{{ x } : x ∈ X } is a cover of (cid:98) X , α is a cover too. By Corollary 7, α (cid:0) { β j } , { W n j } (cid:1) is a canonical cover.Now, given a metrizable compactification pack ( X, (cid:98) X, (cid:101) X ) , we relate the dimension of X with the order of the canonical covers in (cid:98) X . Definition 14.
Let Z be a set, suppose x ∈ Z , V ⊂ Z , E ⊂ Z × Z and that α is a familyof subsets of Z . Then: • mult x α = { U ∈ α : x ∈ V }• mult V α = { U ∈ α : V ∩ U (cid:54) = ∅ }• mult E α = sup { mult E x α : x ∈ Z } Definition 15.
Let ( Z, E ) be a coarse space. We say that asdim( Z, E ) ≤ n if ( Z, E ) satisfies any of the following equivalent properties:a) For every uniform cover β there exists a uniform cover α (cid:31) β with mult α ≤ n + 1 .b) For every controlled set E there exists a uniform cover α such that mult E α ≤ n + 1 . The equivalence of the properties above is given in [11] or [12] (Chapter 3.2 or [11],pags 35-39).From [11] or [12] we take the following theorem (Theorem 5.5 or [11], pag. 59), adaptedto our language: 11 heorem 16 (Grave’s theorem) . If ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack, then asdim( (cid:98) X, E ) = dim X + 1 Lemma 17. If α is a family of subsets of a set Z and E, F ⊂ Z , then: mult F E ( α ) ≤ mult E − ◦ F α mult E ( α ) ≤ mult E − α Proof.
Let U ⊂ Z . From (2) and (4) we get the equivalences: U ∩ ( E − ◦ F ) x (cid:54) = ∅ ⇔ U ∩ E − ( F x ) (cid:54) = ∅ ⇔ E ( U ) ∩ F x (cid:54) = ∅ .Then, mult ( E − ◦ F ) x α = { U ∈ α : U ∩ ( E − ◦ F ) x (cid:54) = ∅ } = { U ∈ α : E ( U ) ∩ F x (cid:54) = ∅ } ≥ { V ∈ E ( α ) : V ∩ F x (cid:54) = ∅ } = mult F x E ( α ) . Taking supreme over x we get mult E − ◦ F α ≥ mult F E ( α ) .The second inequality is deduced from the first one taking into account that mult E ( α ) =mult ∆ E ( α ) . Proposition 18.
Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack and consider ( (cid:98) X, E ) .Let n ∈ N ∪ { } . Then,a) asdim( (cid:98) X, E ) ≤ n .b) For every uniform cover β of (cid:98) X there exists an open, locally finite and uniform cover α of (cid:98) X such that β ≺ α and mult α ≤ n + 1 .are equivalent.Proof. Taking into account Corollary 7, it is obvious that b) implies a). Let us see thereciprocal.Consider a uniform cover β of (cid:98) X . Let E be an open, symmetric and controlled neighbor-hood of the diagonal. Since asdim( (cid:98) X, E ) ≤ n , there exists a uniform cover γ of (cid:98) X such that mult D ( β ) ◦ E ( γ ) ≤ n + 1 . Let α = E ◦ D ( β )( γ ) . Then, mult E α = mult E ( E ◦ D ( β )( γ )) ≤ mult E D ( β )( γ ) ≤ mult D ( β ) ◦ E ( γ ) ≤ n + 1 . Particulary, mult α ≤ mult E α ≤ n + 1 .Clearly, α is uniform. To prove that α is open, fix V ∈ α and take U ∈ γ such that V = E ( D ( β )( U )) . Since V = (cid:83) x ∈D ( β )( U ) E x (see (3)) and each E x is open, we have that V is open. Moreover, for every x ∈ (cid:98) X , mult E x α ≤ mult E α ≤ n + 1 < ∞ and we get that α is locally finite.Finally, fix W ∈ β and take U ∈ γ with W ∈ U (cid:54) = ∅ . Then, by (1), W ⊂ (cid:83) W (cid:48) ∈ βU ∩ W (cid:48) (cid:54) = ∅ W (cid:48) = β ( U ) = D ( β )( U ) ⊂ E ◦ D ( β )( U ) ∈ α . Thus, β ≺ α .Therefore, α is the desired cover. Theorem 19.
Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack and consider ( (cid:98) X, E ) .Let n ∈ N ∪ { } . Then, ) dim X ≤ n .b) asdim( X, E ) ≤ n + 1 .c) For every uniform cover β of (cid:98) X there exists a canonical cover α (cid:31) β which multiplicityis less than or equal to n + 2 .d) For every canonical cover β of (cid:98) X there exists canonical cover α (cid:31) β which multiplicityis less than or equal to n + 2 .are equivalent.Proof. Take into account Corollary 7. The equivalence between a) and b) is Grave’stheorem. b) implies c) due to Proposition 18. The implication c) ⇒ d) is obvious. d)implies b) because of Proposition 13.Implications a) ⇒ b), a) ⇒ c) and a) ⇒ d) are also a consequence of Proposition 49. Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack. If (cid:101) X is the Hilbert cube Q or of thefinite dimensional cube [0 , n , Theorem 19 proves (B) partially, but we have to show thatthe multiplicity of every canonical cover is greater than dim X + 2 , and not just for thebigger ones, as stated.In general, just one canonical cover can not say anything about the dimension of X .For example, suppose that (cid:98) X is countable —to be more specific, let Y be a metriz-able compact set, let { y n } be a dense sequence in Y and put X = Y × { } , (cid:98) X = (cid:0)(cid:83) n ∈ N { y , · · · , y n } × { n } (cid:1) and (cid:101) X = X ∪ (cid:98) X , all of them with the topology induced by Y × [0 , —. {{ x } : x ∈ (cid:98) X } is a canonical cover of (cid:98) X whose multiplicity is , indepen-dently on the dimension of X .For proving (A), we need the special topological properties of the Z-sets of Q or [0 , n .By this reason, we will define the cylindrical subsets , a class of subsets with involve theZ-sets of the Hilbert cube or the finite dimensional cube, which have the properties weneed to prove (A). Definition 20.
A subset X of a topological space (cid:101) X is said to be cylindrical if there existsan embedding j : X × [0 , (cid:44) → (cid:101) X such that j ( x,
0) = x for every x ∈ X . Remark 21. • If X has a tubular neighborhood in (cid:101) X , then X is cylindrical in (cid:101) X , but the reciprocalis false (take for example X = { } and (cid:101) X = [ − , ). Definition 22.
We say that a compactification pack ( X, (cid:98) X, (cid:101) X ) is cylindrical if X is acylindrical subset of (cid:101) X . Lemma 23.
Every Z-set of the Hilbert cube is a cylindrical subset. roof. Anderson’s theorem (see [6], Theorem II-11.1 or [1]) states that every homeomor-phism between two Z-sets of Q can be extended to a homeomorphism of Q onto itself.Let X be a Z-set of Q . Since X × { } and X are Z-sets of Q × [0 , ≈ Q and Q respectively and g : X × { } → X , ( x, → x is a homeomorphism, there exists ahomeomorphism h : Q × [0 , → Q which extends g . Particulary, h | X × [0 , : X × [0 , → Q is and embedding and X is a cylindrical subset of Q . Remarks 24. • If h is the function in Lemma 23’s proof and f = h − , then f : Q → Q × [0 , is ahomeomorphism such that f ( X ) ⊂ Q × { } . Since the reciprocal is obvious, we havean easy proof of the known result in infinite dimensional topology: A closed subset X of Q is a Z-set of Q if and only if there exists a homeomorphism f : Q → Q × [0 , such that f ( X ) ⊂ Q × { } . • In Q , to be cylindrical is stronger than to be nowheredense (see (cid:0)(cid:8) n : n ∈ N (cid:9) ∪ { } (cid:1) × Q in [0 , × Q ) and weaker than to be a Z-set (see (cid:8) (cid:9) × Q in [0 , × Q ).Using example VI 2 of [13], it follows easily that: Lemma 25.
Let n ∈ N . The Z-sets of [0 , n are precisely the closed subsets of [0 , n \ (0 , n Lemma 26.
Let n ∈ N . Every Z-set of [0 , n is a cylindrical subset.Proof. Consider the copy [ − , n of [0 , n . Suppose that X is a Z-set of [ − , n . Accord-ing with Lemma 25, X ⊂ [ − , n \ ( − , n . Consider the continuous maps j : X × [0 , → X × (cid:2) , (cid:3) , ( x, t ) → (cid:0) x, − t (cid:1) and j : X × (cid:2) , (cid:3) → [ − , n , ( x, t ) → t · x . It is easy tocheck that j ◦ j : X × [0 , → [ − , n is an embedding such that j ◦ j ( x,
0) = x forevery x ∈ X . Lemma 27.
Let X be a compact metric space with dim X ≥ n < ∞ . Then, there existsan ε > such that every open and finite cover α of X × [0 , witha) mesh { π ( U ) : U ∈ α } < ε (where π : X × [0 , → X is the projection).b) there does not exist any U ∈ α which intersect both X × { } and X × { } .satisfies mult α ≥ n + 2 .Proof. Remember that if X is a compact Hausdorff space, then dim X × [0 ,
1] = dim X + 1 .It is a corollary of [14], page 194. Also, from Theorem 7, Theorem 8 or Theorems 4-6 of[18].Consider on X × [0 , the supremum metric. Since dim X × [0 ,
1] = dim X + 1 ≥ n + 1 ,there exists ε > such that, for every open cover β of X × [0 , with mesh β < ε , we have mult β ≥ n + 2 . 14et α be an open cover of X satisfying properties a) and b) for ε . Take k ∈ N with k < ε . Let us construct on X × [0 , k ] a cover γ like in the figure 3.Figure 3. Cover α of X × [0 , and cover γ of X × [0 , k ] Let α (cid:48) be the symmetric cover of α on X × [0 , given by α (cid:48) = { φ ( U ) : U ∈ α } ,where φ : X × [0 , → X × [0 , is the symmetry φ ( x, t ) = ( x, − t ) . Pull forward thecover α to the intervals [2 j, j + 1] , for j = 0 , . . . , k − , by means of the translations f j : X × R → X × R , f j ( x, t ) = ( x, t + 2 j ) and pull forward the cover α (cid:48) to the intervals [2 j + 1 , j + 2] , for j = 0 , . . . , k − , by means of the translations f (cid:48) j : X × R → X × R , f (cid:48) j ( x, t ) = ( x, t + 2 j + 1) . Let γ be the cover of X × [0 , k ] given by the union of thosecovers joining every pulled U ∈ α which meets X × { i } , for i = 1 , . . . , k − , with theirreflections on the pulled subsets of α (cid:48) .More accurately, γ = { f j ( U ) : U ∈ α, ≤ j ≤ k − , f j ( U ) ∩ X × { i } = ∅ ∀ i = 1 , . . . , k − }∪{ f (cid:48) j ◦ φ ( U ) : U ∈ α, ≤ j ≤ k − , f (cid:48) j ◦ φ ( U ) ∩ X × { i } = ∅ ∀ i = 1 , . . . , k − }∪{ f j ( U ) ∪ f (cid:48) j ◦ φ ( U ) : U ∈ α, U ∩ X × { } (cid:54) = ∅ , ≤ j ≤ k − }∪{ f (cid:48) j ◦ φ ( U ) ∪ f j +1 ( U ) : U ∈ α, U ∩ X × { } (cid:54) = ∅ , ≤ j ≤ k − } Let γ be the cover of X × [0 , given by γ = { ψ ( U ) : U ∈ γ } , where ψ : X × [0 , k ] → X × [0 , is the homothety ψ ( x, t ) = (cid:0) x, k t (cid:1) . It is easy to check that γ is an open coverof X × [0 , with mesh γ < ε which has the same multiplicity as α . By definition of ε , weget mult α = mult γ ≥ n + 2 . Remark 28. • We get an easy generalization of Lemma 27 by changing “there exists ε > ” by “thereexists an open and finite cover of X α ” and “ mesh π ( α ) < ε ” by “ π ( α ) ≺ α ”. Hint:if X and Y are compact spaces and α is an open and finite cover of X × Y , thenthere exists two open and finite covers β and β of X and Y respectively such that { U × V : U ∈ β , V ∈ β } ≺ α .The following result is based on Theorem 5.9 of [11] (pag. 59):15 emma 29. Let X be a compact metric space. Consider the compactification pack ( X ×{ } , X × (0 , , X × [0 , and consider ( X × (0 , , E ) . If α is an open and uniform coverof X × (0 , , then dim X ≤ mult α − . Particulary, it happens if α is a canonical cover.Proof. If mult α = ∞ , the result is obvious. Suppose now that mult α = n < ∞ . To geta contradiction, assume that dim X ≥ n − . By Lemma 27, there exists ε > such thatevery open cover β of X × [0 , satisfying properties a) and b) of that lemma, also satisfies mult β ≥ n + 1 .Take into account the characterization of uniform covers of Proposition 11. Consideron X × [0 , the supremum metric. Since α is uniform, there exists a compact subset of X × (0 , such that diam U < ε for every U ∈ α with K ∩ U = ∅ . Take δ > such that K ⊂ X × ( δ , .Since α is proper, α ( X × { δ } ) is relatively compact. Take δ > such that α ( X ×{ δ } ) ⊂ X × ( δ , . Let γ = { U ∩ X × [ δ , δ ] : U ∈ α } and consider the cover γ of X × [0 , given by γ = φ − ( γ ) , where φ : X × [0 , → X × [ δ , δ ] is the homeomorphism φ ( x, t ) = (cid:0) x, tδ + (1 − t ) δ (cid:1) .Clearly, γ is an open cover of X × [0 , such that mult γ ≤ mult α ≤ n , mesh π ( γ ) < ε and no V ∈ γ intersects both X × { } and X × { } . By Lemma 27, mult γ ≥ n + 1 , incontradiction with mult γ = mult γ ≤ mult α ≤ n . Hence, dim X ≤ n − . Proposition 30.
Let ( X, (cid:98) X, (cid:101) X ) be a metrizable cylindrical compactification pack. If α isan open and uniform cover of ( (cid:98) X, E ) , then dim X ≤ mult α − .Proof. Consider the compactification pack ( X × { } , X × (0 , , X × [0 , and suppose E (cid:48) = E ( X × { } , X × (0 , , X × [0 , .Let d be a metric on (cid:101) X and let j : X × [0 , → (cid:101) X be an embedding such that j ( x,
0) = x for every x ∈ X . Consider on X × [0 , the metric d (cid:48) given by d (cid:48) ( a, b ) = d( j ( a ) , j ( b )) .Let β = j | − X × (0 , ( α ) . From the continuity of j we get that β is an open cover over X × [0 , and, from the inyectivity of j , that mult β ≤ mult α .Figure 4. A part of a uniform cover α or ( (cid:98) X, E ) ,the induced cover in j ( X × (0 , and β = j | − X × (0 , ( α ) β is uniform for E (cid:48) . Let ε > . Since α is uniform for E , there existsa compact subset K of (cid:98) X such that d( x, y ) < ε whenever ( x, y ) ∈ D ( α ) \ K × K .Let K (cid:48) = j − ( K ) . Observe that K (cid:48) ⊂ j − ( (cid:98) X ) = X × (0 , . Moreover, K (cid:48) is compact,because it is a closed subset of X × [0 , . Finally, for every ( a, b ) ∈ D ( β ) \ K (cid:48) × K (cid:48) , wehave that ( j ( a ) , j ( b )) ∈ D ( α ) \ K × K and, consecuently, d (cid:48) ( a, b ) = d( j ( a ) , j ( b )) < ε .Since β is an open and uniform cover of ( X × (0 , , E (cid:48) ) , Lemma 29 shows that dim X ≤ mult α − . Proposition 31.
Let ( X, (cid:98) X, (cid:101) X ) be a cylindrical metrizable compactification pack (in par-ticular, if (cid:101) X is Q or [0 , n , with n ∈ N , and X is a Z-set of (cid:101) X ). Consider ( (cid:98) X, E ) .Then, the following properties are equivalent:a) dim X ≤ n .b) asdim( (cid:98) X, E ) ≤ n + 1 .c) For every uniform cover β , there exists a canonical cover α such that β ≺ α and mult α ≤ n + 2 .d) For every canonical cover β , there exists a canonical cover α such that β ≺ α and mult α ≤ n + 2 .e) There exists a canonical cover α with mult α ≤ n + 2 .f ) There exists an open and uniform cover α of ( X, (cid:98) X, (cid:101) X ) with mult α ≤ n + 2 .Proof. The equivalences between a),b),c) and d) are given in Proposition 19. The impli-cations d) ⇒ e) ⇒ f) are obvious. f) implies a) because of Proposition 30. Let us start with the following lemma to get a corollary from Proposition 29:
Lemma 32. If α and β are families of subsets of a set Z and there is a surjective map φ : α (cid:16) β such that U ⊃ φ ( U ) for every U ∈ α , then mult β ≤ mult α .Proof. mult β = sup { B : B ⊂ β, (cid:84) V ∈ B V (cid:54) = ∅ } ≤ sup { φ − ( B ) : B ⊂ β, (cid:84) U ∈ φ − ( B ) U (cid:54) = ∅ } ≤ sup { A : A ⊂ α, (cid:84) U ∈ A U (cid:54) = ∅ } = mult α . Corollary 33.
Let ( X, (cid:98) X, (cid:101) X ) be a cylindrical metrizable compactification pack and con-sider ( (cid:98) X, E ) . Then,a) If α is a uniform cover (cid:98) X which has an open refinement, then mult α ≥ asdim( (cid:98) X, E )+1 .b) If E is an open and controlled neighborhood of the diagonal of (cid:98) X × (cid:98) X , then mult E α ≥ asdim( (cid:98) X, E ) + 1 for every uniform cover α of (cid:98) X . roof. To prove a), take an open refinement β of α . Let T be the topology of (cid:98) X andconsider the map φ : α → T given by φ ( V ) = (cid:83) U ∈ βU ⊂ V U . Observe that U ⊃ φ ( U ) for each U . Let γ = φ ( α ) . By Lemma 32, mult γ ≤ mult α .Since γ ≺ α , we have that γ is uniform. Moreover, (cid:91) W ∈ γ W = (cid:91) V ∈ α φ ( V ) = (cid:91) V ∈ α (cid:91) U ∈ βU ⊂ V U = (cid:91) U ∈ β U = (cid:98) X Since γ is an open and uniform cover of (cid:98) X , Proposition 29 and Grave’s theorem show that mult α ≥ mult γ ≥ dim X + 2 = asdim( (cid:98) X, E ) + 1 .Now, let us see b). E ( α ) is uniform because E is controlled and α , uniform. By (3), E ( α ) = { E ( V ) : V ∈ α } = { (cid:83) x ∈ V E x : V ∈ α } . Since each E x is open, we get that E ( α ) is open and, since α ≺ E ( α ) , that E ( α ) is a cover of (cid:98) X .From Lemmas 17 and 30 and Grave’s theorem, we get mult E α ≥ mult E ( α ) ≥ dim X +2 ≥ asdim( (cid:98) X, E ) + 1 .Let ( X, (cid:98) X, (cid:101) X ) be a cylindrical and metrizable compactification pack and consider ( (cid:98) X, E ) . Corollary 33 means that, from the asymptotic dimensional point of view, anopen and uniform cover α or a controlled and open neighborhood of the diagonal are bigenough.This observation suggests a question. In the general case when ( X, (cid:98) X, (cid:101) X ) is not neces-sary cylindrical, when is a cover α or a controlled set E big enough from the asymptoticdimensional point of view? The following results will answer this question. Proposition 34.
Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack and suppose that d is a metric on (cid:101) X . If k = sup x ∈ (cid:101) X d( x, X ) , then the map h : (0 , k ] → (0 , k ] , t → sup x ∈ X d( x, (cid:101) X \ B( X, t )) is increasing and satisfies lim t → h ( t ) = 0 .Proof. If t ≤ t (cid:48) , then (cid:101) X \ B( X, t ) ⊃ (cid:101) X \ B( X, t (cid:48) ) and, for every x ∈ X , d( x, (cid:101) X \ B( X, t )) ≤ d( x, (cid:101) X \ B( X, t (cid:48) )) . Taking supreme over x , we get h ( t ) ≤ h ( t (cid:48) ) .Fix ε > . Since X is compact and X ⊂ (cid:83) x ∈ X B (cid:0) X, ε (cid:1) , there exists x , . . . , x r ∈ X such that X ⊂ (cid:83) rj =1 B (cid:0) x j , ε (cid:1) . For every j , pick a point y j ∈ (cid:98) X ∩ B (cid:0) x j , ε (cid:1) . Let δ =min ≤ j ≤ r d( y j , X ) .Fix t < δ . For every x ∈ X , there is j = 1 , . . . , r such that x ∈ B (cid:0) x j , ε (cid:1) . Observe that y j ∈ (cid:101) X \ B( X, t ) and thus, d( x, (cid:101) X \ B( X, t )) ≤ d( x, y j ) ≤ d( x, x j ) + d( x j , y j ) < ε + ε = ε .Taking supreme over x , we get h ( t ) ≤ ε .The following proposition implies Corollary 36. This corollary has been proved inde-pendently by Grave in [11] or [12], by us in [16] and [17] and by Mine and Yamashitain [15]. We add this proposition here because we need the explicit functions used there.Moreover, we get an easy proof of Corollary 36.18 roposition 35. Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack. Consider the com-pactification pack ( X ×{ } , X × (0 , , X × [0 , and the coarse structures E = E ( X, (cid:98) X, (cid:101) X ) and E (cid:48) = E ( X × { } , X × (0 , , X × [0 , . Let d be a metric on (cid:101) X . Consider the metric d = k d on (cid:101) X , where k = sup x ∈ (cid:101) X d ( x, X ) .Then, there exist an f : ( (cid:98) X, E ) → ( X × (0 , , E (cid:48) ) and a g : ( X × (0 , , E (cid:48) ) → ( (cid:98) X, E ) satisfying • For every x ∈ (cid:98) X f ( x ) = ( z, t ) , with t = d( x, X ) , z ∈ X and d( x, z ) = t . • For every ( z, t ) ∈ X × (0 , , g ( z, t ) = y with y ∈ (cid:101) X \ B( X, t ) and d( y, z ) = d( z, (cid:101) X \ B( X, t )) .in which case, they are coarse equivalences, the one inverse of the other.Proof. Since sup x ∈ (cid:101) X d( x, X ) = 1 , such f and g do exist. To check their coarse equiva-lentness, it suffices to show that they satisfice property c) of Proposition 1. Consider on X × [0 , the metric d (cid:48) (( x, t ) , ( z, s )) = d( x, z ) + | t − s | .Fix E ∈ E and let us see that f × f ( E ) ∈ E (cid:48) . Let ε > and suppose that K is acompact subset of (cid:98) X such that d( x, x (cid:48) ) < ε whenever ( x, x (cid:48) ) ∈ E \ K × K .Let δ = min { ε , d( K, X ) } and consider K (cid:48) = X × [ δ, . Pick (( z, t ) , ( z (cid:48) , t (cid:48) )) ∈ f × f ( E ) \ K (cid:48) × K (cid:48) and take ( x, x (cid:48) ) ∈ E such that f ( x ) = ( z, t ) and f ( x (cid:48) ) = ( z (cid:48) , t (cid:48) ) . Suppose,without loss of generality, that t ≤ t (cid:48) .Neither ( z, t ) ∈ K (cid:48) nor ( z (cid:48) , t (cid:48) ) ∈ K (cid:48) , that is, either t < δ or t (cid:48) < δ . Then, d( x, X ) = t < δ ≤ d( K, X ) and thus, x (cid:54)∈ K . Hence ( x, x (cid:48) ) ∈ E \ K × K and d( x, x (cid:48) ) < ε . Moreover, t (cid:48) = d( x (cid:48) , X ) ≤ d( x (cid:48) , z ) ≤ d( x (cid:48) , x ) + d( x, z ) = d( x (cid:48) , x ) + t < d( x (cid:48) , x ) + δ . Therefore, d (cid:48) (( x, t ) , ( z, s )) = d( z, z (cid:48) ) + | t − t (cid:48) | ≤ d( z, x ) + d( x, x (cid:48) ) + d( x (cid:48) , z (cid:48) ) + t (cid:48) − t = t + d( x, x (cid:48) ) + t (cid:48) + t (cid:48) − t = d( x, x (cid:48) ) + 2 t (cid:48) ≤ x, x (cid:48) ) + 2 δ < ε + 2 ε = ε and we get f × f ( E ) ∈ E (cid:48) .Fix F ∈ E (cid:48) and let us see that ( f × f ) − ( F ) ∈ E . Let ε > and suppose that K (cid:48) isa compact subset of X × (0 , such that d (cid:48) (( z, t ) , ( z (cid:48) , t (cid:48) )) < ε whenever (( z, t ) , ( z (cid:48) , t (cid:48) )) ∈ F \ K (cid:48) × K (cid:48) . Take δ such that K (cid:48) ⊂ X × [ δ , . Put δ = min { δ , ε } and put K = (cid:101) X \ B (
X, δ ) . Pick ( x, x (cid:48) ) ∈ ( f × f ) − ( F ) \ K × K and take (( z, t ) , ( z (cid:48) , t (cid:48) )) ∈ F such that f ( x ) = ( z, t ) and f ( x (cid:48) ) = ( z (cid:48) , t (cid:48) ) . Suppose, without loss of generality, that t ≤ t (cid:48) .Neither x ∈ K nor x (cid:48) ∈ K , that is, either t = d( x, X ) < δ or t (cid:48) = d( x (cid:48) , X ) < δ . Then, t < δ ≤ δ and thus ( z, t ) (cid:54)∈ K (cid:48) . Hence, (( z, t ) , ( z (cid:48) , t (cid:48) )) ∈ E \ K (cid:48) × K (cid:48) and d (cid:48) (( z, t ) , ( z (cid:48) , t (cid:48) )) < ε . Therefore, d( x, x (cid:48) ) ≤ d( x, z ) + d( z, z (cid:48) ) + d( z (cid:48) , x (cid:48) ) = t + d( z, z (cid:48) ) + t (cid:48) = 2 t + d( z, z (cid:48) ) + | t − t (cid:48) | < δ + d (cid:48) (( z, t ) , ( z (cid:48) , t (cid:48) )) < ε + ε = ε ( f × f ) − ( F ) ∈ E .Let G = { (( z, t ) , f ◦ g ( z, t )) : ( z, t ) ∈ X × (0 , } and let us see that G ∈ E (cid:48) . Fix ε > .Consider the function h : (0 , → (0 , , t → sup x ∈ X d( x, (cid:101) X \ B( X, t )) . By Proposition 34, lim t → h ( t ) = 0 , so there exists δ > such that h ( t ) < ε when t < δ .Let K = X × [ δ, and pick (( x, t ) , ( z, s )) ∈ G \ K × K . Then, either t < δ or s < δ .Observe that ( z, s ) = f ◦ g ( x, t ) . Put y = g ( x, t ) , so we have ( z, s ) = f ( y ) . Since y ∈ (cid:101) X \ B( X, t ) , it follows that s = d( y, X ) ≥ t and thus, t < δ . Therefore, d (cid:48) (( x, t ) , ( z, s )) = d( x, z ) + | t − s | ≤ d( x, y ) + d( y, z ) + s − t = d( x, y ) + 2d( y, z ) − t < d( x, y ) + 2d( y, X ) ≤ x, y ) =3d( x, (cid:101) X \ B( X, t )) ≤ h ( t ) < ε = ε and we get G ∈ E (cid:48) . Therefore, f is a coarse equivalence and g is its coarse inverse. Corollary 36. If ( X , (cid:98) X , (cid:101) X ) and ( X , (cid:98) X , (cid:101) X ) are two metrizable compactification packssuch that X and X are homeomorphic, then ( (cid:98) X , E ) and ( (cid:98) X , E ) are coarse equivalentwhere, for i = 1 , , E i = E ( X i , (cid:98) X i , (cid:101) X i ) .Proof. For i = 1 , , consider the compactification pack ( X i × { } , X i × (0 , , X i × [0 , and suppose E (cid:48) i = E ( X i × { } , X i × (0 , , X i × [0 , .Let h : X → X be a homeomorphism and let d be a metric on X . Consider on X × [0 , and X × [0 , the metrics d and d respectively, given by: d (( x, t ) , ( y, s )) = d( x, y ) + | t − s | d (( x, t ) , ( y, s )) = d( h − ( x ) , h − ( y )) + | t − s | Using d and d , it is easy to check that the map h (cid:48) : X × (0 , → X × (0 , , ( x, t ) → ( h ( x ) , t ) satisfies property d) of Proposition 1 and hence, h (cid:48) is a coarse equivalence.Finally, from Proposition 35, we get ( (cid:98) X , E ) ≈ ( X × (0 , , E (cid:48) ) ≈ ( X × (0 , , E (cid:48) ) ≈ ( (cid:98) X , E ) Proposition 37.
Let ( X, (cid:98) X, (cid:101) X ) be a compactification pack, let d be a metric on (cid:101) X , let E ⊂ (cid:98) X × (cid:98) X and let k = sup x ∈ (cid:101) X d( x, X ) . Then, E ∈ E if and only if there exists φ : (0 , k ] → R + with lim t → φ ( t ) = 0 such that E ⊂ { ( x, y ) ∈ X : d( x, y ) < φ (min { d( x, X ) , d( y, X ) } ) } Proof.
Suppose such φ exists. Fix ε > and take δ > such that φ ( t ) < ε for every t < δ .Let K = (cid:101) X \ B( X, δ ) and pick ( x, y ) ∈ E \ K × K . Then, neither x ∈ K nor y ∈ K . In20ny case, min { d( x, X ) , d( y, X ) } < δ . Hence, d( x, y ) ≤ φ (min { d( x, X ) , d( y, X ) } ) < ε and E ∈ E .Now, assume that E ∈ E . For every t ∈ (0 , k ] , let K t = (cid:101) X \ B( X, t ) and let φ ( t ) = t + sup { d( x, y ) : ( x, y ) ∈ E \ K t × K t } (if E \ K t × K t is empty, put φ ( t ) = t ) . Let us seethat lim t → φ ( t ) = 0 . Fix ε > , consider a compact subset K of (cid:98) X such that d( x, y ) < ε whenever ( x, y ) ∈ E \ K × K and set δ = d( x, K ) . Fix t < δ and pick ( x, y ) ∈ E \ K t × K t .Since K ⊂ K t and hence ( x, y ) ∈ E \ K × K , we have that d( x, y ) < ε , t < ε and φ ( t ) < ε .Pick a point ( x, y ) ∈ E and put t = min { d( x, X ) , d( y, Y ) } . Since ( x, y ) ∈ E \ K t × K t ,we have that d( x, y ) < φ ( t ) = φ (min { d( x, X ) , d( y, X ) } ) . Proposition 38.
Let ( X, (cid:98) X, (cid:101) X ) be a compactification pack, let d be a metric on (cid:101) X , let E ⊂ (cid:98) X × (cid:98) X and let k = sup x ∈ (cid:101) X d( x, X ) . Then, E is a neighborhood of the diagonal if andonly if there exists an increasing function λ : (0 , k ] → R + such that { ( x, y ) ∈ X : d( x, y ) < λ (min { d( x, X ) , d( y, X ) } ) } ⊂ E Proof.
Suppose such λ exists. Let λ : (0 , k ] → R + be an increasing and continuousmap such that λ ( t ) ≤ λ ( t ) for every t . For example, we may define λ as follows: set λ (cid:0) kn (cid:1) = λ (cid:0) kn +1 (cid:1) for every n ∈ N and extend λ linearly to every interval (cid:2) kn , kn +1 (cid:3) .Let F = ψ − ( R + ) , where ψ is the continuous function ψ : (cid:98) X × (cid:98) X → R , ( x, y ) → λ (min { d( x, X ) , d( y, X ) } ) − d( x, y ) . Then, F is an open subset of (cid:98) X × (cid:98) X containing thediagonal such that F ⊂ { ( x, y ) ∈ X : d( x, y ) < λ (min { d( x, X ) , d( y, X ) } ) } ⊂ E .Now, suppose E is a neighborhood of the diagonal. Consider the supremum metric d ∞ on (cid:101) X × (cid:101) X . Take an open subset F ⊂ E containing the diagonal. By the closenessof (cid:98) X × (cid:98) X \ F , we may define the map f : X → R + , x → d ∞ (( x, x ) , (cid:98) X × (cid:98) X \ F ) and it iscontinuous. For every k ∈ (0 , k ] , K t is compact and then, f has a minimum in K t , so wemay define the map λ : (0 , k ] → R + , t → min t ∈ K t f ( t ) .For every t ≤ t (cid:48) , K t ⊃ K t (cid:48) , hence λ ( t ) ≤ λ ( t (cid:48) ) and we get that λ is increasing. Let ( x, y ) ∈ (cid:98) X × (cid:98) X be such that d( x, y ) < λ (min { d( x, X ) , d( y, X ) } ) . Suppose, without loss ofgenerality, that d( x, X ) ≤ d( y, X ) and put t = d( x, X ) . Since x ∈ K t , we have d ∞ (( x, x ) , ( x, y )) < λ ( t ) ≤ d ∞ (( x, x ) , (cid:98) X × (cid:98) X \ F ) Then, ( x, y ) (cid:54)∈ (cid:98) X × (cid:98) X \ F and we conclude that ( x, y ) ∈ F ⊂ E . Lemma 39.
Let f : Z → Z (cid:48) be a map between two sets, let E ⊂ Z × Z , and let α be afamily of subsets of Z (cid:48) . Then, mult E f − ( α ) ≤ mult f × f ( E ) α .Proof. Fix x ∈ Z and take U ∈ α such that f − ( U ) meets E x . Then, by (6), ∅ (cid:54) = f ( f − ( U ) ∩ E x ) = U ∩ f ( E x ) ⊂ U ∩ (cid:83) z ∈ f − ( f ( x )) f ( E z ) = U ∩ ( f × f ( E )) f ( x ) .Hence, mult E x f − ( α ) = { f − ( U ) : U ∈ α, f − ( U ) ∩ E x (cid:54) = ∅ } ≤ { U : U ∈ α, U ∩ ( f × f ( E )) f ( x ) (cid:54) = ∅ } ≤ mult ( f × f ( E )) f ( x ) α ≤ mult f × f ( E ) α . Taking supreme over x we get the inequality. 21 roposition 40. Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack, let d be a metricon (cid:101) X , let k = sup x ∈ (cid:101) X d( x, X ) and let λ : (0 , k ] → R + be an increasing and continuousfunction such that lim t → λ ( t ) = 0 .Consider the maps h : R + → (0 , k ] and φ : (0 , k ] → R + given by: h ( t ) = (cid:40) sup x ∈ X d( x, (cid:101) X \ B( X, t ) if t ≤ kk if t ≥ kφ ( t ) = h ( t ) + λ ( t ) + h ( t + λ ( t )) and consider the set: E d ,λ = { ( x, y ) ∈ (cid:98) X × (cid:98) X : d( x, y ) < φ (min { d( x, X ) , d( y, X ) } ) } Then, E d ,λ is a controlled subset of ( (cid:98) X, E ) such that mult E d ,λ α ≥ dim X + 2 for everyuniform cover α of ( (cid:98) X, E ) Proof.
By Proposition 34, φ ( t ) → when t → . Hence, from Proposition 37, we get E d ,λ ∈ E . Let α be a uniform cover of ( (cid:98) X, E ) .Consider the compactification pack ( X × { } , X × (0 , , X × [0 , and the coarse space ( (cid:98) X, E (cid:48) ) , where E (cid:48) = E ( X × { } , X × (0 , , X × [0 , . Consider on (cid:101) X the metric d = k d and, on X × [0 , , the metric d ∞ (( x, t ) , ( x (cid:48) , t (cid:48) )) = max { d( x, x (cid:48) ) , | t − t (cid:48) |} .By Proposition 35, there is a coarse equivalence g : X × (0 , → (cid:98) X such that forevery ( x, t ) , g ( x, t ) = y where y ∈ (cid:101) X \ B d ( X, t ) is such that d( x, y ) = d( x, (cid:101) X \ B d ( X, t )) .Consider the function λ : (0 , → R + such that for every t , λ ( t ) = 1 k λ ( kt ) (10)and consider the set F = (cid:8) (( x, t ) , ( x (cid:48) , t (cid:48) )) ∈ ( X × (0 , : d ∞ (( x, t ) , ( x (cid:48) , t (cid:48) )) < λ (min { t, t (cid:48) } ) (cid:9) .Since, for each ( x, t ) , we have t = d ∞ (( x, t ) , X × { } ) , Proposition 37 shows F ∈ E and Proposition 38, that F is a neighborhood of the diagonal. Let F ⊂ F be a controlledand open neighborhood of the diagonal and let β = F ( g − ( α )) .Let us see that:i) β is an open and uniform cover of ( X × (0 , , E (cid:48) ) .ii) g × g ( F ) ⊂ E d ,λ .For each V ∈ β , V = F ( U ) = (cid:83) x ∈ U F x for some U ∈ g − ( α ) . Since each F x is open, V is open. Moreover, g − ( α ) ≺ β and hence β is a cover of X × (0 , . Finally, by Corollary3, β = F ( g − ( α )) is uniform for E (cid:48) and we get i).Let us consider: • the map h : R + → (0 , such that h ( t ) = sup x ∈ X d( x, (cid:101) X \ B d ( X, t )) when t ≤ and h ( t ) = 1 when t ≥ the map φ : (0 , → R + , given by φ ( t ) = h ( t ) + λ ( t ) + h ( t + λ ( t )) • the set E d ,λ = { ( x, y ) ∈ X : d( x, y ) < φ (min { d( x, X ) , d( y, X ) } ) } By Proposition 34, h and φ are increasing. It is easy to check that, for every t , h ( t ) = k h ( kt ) and φ ( t ) = k φ ( kt ) . Using those equalities and (10), it follows easily that E d ,λ = E d ,λ . Then, to prove ii), it suffices to show that g × g ( F ) ⊂ E d ,λ .Pick ( y, y (cid:48) ) ∈ g × g ( F ) and take (( x, t ) , ( x (cid:48) , t (cid:48) )) ∈ F such that g ( x, t ) = y and g ( x (cid:48) , t (cid:48) ) = y (cid:48) . Suppose, without loss of generality, that t ≤ t (cid:48) . Observe that t (cid:48) = t + | t − t (cid:48) | Let ( X, (cid:98) X, (cid:101) X ) be a compactification pack and consider ( (cid:98) X, E ) . Let E bethe controlled set E d ,λ of Proposition 40. Then:a) mult E α ≥ asdim( (cid:98) X, E ) + 1 for every uniform cover α of (cid:98) X .b) mult α ≥ asdim( (cid:98) X, E ) + 1 for every uniform cover α of (cid:98) X such that K ( E ) ≺ α .Proof. Let α be a uniform cover of (cid:98) X . By Proposition 40 and Grave’s theorem, mult E α ≥ dim X + 2 = asdim( (cid:98) X, E ) + 1 , so we get a).To see b), suppose K ( E ) ≺ α . For every U ∈ α , let V U = { x ∈ (cid:98) X : E x ⊂ U } and let γ = { V U : U ∈ α } . Observe that, by (3), E ( V U ) = (cid:83) x ∈ V U E x ⊂ U for every U . Hence, γ ≺ α and we get that γ is uniform. For all x ∈ (cid:98) X , there is U ∈ α such that E x ⊂ U and,consecuently, x ∈ V U . Then, γ is a cover of (cid:98) X .Fix x ∈ (cid:98) X and let U ∈ α be such that E x ∩ V U (cid:54) = ∅ . Let y ∈ E x ∩ V U . Since E is symmetric, x ∈ E y and, since y ∈ V U , we have that E y ⊂ U . Then, x ∈ U . Hence,23 ult E x γ = { V U : U ∈ α, E x ∩ V U (cid:54) = ∅ } ≤ { U : U ∈ α, x ∈ U } = mult x α . Takingsupreme over x and applying a) we get: mult E γ ≤ mult α ≤ asdim( (cid:98) X, E ) + 1 Corollary 41 means that, from the point of view of the asymptotic dimension, the set E d ,λ of Proposition 40 and the cover K ( E d ,λ ) are big enough. Proposition 42. Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack and consider ( (cid:98) X, E ) .Then, the following are equivalent:a) dim X ≤ n .b) asdim( (cid:98) X, E ) ≤ n + 1 .c) For every uniform cover β there exists a canonical cover α such that β ≺ α and mult α ≤ n + 2 .d) For every canonical cover β there exists a canonical cover α such that β ≺ α and mult α ≤ n + 2 .e) There exists a uniform cover α (a canonical cover, respectively) such that mult E α ≤ n + 1 , where E is the subset E d ,λ of Proposition 40.Proof. It is a consequence of Corollary 7 and Propositions 19 and 40. Let ( X, (cid:98) X, (cid:101) X ) be a compactification pack. In order to prove Grave’s theorem, we have tosee that: asdim( (cid:98) X, E ) ≥ dim X + 1 (12) asdim( (cid:98) X, E ) ≤ dim X + 1 (13) Proof 1 of (12). It is a consequence of Proposition 40.But it is not the natural way to prove (12). The natural way (more or less the wayused by Grave applying other result instead of Proposition 29) is the following: Proof 2 of (12). Consider ( X × (0 , , E (cid:48) ) , where E (cid:48) = E ( X × { } , X × (0 , , X × [0 , ).Since X and X × { } are homeomorphic, by Corollary 36, it follows that ( (cid:98) X, E ) and ( X × (0 , , E (cid:48) ) are coarse equivalent. Therefore, asdim( (cid:98) X, E ) = asdim( X × (0 , , E (cid:48) ) ≥ dim X + 1 , where the last inequality is given by Propositions 13 and 29.24n Theorem 2.9 of [5] (pag. 3712), the authors defined a canonical cover similar to α ( { β n } , { W n } ) of Lemma 12. They used it to prove that if X has finite dimension, thenthere exists a canonical cover with finite multiplicity. They bounded the multiplicity ofthat canonical cover by X + 2 .The construction of that canonical cover induces an implicit problem in the Ttopolog-ical Dimension Theory: To decrease the bound of the multiplicite of that canonical coverto the minimum (that is dim X + 2 ) we need the sequence { β n } to satisfice some specialdimensional properties. Does this special sequence exists?We solved that topological problem in dimension theory in [16] —we quote it in The-orem 48—. With this techniques, we will be able to give a new proof of (13) and, at thesame time, define a canonical cover with minimal multiplicity. Definition 43. Let α , . . . , α m be families of subsets of a set Z . The common multiplicityof α , . . . , α m is: mult( α , · · · , α m ) = sup x ∈ Z mult x α + · · · + mult x α m =sup (cid:40) m (cid:88) i =1 A i : A i ⊂ α i ∀ i, m (cid:92) i =1 (cid:92) U ∈ A i U (cid:54) = ∅ (cid:41) Remark 44. • mult( α , · · · , α m ) is a multiplicity greater than or equal to mult( α ∪ · · · ∪ α m ) whichis equal when α , . . . , α r are pairwise disjoint. Proposition 45. Let ( (cid:101) X, d) be a metric space and consider X ⊂ (cid:101) X . Consider the topology T of (cid:101) X attached to d and the map v : T | X → T , U → { x ∈ (cid:101) X : d( x, U ) < d( x, X \ U ) } (assuming that d( x, ∅ ) = ∞ for every x ∈ (cid:101) X ). Then:a) For every U ∈ T | X , v ( U ) ∩ X = U .b) v ( X ) = (cid:101) X and v ( ∅ ) = ∅ .c) For every U , U ∈ T | X , U ⊂ U if and only if v ( U ) ⊂ v ( U ) .d) For every U , . . . , U r ∈ T | X , U ∩ · · · ∩ U r (cid:54) = ∅ if and only if v ( U ) ∩ · · · ∩ v ( U r ) (cid:54) = ∅ .e) For every U ∈ T | X , U = ∅ if and only if v ( U ) = ∅ f ) For every U , U ∈ T | X , v ( U ∩ U ) = v ( U ) ∩ v ( U ) .Proof. a)-c) are easy to check. e) is a consequence of a) and b). d) is a consequence of e)and f). It suffices to prove f).Since U ∩ U ⊂ U , c) shows that v ( U ∩ U ) ⊂ v ( U ) . By the same reason, v ( U ∩ U ) ⊂ v ( U ) . Hence, v ( U ∩ U ) ⊂ v ( U ) ∩ v ( U ) . 25ix x ∈ v ( U ) ∩ v ( U ) . Then, d( x, U ) < d ( x, X \ U ) and d( x, U ) < d ( x, X \ U ) .Choose ε > such that d( x, X \ U ) − d( x, U ) > ε and d( x, X \ U ) − d( x, U ) > ε .Take y ∈ X such that d( x, y ) < d( x, X ) + ε . For every z ∈ X \ U , we have d( y, z ) ≥ d( x, z ) − d( x, y ) > d( x, X \ U ) − (cid:0) d( x, X ) + ε (cid:1) ≥ d( x, X \ U ) − d( x, U ) − ε > ε − ε = ε .Then, d( y, X \ U ) > ε and hence, y ∈ U . By the same reason, y ∈ U . Therefore: y ∈ X and d( y, x ) < d( x, X ) + ε ⇒ y ∈ U ∩ U (14)From (14) we deduce d (cid:0) x, X \ ( U ∩ U ) (cid:1) ≥ d( x, X ) + ε (15)Fix δ > and take y (cid:48) ∈ X such that d( x, y (cid:48) ) < d( x, X ) + min (cid:8) δ, ε (cid:9) . By (14), y (cid:48) ∈ U ∩ U . Hence, d( x, X ) ≤ d( x, U ∩ U ) < d( x, X ) + δ for every δ > and we get d( x, U ∩ U ) = d( x, X ) . We conclude from (15) that d( x, U ∩ U ) < d (cid:0) x, X \ ( U ∩ U ) (cid:1) , hence that x ∈ v ( U ∩ U ) and finally that v ( U ) ∩ v ( U ) ⊂ v ( U ∩ U ) . Remarks 46. • The function v defined above is called “Ext” in [21], pag 125. • The function described in Proposition 2.7 of [5], pag 3711, satisfies properties a)-d)of proposition . Proposition 47. Let ( X, (cid:98) X, (cid:101) X ) be a metrizable compactification pack, let T be the topologyof (cid:101) X and consider ( (cid:98) X, E ) .Suppose that { W n } ∞ n =0 is a sequence of open neighborhoods of X such that W = (cid:101) X , W ⊃ W ⊃ W ⊃ W ⊃ W ⊃ . . . and (cid:84) ∞ n =0 W n = X .Suppose that { α n } ∞ n =0 is a family of open covers of X and let, for every n , β n = { v ( U ) : U ∈ α n } , where v : T | X → T is a map satisfying properties a)-d) of Proposition 46.Consider the cover α ( { β n } , { W n } ) defined in Proposition 12. Then,a) mult α ( { β n } , { W n } ) ≤ sup n ∈ N ∪{ } mult( α n , α n +1 ) .b) If mesh α i → , then lim m,n → mesh { V ∩ W m : V ∈ β n } = 0 and, consequently, α ( { β n } , { W n } ) is a uniform cover of ( (cid:98) X, E ) .Proof. For short, denote α ( { β n } , { W n } ) by α . Property a) of Proposition 46 states that,for every n , v : α n → β n is bijection and property d) states that, for every n , . . . , n r : mult( β n , · · · , β n r ) = sup (cid:40) r (cid:88) k =1 B k : B k ⊂ β n k ∀ k, r (cid:92) k =1 (cid:92) V ∈ B k V (cid:54) = ∅ (cid:41) = up (cid:40) r (cid:88) k =1 { v ( U ) : U ∈ A k } : A k ⊂ α n k ∀ k, r (cid:92) k =1 (cid:92) U ∈ A k v ( U ) (cid:54) = ∅ (cid:41) =sup (cid:40) r (cid:88) k =1 A k : A k ⊂ α n k ∀ k, r (cid:92) k =1 (cid:92) U ∈ A k U (cid:54) = ∅ (cid:41) = mult( α n , · · · , α n r ) (16)Fix x ∈ (cid:98) X . It is easy to check that if G ∈ α with x ∈ G , then G = V ∩ ( W k \ W k − ) ,where V ∈ β k , k ∈ { N, N + 1 } and N = max { n ∈ N : x ∈ W n } . Then, mult x α = { G ∈ α : x ∈ G } ≤ { V ∩ ( W N \ W N − ) : V ∈ β N , x ∈ V } + { V ∩ ( W N +1 \ W N − ) : V ∈ β N , x ∈ V } ≤ { V : V ∈ β N , x ∈ V } + { V : V ∈ β N , x ∈ V } ≤ mult( β N , β N +1 ) =mult( α N , α N +1 ) ≤ sup n ∈ N ∪{ } mult( α n , α n +1 ) . Taking supreme over x we get a).Assume mesh α n → . Suppose mesh { V ∩ W m : V ∈ β n } (cid:54)→ when m, n → ∞ .Then, there exists ε > and two subsequences { n k } and { m k } such that mesh { V ∩ W m k : V ∈ β n k } > ε . For every k , take U k ∈ α k with diam v ( U k ) ∩ W m k > ε and take x k , y k ∈ v ( U k ) ∩ W m k with d( x k , y k ) > ε .Since (cid:101) X is compact, we may suppose, by taking subsequences if necessary, that x k → x and y k → y for some x, y ∈ (cid:101) X . Then, d( x, y ) = lim d( x k , y k ) ≥ ε (17)For every i and every k ≥ i , we have x k , y k ∈ W m k ⊂ W k ⊂ W i and hence, x, y ∈ W i .Thus, x, y ∈ (cid:84) i ∈ N W i = X .Let N ∈ N with mesh α N < ε . Choose U x , U y ∈ α N such that x ∈ U x and y ∈ U y .Observe that v ( U x ) and v ( U y ) are two neighborhoods of x and y respectively. Since mesh α n k → , x k → x and y k → y , it follows that there is k (cid:48) such that mesh α k (cid:48) < ε , x k (cid:48) ∈ v ( U x ) and y k (cid:48) ∈ v ( U y ) . Since v ( U k (cid:48) ) ∩ v ( U x ) ⊃ { x } (cid:54) = ∅ and v ( U k (cid:48) ) ∩ v ( U y ) ⊃ { y } (cid:54) = ∅ ,we have that U k (cid:48) ∩ U x (cid:54) = ∅ and U k (cid:48) ∩ U y (cid:54) = ∅ . Take x (cid:48) ∈ U k (cid:48) ∩ U x and y (cid:48) ∈ U k (cid:48) ∩ U y to get: d( x, y ) ≤ d( x, x (cid:48) ) + d( x (cid:48) , y (cid:48) ) + d( y (cid:48) , y ) ≤ diam U x + diam U k (cid:48) + diam U y ≤ mesh α N + mesh α n k (cid:48) + mesh α N < ε = ε in contradiction with (17).Then, lim m,n →∞ mesh { V ∩ W m : V ∈ β n } = 0 and hence, by Proposition (12), α is auniform cover.From [16] we take the following result: Theorem 48. Let ( X, d) be a compact metric space with dim X ≤ n < ∞ and suppose { ε i } ∞ i =1 ⊂ R + . Then, there exists a sequence of open and finite covers of X { α i } ∞ i =0 suchthat:a) α = { X } and mesh α i < ε i for every i ∈ N . ) mult( α i , α i +1 ) ≤ n + 2 for every i ∈ N ∪ { } Proof. It is a consequence of Theorems 74, 81, 104 or 154 of [16]. Proposition 49. Let ( X, (cid:98) X, (cid:101) X ) be a compactification pack, consider ( (cid:98) X, E ) and let γ bea uniform cover of (cid:98) X . Then, there exists an open, locally finite and uniform cover α (i.e. a canonical cover) such that γ ≺ α and mult α ≤ dim X + 2 .Moreover, given a sequence of open subsets of (cid:101) X { W i } ∞ i =0 with W = (cid:101) X , W ⊃ W ⊃ W ⊃ W ⊃ W ⊃ . . . and (cid:84) ∞ i =0 W i = X , we can construct such α by letting α = α ( { β k } , { W i k } ) (as defined in Proposition 12), where { β k } ∞ k =0 is a sequence of openand finite families of subsets of (cid:98) X such that β = { (cid:101) X } , X ⊂ (cid:83) V ∈ β i V for every i and lim i,j → mesh { V ∩ W j : V ∈ β i } = 0 and { i k } ∞ k =0 is a subsequence with i = 0 .Proof. Let { W i } ∞ i =0 be as above (for example, consider W = (cid:101) X and, for every i ∈ N , W i = B (cid:0) X, k i (cid:1) , where k = sup x ∈ (cid:101) X d( x, X ) ).If dim X = ∞ , this proposition is a consequence of Propositions 13 and 12. If dim X = n < ∞ , by Theorem 12, there exist a sequence of open and finite covers { α i } ∞ i =0 of X with α = { X } , lim i →∞ mesh α i = 0 and mult( α i , α i +1 ) ≤ n + 2 for every i ∈ N ∪ { } .Figure 5. Covers { α i } ∞ i =0 and { α i ∪ α i +1 } ∞ i =0 of [0 , with mult α i ≤ and mult( α i , α i +1 ) ≤ ∀ i ∈ N ∪ { } Let T be the topology of (cid:101) X , consider a map v : T | X → T satisfying properties a)-d)of Proposition 46 and suppose β i = { v ( U ) : U ∈ α i } for every i ∈ N ∪ { } .By Proposition 47, lim i,j →∞ mesh { V ∩ W j : V ∈ β i } = 0 and, by Lemma 12, there28xists a subsequence { i k } ∞ k =0 with i = 0 such that γ ∪ x ∈ (cid:98) X ≺ α ( { β k } , { W i k } ) .Figure 6. Covers { α i } ∞ i =0 of [0 , of figure 5and cover α ( { β k } , { W i k } ) of [0 , × (0 , γ ∪ x ∈ (cid:98) X is a cover of (cid:98) X , so it is α ( { β k } , { W i k } ) . According to 47, α ( { β k } , { W i k } ) is also an open, uniform and locally finite cover of (cid:98) X such that mult α ( { β k } , { W i k } ) ≤ sup i ∈ N ∪{ } mult( α i , α i +1 ) ≤ n + 2 . Particulary, it is a canonical cover of ( X, (cid:98) X, (cid:101) X ) (seeLemma 7). Proof of (13). It is a consequence of Proposition 49. References [1] Anderson, R. D. Topological properties of the Hilbert cube and the infinite product ofopen intervals Trans. Amer. Math. Soc. 126 (1967) 200-216[2] G. Bell, A. Dranishnikov, Asymptotic dimension in Bedlewo . Topology Proc. 38(2011), 209-236[3] G. Bell, A. N. Dranishnikov, Asymptotic dimension . Topology Appl. 155 (2008), no.12, 1265-1296[4] E. Cuchillo Ibáñez, J. Dydak, A. Kodama y M. A. Morón, C coarse geometry ofcomplements of Z-sets in the Hilbert cube . Trans. Amer. Math. Soc. 360 (2008), no.10, 5229–5246[5] E. Cuchillo Ibáñez, M. A. Morón, Canonical covers and dimension of Z-sets in theHilbert cube . Proc. Amer. Math. Soc. 136 (2008), no. 10, 3709–3716[6] T. A. Chapman, Lectures on Hilbert Cube Manifolds . American Mathematical society,1976. Regional Conference Series in Mathematics Vol. 28.297] A. N. Dranishnikov, Asymptotic topology . Russian Math. Surveys 55 (2000), no. 6,1085–1129[8] A. N. Dranishnikov, J. Smith On asymptotic Assouad-Nagata dimension. TopologyAppl. 154 (2007), no. 4, 934-952.[9] James Dugundji, An extension of Tietze’s theorem , Pacific Journal of Mathematics 1(1952) 353-367. MR0044116[10] J. Dydak and S. Hoffland, An alternative definition of coarse structures . TopologyAppl. 155 (2008), no. 9, 1013–1021[11] Bernd Grave, Coarse geometry and asymptotic dimension . Phd Thesis.[12] Bernd Grave Asymptotic dimension of coarse spaces . New York J. Math. 12 (2006)[13] Hurewictz-Wallman, Dimension Theory . Princeton University Press, Princeton, NJ,1941.[14] W. Hurewicz, Sur la dimension des produits Cartésiens , Annals of Mathematics (2),vol. 36 (1935), pp. 194-197[15] Kotaro Mine, Atsushi Yamashita C coarse structures and smirnov compactifications Preprint (2011) arXiv:1106.1672[16] Jesús P. Moreno Damas, Geometría de recubrimientos: Dimensión Topológica y Es-tructuras Coarse C PhD thesis. Universidad Complutense de Madrid, 2012.[17] Jesús P. Moreno Damas, Propiedades de la geometría C a gran escala con com-pactificación de Higson metrizable Trabajo para la obtención del DEA. UniversidadComplutense de Madrid, 2007.[18] K. Morita, On the Dimension of Product Spaces , American Journal of Mathematics,Vol. 75, No. 2, 205-223, Apr. 1953[19] John Roe, Lectures on Coarse Geometry . American Mathematical Society, 2003. Uni-versity Lecture Series Vol. 31.[20] John Roe, Corrections to Lectures on Coarse Geometry Theory of Retracts . Wayne State University Press, 1965[22] Nick Wright, C Coarse Geometry . PhD thesis. Penn State University 2002.[23] Wright, Nick C coarse geometry and scalar curvaturecoarse geometry and scalar curvature