- Email: [email protected]
Coarse metric and uniform metric
Topology and its Applications 260 (2019) 1–12
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Coarse metric and uniform metric Chi-Keung Ng Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
a r t i c l e
i n f o
Article history: Received 7 November 2018 Received in revised form 15 March 2019 Accepted 16 March 2019 Available online 20 March 2019 MSC: primary 51K05, 54E15, 54E35 Keywords: Pseudo metric spaces Coarse spaces Uniform spaces
a b s t r a c t We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the coarse structure E on a set X is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set P0 (X) of non-empty subsets of X and show that it induces the Hausdorff coarse structure on P0 (X). On the other hand, we define the notion of pseudo uniform metric. Each pseudo uniform metric induces a uniform structure on the underlying space. In the reverse direction, we show that a uniform structure U on a set X is induced by a map d from X × X to a partially orderedset (with no requirement on d) if and only if U admits a base B such that B ∪ { U} is closed under arbitrary intersections. In this case, U is actually defined by a pseudo uniform metric. We also show that a uniform structures U comes from a pseudo uniform metric that takes values in a totally ordered set if and only if U admits a totally ordered base. Finally, a valuation ring will produce an example of a coarse and pseudo uniform metric that take values in a totally ordered set. © 2019 Published by Elsevier B.V.
1. Introduction and notations The notion of coarse spaces was first introduced in [6]. It can be regarded as an abstract framework for the study of large scale properties of metric spaces. A throughout account for coarse spaces can be found in [12] (see also [1,3,4,7–10,13–16] for some information on coarse structure). On the other hand, coarse structure can be deemed as an opposite to uniform structure (see e.g. [5] or [11]). Obviously, pseudo metric is a common source of examples for both of them. In fact, uniform spaces are generalizations of pseudo metric spaces and coarse structures were first studied for metric spaces (see [6]). It is known that a coarse structure (respectively, a uniform structure) is defined by a pseudo metric if and only if it has a countable base (see e.g. [12, Theorem 2.55] for the case of coarse structures and [11, Corollary I.4.4] for the case of E-mail address: [email protected]. https://doi.org/10.1016/j.topol.2019.03.017 0166-8641/© 2019 Published by Elsevier B.V.
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
2
uniform structures). The aim of this article is to introduce two general notions of metric, and to study coarse structures and uniform structures associated respectively with them. For a partially ordered set I with a smallest element 0I , we denote by I∞ the extension of I by adjoining a new element ∞ that is larger than all elements in I. The most general form of “metric” on a set X is simply a map d : X × X → I∞ . If the map d satisfies d(x, y) = d(y, x) as well as d(x, x) = 0I (x, y ∈ X), and it also fulfills certain growth condition as in Definition 2(a) (respectively, descent condition as in Definition 13(b)), then d is called a coarse metric (respectively, uniform metric). It will be obvious that a coarse metric (respectively, uniform metric) will induce a coarse structure (respectively, uniform structure) on the underlying space. A natural question is how to characterize those coarse structures and uniform structures coming from such generalized notions of metric. In fact, it is not hard to check that any coarse space actually comes from a coarse metric. Furthermore, there is a bijective correspondence between the collection of coarse structures on a set X and the collection of coarse metrics on X that are “saturated” (Theorem 5). Through the correspondence of coarse structures and coarse metrics, one can rephrase some terminologies in coarse spaces back in metric terms, which makes them easier to understand, and hopefully easier to manipulate. For example, a coarse space is coarsely connected if and only if one (and hence all) of its defining coarse metrics does not take the value ∞. A list of other translations can be found in Propositions 8 and 9. Philosophically, the above correspondence tells us that by studying the coarse structure of a metric space, one actually “forgets” the triangle inequality and “remembers” only the growth condition (as in Definition 2(a)). On the other hand, we will show that a coarse structure has a totally ordered base if and only if it is defined by a coarse metric taking values in a totally ordered set (Corollary 12(a)). Furthermore, we investigate the relation between the Hausdorff coarse structure on the set of non-empty subsets of a coarse space (Definition 10) and the Hausdorff coarse metric on the same collection of subsets induced by the coarse metric defining the original coarse space (Proposition 11). In the case of uniform structures, the correspondence is not as perfect. We will show in Section 3 that for a uniform structure U, there is a map d : X × X → I∞ (without any further requirement on d) such that
Dα : α ∈ I \ {0I }
forms a base for U, where Dα := {(x, y) : d(x, y) ≤ α},
(1.1)
if and only if U admits a base B with B ∪ { U} being closed under arbitrary intersections. In this case, U is actually defined by a pseudo uniform metric (Theorem 17). In particular, if U admits a totally ordered base, then it is defined by a pseudo uniform metric (Corollary 18). In Section 4, we give a mild condition, under which a coarse metric will become a pseudo uniform metric (Proposition 19). We will close this short article by giving an example of a coarse and pseudo uniform metric coming from a valuation ring (Example 20). In the remainder of this section, let us set some notation. Suppose that I and I∞ are as in the second paragraph of this Introduction. • The smallest element in I, if it exists, is called the zero of I and will always be denoted by 0I . • I is called an upward directed set if for any α, β ∈ I, there exists γ ∈ I with α ≤ γ and β ≤ γ.
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
3
• If J is another partially ordered set, and Λ : I → J is a map, then we set Λ∞ : I∞ → J∞ to be the extension of Λ such that Λ∞ (∞) = ∞.
(1.2)
A partially ordered set I is said to be meet-complete if every non-empty subset S of I admits a greatest lower-bound inf S in I. Furthermore, I is called a complete lattice if it is meet-complete and each non-empty subset of I has a least upper-bound in I. For a set X, we use P0 (X) to denote the collection of all non-empty subsets of X and use ΔX to denote the diagonal in X × X; namely, ΔX := {(x, x) ∈ X × X : x ∈ X}. Moreover, for any A, B ⊆ X × X, we put A−1 := {(y, x) : (x, y) ∈ A} as well as A ◦ B := (x, z) ∈ X × X : there exists y ∈ X such that (x, y) ∈ A and (y, z) ∈ B . In this case, A−1 is called the inverse of A and A ◦ B is called the product of A and B. Furthermore, if S ⊆ X, we denote B[S] := {y ∈ X : (x, y) ∈ B, for some x ∈ S}, and B[x] := B[{x}]. Definition 1. Let I be a partially ordered set with a zero, and d : X × X → I∞ be a map. (a) We set D(z, α) := Dα [z]
(z ∈ X; α ∈ I)
(where Dα is as in (1.1)), and call D(z, α) the ball of radius α with center z. Moreover, we denote Bd := Dα : α ∈ I \ {0I } . (b) Suppose that d satisfies (D1) d(x, x) = 0I for any x ∈ X; (D2) d(x, y) = d(y, x) for any x, y ∈ X. Then d is called a semi-I-metric. 2. Coarse I-metric In this section, we define and study coarse metrics and relate them to coarse structures. Recall that if E ⊆ P0 (X × X) such that ΔX ∈ E, and E is closed under the formation of subsets, inverses, products and finite unions, then E is called a coarse structure on X. In this case, (X, E) (or simply X) is called a coarse space. A subcollection B ⊆ E is called a base for E if every element in E is contained in an element of B. Definition 2. Let I be a upward directed set with a zero, and d : X × X → I∞ be a semi-I-metric. (a) Suppose that there is a function Φ : I → I such that d(x, z) ≤ Φ(α) whenever α ∈ I and x, y, z ∈ X satisfy d(x, y) ≤ α and d(y, z) ≤ α; in other words,
4
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
Dα ◦ Dα ⊆ DΦ(α)
(α ∈ I).
Then d is called a coarse I-metric on X, and (X, I, d) is a called a coarse metric space. (b) A coarse I-metric d is said to be saturated if for each α, β ∈ I, • the inclusion Dα ⊆ Dβ implies α ≤ β; • for any subset S ⊆ Dα with S −1 = S and ΔX ⊆ S, one can find γ ∈ I such that S = Dγ . (c) Suppose that I is another upward directed set with a zero and d is a coarse I -metric on X. We say that d is coarsely dominated by d (and denote this by d d ) if there is a map Γ : I → I such that d(x, y) ≤ Γ∞ (d (x, y)) for every x, y ∈ X (see (1.2)). If d d and d d, then we say that d is coarsely equivalent to d and denote this by d ∼ d . Remark 3. Let d be a coarse I-metric. (a) If Φ is a map satisfying the requirement in Definition 2(a), then it is not hard to see that d(x, y) ≤ Φ(d(x, y)) (x, y ∈ X). Moreover, as I is upward directed, one can always find a map Φ with α ≤ Φ(α) (α ∈ I) that satisfies the requirement of Definition 2(a). (b) Suppose, in addition, that I is meet-complete. Then clearly, α∈S Dα ⊆ Dinf S for S ⊆ I. We set I(α) := {β ∈ I : Dα ◦ Dα ⊆ Dβ }
(α ∈ I),
ˆ ˆ is increasing. If x, y, z ∈ X satisfying d(x, y) ≤ α and and define Φ(α) := inf I(α). It is obvious that Φ ˆ d(y, z) ≤ α, then d(x, z) ≤ β for any β ∈ I(α), and hence d(x, z) ≤ Φ(α). Thus, in the case when I is meet-complete, we can always find an increasing map Φ satisfying the requirement in Definition 2(a). Let us begin will the following easy fact. Lemma 4. Let T be a subcollection of P0 (X × X). We set Ts := {(A ∩ A−1 ) ∪ ΔX : A ∈ T} as well as T :=
S : ∅ = S ⊆ Ts .
The collection T is closed under arbitrary intersections (i.e. the intersection of any subset of T belongs to T). Moreover, if T is totally ordered, then so is T. Proof. The first claim is obvious. Suppose now that T is totally ordered. Then Ts is also totally ordered. Consider C, D ⊆ Ts . Then either there exists C0 ∈ C such that C0 ⊆ D for every D ∈ D, or for each C ∈ C, one can find D ∈ D with D ⊆ C. In the first case, we have C ⊆ C0 ⊆ D. In the second case, we know that D ⊆ C. 2 Theorem 5. Let X be a set. (a) Suppose that I is a upward directed set with a zero and d is a coarse I-metric on X. Then Bd ∪ {D0I } (see Definition 1(a)) is a base for a coarse structure Ed on X. Moreover, if I is another upward directed set with a zero and d is a I -metric on X, then d is coarsely equivalent to d if and only if Ed = Ed . (b) Let E be a coarse structure on X. There is a unique upward directed set IE with a zero such that one can find a (necessarily unique) saturated coarse IE -metric dE with E = EdE . In this case, IE ∞ is a complete lattice. (c) Let B be a base for a coarse structure E on X, and B be as in Lemma 4. Then B is a meet-complete upward directed set, and there is a coarse B-metric dB such that E = EdB .
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
5
Proof. (a) For any α, β ∈ I, if γ ∈ I satisfying α ≤ γ and β ≤ γ, then Dα ◦ Dβ ⊆ DΦ(γ) . This gives the first statement. For the second statement, we note d d if and only if Ed ⊆ Ed . (b) As E is closed under the formation of finite unions and subsets, we see that IE := {E ∈ E : E −1 = E; ΔX ⊆ E}
(2.1)
is a meet-complete lattice (under inclusion), and it contains ΔX as its smallest element (i.e. zero). Moreover, since every subset of IE that has an upper bound in IE has a least upper bound in IE , we know that IE ∞ is a complete lattice. Set E
d (x, y) :=
ΔX ∪ {(x, y), (y, x)}
when (x, y) ∈ E for some E ∈ E
∞
otherwise.
E E E E E Obviously, dE : X × X → IE ∞ is a semi-I -metric. If we put Φ (E) := E ◦ E ∈ I for each E ∈ I , then Φ will satisfy the requirement in Definition 2(a), and dE is a coarse IE -metric. Furthermore, as
E = {(x, y) ∈ X × X : dE (x, y) ≤ E}
(E ∈ IE ),
(2.2)
we see that dE is saturated and that E = EdE . Suppose now that I is another upward directed set with a zero and d is a saturated coarse I-metric on X with E = Ed . Then the saturation assumption of d implies that α → Dα is an order isomorphism from I onto IE . Furthermore, it follows from the definitions and (2.2) that for any u, v ∈ X and β ∈ I, one has d(u, v) ≤ β
if and only if dE (u, v) ≤ Dβ .
(2.3)
For every x, y ∈ X, it is not hard to verify, through (2.3), that dE (x, y) = Dd(x,y) . In other words, d is the same as dE under the order isomorphism α → Dα . (c) The meet-completeness of B follows from Lemma 4. Suppose that S, T ∈ P0 (Bs ), where Bs is as in Lemma 4. Take any S ∈ S and T ∈ T. As Bs is a base for E, there exists B ∈ Bs with S ∪ T ⊆ B. Then B ∈ B and ( S) ∪ ( T) ⊆ B. This shows that B is upward directed. Let us define dB (x, y) := here we use the convention that
{D ∈ B : (x, y) ∈ D}
(x, y ∈ X);
(2.4)
∅ = ∞. We also set
ΦB (B) :=
{D ∈ B : B ◦ B ∈ D}
(B ∈ B).
Clearly, ΦB satisfies the requirement in Definition 2(a), and dB is a coarse B-metric. Moreover, as B = {(x, y) ∈ X × X : dB (x, y) ≤ B} for any B ∈ B, we know that E = EdB . 2 Obviously, in the case when I = {0I } and d is a coarse-I-metric, then Bd is a base for Ed . On the other E hand, although IE ∞ in part (b) above is a complete lattice, it does not mean that I ∪ {X} is closed under E E arbitrary unions. In fact, if S ⊆ I such that S ∈ / I , then the least upper bound of S in IE ∞ is ∞. Remark 6. The maps ΦE and ΦB in the proof of parts (b) and (c) of Theorem 5 are increasing and satisfy the requirement in Definition 2(a) for dE and dB , respectively. Moreover, one has E ⊆ ΦE (E) (respectively, B ⊆ ΦB (B)) for every E ∈ IE (respectively, B ∈ B).
6
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
The following example tells us that if d is a coarse I-metric and J is a upward directed set containing I, then the coarse structure induced by d when d is considered as a coarse J-metric may not be the same as the one when d is considered as a coarse I-metric. Example 7. Let (R, d1 ) be the Euclidean metric space. Clearly, d1 is a R+ -metric and the coarse structure generated by this R+ -metric is the usual one. However, if we set J = R+ ∞ , then d1 is also a J-metric, but the coarse structure generated by this J-metric is the “trivial one”, because R = D∞ is a controlled set (here ∞ is the largest element in J, but not the largest element in J∞ ). One can express many concepts concerning coarse structures in terms of metric, which seem easier to understand and handle. Let us list some of them in the following. Proposition 8. Let (X, E) and (Y, F) be two coarse spaces. Suppose that dX (respectively, dY ) is a coarse I-metric on X (respectively, coarse J-metric on Y ) that induces the underlying coarse structure. Let f, g : X → Y be two maps. (a) (X, E) is coarsely connected if and only if the largest element ∞ ∈ I∞ does not belong to dX (X × X). (b) B ⊆ X is bounded if and only if one can find (x, α) ∈ X × I with B ⊆ D(x, α). (c) f is bornologous if and only if dY ◦ (f × f ) dX . (d) f is proper if and only if there is a map Υ : Y × J → X × I with f −1 (D(y, β)) ⊆ D(Υ(y, β)), for each (y, β) ∈ Y × J. (e) f is effectively proper if and only if dX dY ◦ (f × f ). (f) f and g are close if and only if there exists β ∈ J with dY (f (x), g(x)) ≤ β, for any x ∈ X. Proof. (a) Recall that (X, E) is coarsely connected if and only if {(x, y)} ∈ E, for every x, y ∈ X. Clearly, this is equivalent to dX (x, y) ∈ I, for every x, y ∈ X. (b) Recall that B is bounded if and only if B ⊆ E[x] for some E ∈ E and x ∈ X. The equivalence in the statement is more or less trivial. (c) Recall that f is bornologous if and only if (f × f )(E) ⊆ F; equivalently, for every α ∈ I, one can find Γ(α) ∈ J such that (f × f )(Dα ) ⊆ DΓ(α) . Now, for a map Γ : I → J, the condition (f × f )(Dα ) ⊆ DΓ(α) is equivalent to the requirement as in Definition 2(c) for dY ◦ (f × f ) dX . (d) Recall that f is proper if and only if f −1 (B) is bounded for any bounded set B ⊆ Y . Thus, this part follows directly from part (b). (e) Recall that f is effectively proper if and only if (f × f )−1 (F) ⊆ E. This is the same as saying that for β ∈ J, one find Γ(β) ∈ I such that (f × f )−1 (Dβ ) ⊆ DΓ(β) , or equivalently, dX (x, y) ≤ Γ(β) whenever dY (f (x), f (y)) ≤ β. Now, a map Γ : J → I satisfies the above displayed statement if and only if it satisfies the requirement in Definition 2(c) for dY ◦ (f × f ) dX . (f) Recall that f and g are close if and only if {(f (x), g(x)) : x ∈ X} ∈ F, which is obviously the same as the requirement in the statement of part (f). 2 Notice that in Example 7, although the range of d1 take the value “∞”, it is the largest element in J = R+ ∞ , but not the largest element of J∞ . Therefore, the resulting coarse structure is connected. On the other hand, we can “reverse-engineer” some concepts in coarse structure back to metric space terms. The following is such an example. Let us recall from [12, Definition 3.9] that a coarse space (X, E) is said to have bounded geometry if one can find E ∈ E containing ΔX such that E −1 = E and
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
supx∈X max capE ((F ◦ E)[x]), capE ((F −1 ◦ E)[x]) < ∞
7
(F ∈ E),
(2.5)
where capE (S) := sup{m ∈ N : there exist y1 , ..., ym ∈ S with (yi , yj ) ∈ / E when i = j}. Clearly, capE (S) ≤ capE (S) if E ⊆ E and capE (S) ≤ capE (S ) if S ⊆ S . Moreover, one has F ⊆ F ◦E ∈ E. Consequently, one may replace (2.5) with supx∈X capE (F [x]) < ∞, for every F ∈ E with F −1 = F . Thus, in the case when E is defined by a coarse I-metric, (X, E) has bounded geometry if and only if one can find α1 ∈ I satisfying supx∈X capDα1 (D(x, α)) < ∞
(α ∈ I).
(2.6)
Proposition 9. Let (X, E) be a coarse space and dX be a coarse I-metric defining E. The following statements are equivalent. B1) (X, E) has bounded geometry. B2) There is α1 ∈ I satisfying: for any α ∈ I, there exists n1 ∈ N such that each ball of radius α contains at most n1 points with their pairwise dX -distances not dominated by α1 (i.e. dX (x, y) α1 ). B3) There is α2 ∈ I satisfying: for any α ∈ I, there exists n2 ∈ N such that for each x ∈ X, the ball D(x, α) contains at most n2 disjoint relative balls of radius α2 (here, relative balls of radius α2 are subsets of the form D(x, α) ∩ D(y, α2 ) for some y ∈ D(x, α)). B4) There is α3 ∈ I satisfying: for any α ∈ I, there exists n3 ∈ N such that each ball of radius α is contained in the union of n3 balls of radius α3 . Proof. (B1) ⇔ (B2) This equivalence is simply a matter of reformulating (2.6). (B2) ⇒ (B3) If D(x, α) contains at most n1 points with their mutual dX -distances not dominated by α1 , then clearly, it cannot contain more than n1 disjoint relative balls of radius α1 . (B3) ⇒ (B2) Suppose that D(x, α) contains at most n2 disjoint relative balls of radius α2 . Let Φ be as in Definition 2(a) and α1 := Φ(α2 ). Then D(x, α) cannot contain more than n2 points with their mutual dX -distance not dominated by α1 . (B1) ⇔ (B4) Let us recall from [12, Definition 3.1(a)] the following definition: entE (S) := inf{n ∈ N : there exist x1 , ..., xn ∈ X with S ⊆ E[x1 ] ∪ · · · ∪ E[xn ]} (where inf ∅ := ∞). It was shown in [12, Proposition 3.2(d)] that capE◦E (S) ≤ entE (S) ≤ capE (S). Therefore, one can replace “cap” by “ent” in the definition of bounded geometry; namely, supx∈X entE (F [x]) < ∞,
for every F ∈ E with F −1 = F.
In other words, (X, E) has bounded geometry if and only if there exists α3 ∈ I such that supx∈X entDα3 (D(x, α)) < ∞
(α ∈ I).
This statement is clearly equivalent to Statement (B4). 2 Using the equivalence of Statements (B1) and (B2), one sees an easy path to the fact that every bounded geometry space is coarse equivalent to a uniformly discrete space. In fact, consider B = {D(xi , α1 ) : i ∈ I} to be a maximal collection of disjoint balls of radius α1 . Since i∈I D(xi , Φ(α1 )) = X, we know that X is
8
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
coarse equivalent to its subspace {xi : i ∈ I} (as in Remark 3(a), we may assume that α1 ≤ Φ(α1 )), and the later is uniformly discrete. We end this section with a discussion of the coarse structure induced on the collection P0 (X) of non-empty subsets of a coarse space X. Definition 10. Let (X, E) be a coarse space. For any E ∈ IE (see (2.1)), we set ˇ := (R, S) ∈ P0 (X) × P0 (X) : R ⊆ E[S] and S ⊆ E[R] . E
(2.7)
ˇ on P0 (X) generated by {E ˇ : E ∈ IE } is called the Hausdorff coarse structure The coarse structure E associated with E. ˇ ˇ : B ∈ Bs } (see Lemma 4) is a base for E. Notice that if B is a base for E, then {B ˇ are defined by coarse metrics. It is natural to ask the relation between By Theorem 5(b), both E and E ˇ In the case when I is meet-complete, one natural guess is the metrics defining E and those defining E. following Hausdorff semi-metric associated with a coarse I-metric d:
ˇ D(s, α) and S ⊆ D(r, α) (R, S ∈ P0 (X)); (2.8) d(R, S) := inf α ∈ I : R ⊆ s∈S
r∈R
again, we use the convention that inf ∅ = ∞. Two natural questions are: ˇ actually a coarse I-metric? (1) is d ˇd? ˇ coincide with E (2) does the coarse structure defined by d We doubt if these two questions have positive answers in general. However, we will consider a situation when they do. Proposition 11. Let I be a meet-complete totally ordered set, and d is a coarse I-metric on a set X. ˇ is a coarse I-metric on P0 (X). (a) d ˇd. ˇ is precisely E (b) The coarse structure induced by d ˇ ˇ ˇ R) = 0I and d(R, S) = d(S, R) for R, S ∈ P0 (X). By Remark 3(b), there is Proof. (a) It is clear that d(R, an increasing map Φ : I → I satisfying the requirement in Definition 2(a). Consider α ∈ I. When α is the ˜ largest element of I (if it exists), then we set Φ(α) := α. When α is not the largest element of I, we fix an ˜ element β(α) ∈ I with α β(α) and set Φ(α) := Φ(β(α)). Let R, S, T ∈ P0 (X). Consider
I(R, S) := δ ∈ I : R ⊆
s∈S
D(s, δ) and S ⊆
r∈R
D(r, δ) .
ˇ ˇ Assume that d(R, S) ≤ α and d(S, T ) ≤ α. Since α = ∞, we know that I(R, S) = ∅ and I(S, T ) = ∅. If ˜ ˇ ˇ α is the largest element of I, then obviously, d(R, T ) ≤ α = Φ(α). Otherwise, since d(R, S) β(α), there exists γ ∈ I(R, S) with γ ≤ β(α) (because I is totally ordered). Similarly, one can find γ ∈ I(S, T ) with γ ≤ β(α). Hence, if γ := max{γ , γ }, then R ⊆ t∈T D(t, Φ(γ)) and T ⊆ r∈R D(r, Φ(γ)). This implies ˜ ˇ that d(R, T ) ≤ Φ(γ) ≤ Φ(α), because Φ is increasing. ˇ d , where D ˇ α : α ∈ I is a base for E ˇ α is defined as in (2.7). Fix α0 ∈ I. (b) As said in the above, D When α0 is the largest element in I (if it exists), one has
ˇ α = (R, S) ∈ P0 (X) × P0 (X) : R ⊆ D 0
s∈S
D(s, β) and S ⊆
ˇ = {(R, S) ∈ P0 (X) × P0 (X) : d(R, S) ≤ α0 }.
r∈R
D(r, β), for some β ∈ I
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
9
Assume that α0 is not the largest element in I. Choose any β ∈ I with α0 β. If R, S ∈ P0 (X) satisfying ˇ S) ≤ α0 , then, as in the argument of part (a), one can find γ ∈ I(R, S) with γ ≤ β. From this, we d(R, ˇ β . Conversely, if (R, S) ∈ D ˇ α , then it is clear that d(R, ˇ know that (R, S) ∈ D S) ≤ α0 . 2 0 We end this section with the following result concerning the case when E has a totally ordered base. Note that part (a) of this result comes from Lemma 4, as well as parts (a) and (c) of Theorem 5. Part (b) is a corollary of Proposition 11 and Theorem 5(c). Corollary 12. Let E be a coarse structure on a set X. (a) E has a totally ordered base B if and only if there is a meet-complete totally ordered set I and a coarse I-metric d with E = Ed . In this case, (I, d) can be chosen to be (B, dB ) (see Theorem 5(c)). ˇ B is a coarse B-metric on P0 (X), and the (b) Suppose that E admits a totally ordered base B. Then d B ˇ coarse structure induced by d is precisely the Hausdorff coarse structure associated with E. 3. Uniform I-metric In this section, we consider the uniform structure that comes from some form of metric. Let us recall that a uniform structure on a set X is a subcollection U ⊆ P0 (X × X) such that for every U, V ∈ U and S ∈ P0 (X × X) with U ⊆ S, one has U ∩ V, U −1 , S ∈ U, ΔX ⊆ U and there exists W ∈ U with W ◦ W ⊆ U . A subcollection B ⊆ U is called a base for U if for every U ∈ U, there exist V ∈ B with V ⊆ U . Definition 13. (a) If I is a partially ordered set with a zero such that I \ {0I } is non-empty and is downward directed, then we say that I is a D-index set. (b) Suppose that I is a D-index set and d is a semi-I-metric. If there is a function Ψ : I \ {0I } → I \ {0I } such that for any β ∈ I \ {0I }, one has d(x, z) ≤ β whenever x, y, z ∈ X satisfying d(x, y) ≤ Ψ(β) and d(y, z) ≤ Ψ(β); i.e., DΨ(β) ◦ DΨ(β) ⊆ Dβ , then d is called a pseudo uniform-I-metric. (c) A uniform I-metric is a pseudo uniform-I-metric d satisfying α∈I\{0I } Dα = ΔX . Remark 14. If I \ {0I } does not have a smallest element, then inf I \ {0I } exists and equals 0I . In this case, a pseudo uniform I-metric is a uniform I-metric if and only if the relation d(x, y) = 0I implies x = y. The following result is more or less trivial. Proposition 15. Let X be a set, and I be a D-index set. Suppose that d is a pseudo uniform I-metric on X. Then Bd (see Definition 1(a)) is a base for a uniform structure Ud on X. If, in addition, d is a uniform I-metric, then the topology induced by Ud is Hausdorff. We say that a uniform structure U is trivial if it is a principal filter (i.e. there exists U0 ∈ U with U = {U ⊆ X × X : U0 ⊆ U }); otherwise, U is said to be non-trivial. It is clear that U is non-trivial if and only if 0U :=
U∈ / U.
If U is a trivial uniform structure and we define d : X × X → R+ by
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
10
d(x, y) := then U = Ud (because
0
when (x, y) ∈ 0U
1
otherwise,
U = D1/2 ).
Lemma 16. Let U be a non-trivial uniform structure. Suppose that A is a base for U satisfying:
{A ∈ A : (x, y) ∈ A} ∈ U,
for every (x, y) ∈ X × X \ 0U .
If B := A \ {0U } (see Lemma 4), then B is a base for U. Proof. Clearly, the collection As as in Lemma 4 is a base for U and we have As ⊆ B (as 0U ∈ / As by the assumption of non-triviality). Therefore, it remains to show that B ⊆ U. Pick any (x, y) ∈ X × X \ 0U . As (y, x) ∈ / 0U , one has {B ∈ As : (x, y) ∈ B} = {A ∩ A−1 : A ∈ A; (x, y) ∈ A} ∩ {A ∩ A−1 : A ∈ A; (y, x) ∈ A}. The hypothesis implies that {B ∈ As : (x, y) ∈ B} ∈ U. Suppose now that D ∈ B. There exists C ⊆ As with D = C. As 0U D, there exists (x, y) ∈ C \ 0U . It then follows from C ⊆ {B ∈ As : (x, y) ∈ B} that and the above gives D ∈ U.
{B ∈ As : (x, y) ∈ B} ⊆ D
2
Theorem 17. Let U be a non-trivial uniform structure on a set X. (a) There exist a partially ordered set I with a zero and a map d : X × X → I∞ such that Bd (see Definition 1(a)) becomes a base for U if and only if U admits a base B such that B ∪ {0U } is closed under arbitrary intersections. In this case, one can choose I to be the meet-complete D-index set JB := Bs ∪ {0U } (see Lemma 4) and d to be a pseudo uniform I-metric. (b) If U admits a base B with B ∪{0U } being closed under arbitrary intersections and the topology induced by U is Hausdorff, then one can find a uniform JB -metric d with U = Ud . Proof. (a) Suppose that such a map d exists. For any (x, y) ∈ X × X \ 0U , as 0U = d(x, y) = 0I
and Dd(x,y) ⊆
Bd , we have
{B ∈ Bd : (x, y) ∈ B}.
Thus, one can apply Lemma 16 to conclude that B := Bd \ {0U } is a base for U. Moreover, we know from Lemma 4 that B ∪ {0U } = Bd is closed under arbitrary intersections. Conversely, suppose that such a base B exists. Then Bs is a base of U such that Bs ∪ {0U } is closed under arbitrary intersections. As U is non-trivial, JB \ {0U } = Bs is downward directed. Define dB (x, y) :=
{B ∈ Bs : (x, y) ∈ B} ∈ JB ∞
(x, y ∈ X)
(we again use the convention that ∅ := ∞). It is clear that dB is a semi-JB -metric (observe that Bs = 0U ). Moreover, when S ∈ JB and x, y ∈ X, one has dB (x, y) ≤ S if and only if (x, y) ∈ S. Thus, {(x, y) ∈ X × X : dB (x, y) ≤ S} = S
(S ∈ JB ).
(3.1)
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
11
Consider any S ∈ Bs ⊆ U. Pick an arbitrary B ∈ Bs with B ◦ B ⊆ S. If x, y, z ∈ X satisfying dB (x, y) ≤ B and dB (y, z) ≤ B, then (x, z) ∈ S, or equivalently, dB (x, y) ≤ S. These show that dB is a pseudo uniform JB -metric. Moreover, (3.1) tells us that U = UdB . (b) It follows from (3.1) that 0U =
B∈Bs
{(x, y) ∈ X × X : dB (x, y) ≤ B}.
From this, we see that dB is a uniform JB -metric if and only if 0U = ΔX , or equivalently, the topology induced by U is Hausdorff. 2 It follows from the proof of part (a) above that we have one more equivalent condition of U being defined by a pseudo uniform I-metric: U admits a base A satisfying the requirement in Lemma 16. Corollary 18. Let U be a non-trivial uniform structure on a set X. There exist a totally ordered set I with a zero and a pseudo uniform I-metric d on X with U = Ud if and only if there is a totally ordered base A of U. In the case, we can take I = A. Proof. Suppose that there is such a pseudo uniform I-metric d. Then, obviously, Dα : α ∈ I \ {0I } is a totally ordered base for U. Conversely, suppose that one can find a totally ordered base A for U. Let (x, y) ∈ X × X \ 0U . As 0U = A, there is A0 ∈ A with (x, y) ∈ / A0 . If B ∈ A contains (x, y), then A0 ⊆ B (as A is totally ordered), and A satisfies the hypothesis of Lemma 16. Hence, B := A \ {0U } is a base for U. Furthermore, Lemma 4 tells us that A is totally ordered and is closed under arbitrary intersections. The conclusion now follows from Theorem 17(a). 2 4. An example Before we give the example concerning valuation rings, we first consider a connection between coarse metrics and pseudo uniform metrics. We recall that a subset S of a partially ordered set I is downward cofinal if for every α ∈ I, there exists β ∈ S such that β ≤ α. The following result is more or less obvious. Proposition 19. Let I be upward directed set which is also a D-index set. Suppose that d is a coarse I-metric on X. If there exists Φ : I → I satisfying the requirement in Definition 2(a) such that Φ(I \ {0I }) is a downward cofinal subset of I \ {0I }, then d is also a pseudo uniform I-metric. Consequently, if I is a upward directed set which is also a D-index set, and d is a coarse I-metric on X such that for every α ∈ I and x, y, z ∈ X, one has d(x, z) ≤ α whenever d(x, y) ≤ α and d(y, z) ≤ α, then d is also a pseudo uniform I-metric. In this case, we called d an pseudo ultra I-metric. The following is an example of a pseudo ultra I-metric with I being a totally ordered set. Example 20. Suppose that R is a (unital) ring and Γ is a totally ordered abelian group. Let Γ0 be the ordered semi-group obtained by adjoining to Γ a new element ω, such that ω is greater than all elements in Γ and that α + β = ω when either α = ω or β = ω. As in [2, Definition VI.3.1], a map ν : R → Γ0 is called a valuation if for any x, y ∈ R, one has (V1) (V2) (V3) (V4)
ν(xy) = ν(x) + ν(y); ν(x + y) ≥ min{ν(x), ν(y)}; ν(1) = 0; ν(0) = ω.
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
12
Under the reverse ordering ≤op , the set Γ0 is a upward directed set with a zero (namely, ω) and is also a D-index set (because it is totally ordered). Let us define dν : R × R → Γ0 by dν (x, y) := ν(x − y)
(x, y ∈ R).
Obviously, Condition (V4) implies Condition (D1) in Definition 1(b). On the other hand, Conditions (V1) and (V3) implies that ν(−x) = ν(x), and this verifies Condition (D2). Using Condition (V2), we know that for every x, y, z ∈ R, we have dν (x, z) = ν(x − z) ≤op max{dν (x, y), dν (y, z)}. Consequently, dν is a pseudo ultra Γ0 -metric. Suppose, furthermore, that R is a division ring. Then for every x ∈ R \ {0}, we learn from Condition (V1) that 0 = ν(xx−1 ) = ν(x) + ν(x−1 ), which implies that ν(x) = ω. Thus, in this case, (x, y) ∈ R × R : dν (x, y) = ω = ΔR . Furthermore, since a totally ordered group can never has a greatest element, we know that (Γ, ≤op ) can never has a smallest element. Therefore, if R is a division ring, then dν is a uniform Γ0 -metric (see Remark 14). Acknowledgement The author is supported by National Natural Science Foundation of China (11471168) and (11871285). References [1] F. Baudier, G. Lancien, Th. Schlumprecht, The coarse geometry of Tsirelson’s space and applications, J. Amer. Math. Soc. 31 (2018) 699–717. [2] N. Bourbaki, Commutative Algebra, Springer-Verlag, Berlin, 1989. [3] X. Chen, Q. Wang, G. Yu Guoliang, The coarse Novikov conjecture and Banach spaces with Property (H), J. Funct. Anal. 268 (2015) 2754–2786. [4] D. Dikranjan, N. Zava, Some categorical aspects of coarse spaces and balleans, Topology Appl. 225 (2017) 164–194. [5] J.R. Isbell, Uniform Spaces, Math. Surveys, vol. 12, Amer. Math. Soc., 1964. [6] N. Higson Nigel, E.K. Pedersen, J. Roe, C ∗ -algebras and controlled topology, K-Theory 11 (1997) 209–239. [7] N. Higson, J. Roe, The coarse Baum-Connes conjecture, in: Novikov Conjectures, Index Theorems and Rigidity, vol. 2, Oberwolfach, 1993, in: London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, 1995, pp. 227–254. [8] N. Higson, J. Roe, G. Yu, A coarse Mayer-Vietoris principle, Math. Proc. Cambridge Philos. Soc. 114 (1993) 85–97. [9] K. Mine, A. Yamashita, Metric compactifications and coarse structures, Canad. J. Math. 67 (2015) 1091–1108. [10] A. Nicas, D. Rosenthal, Coarse structures on groups, Topology Appl. 159 (2012) 3215–3228. [11] W. Page, Topological Uniform Structures, Dover Publ., Inc., New York, 1988. [12] J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, Amer. Math. Soc., 2003. [13] N. Wright, C0 -coarse geometry and scalar curvature, J. Funct. Anal. 197 (2003) 469–488. [14] N. Wright, Simultaneous metrizability of coarse spaces, Proc. Amer. Math. Soc. 139 (2011) 3271–3278. [15] T. Yamauchi, Straight finite decomposition complexity implies property A for coarse spaces, Topology Appl. 231 (2017) 329–336. [16] S. Zhang, Coarse quotient mappings between metric spaces, Israel J. Math. 207 (2015) 961–979.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Coarse metric and uniform metric Chi-Keung Ng Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
a r t i c l e
i n f o
Article history: Received 7 November 2018 Received in revised form 15 March 2019 Accepted 16 March 2019 Available online 20 March 2019 MSC: primary 51K05, 54E15, 54E35 Keywords: Pseudo metric spaces Coarse spaces Uniform spaces
a b s t r a c t We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the coarse structure E on a set X is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set P0 (X) of non-empty subsets of X and show that it induces the Hausdorff coarse structure on P0 (X). On the other hand, we define the notion of pseudo uniform metric. Each pseudo uniform metric induces a uniform structure on the underlying space. In the reverse direction, we show that a uniform structure U on a set X is induced by a map d from X × X to a partially orderedset (with no requirement on d) if and only if U admits a base B such that B ∪ { U} is closed under arbitrary intersections. In this case, U is actually defined by a pseudo uniform metric. We also show that a uniform structures U comes from a pseudo uniform metric that takes values in a totally ordered set if and only if U admits a totally ordered base. Finally, a valuation ring will produce an example of a coarse and pseudo uniform metric that take values in a totally ordered set. © 2019 Published by Elsevier B.V.
1. Introduction and notations The notion of coarse spaces was first introduced in [6]. It can be regarded as an abstract framework for the study of large scale properties of metric spaces. A throughout account for coarse spaces can be found in [12] (see also [1,3,4,7–10,13–16] for some information on coarse structure). On the other hand, coarse structure can be deemed as an opposite to uniform structure (see e.g. [5] or [11]). Obviously, pseudo metric is a common source of examples for both of them. In fact, uniform spaces are generalizations of pseudo metric spaces and coarse structures were first studied for metric spaces (see [6]). It is known that a coarse structure (respectively, a uniform structure) is defined by a pseudo metric if and only if it has a countable base (see e.g. [12, Theorem 2.55] for the case of coarse structures and [11, Corollary I.4.4] for the case of E-mail address: [email protected]. https://doi.org/10.1016/j.topol.2019.03.017 0166-8641/© 2019 Published by Elsevier B.V.
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
2
uniform structures). The aim of this article is to introduce two general notions of metric, and to study coarse structures and uniform structures associated respectively with them. For a partially ordered set I with a smallest element 0I , we denote by I∞ the extension of I by adjoining a new element ∞ that is larger than all elements in I. The most general form of “metric” on a set X is simply a map d : X × X → I∞ . If the map d satisfies d(x, y) = d(y, x) as well as d(x, x) = 0I (x, y ∈ X), and it also fulfills certain growth condition as in Definition 2(a) (respectively, descent condition as in Definition 13(b)), then d is called a coarse metric (respectively, uniform metric). It will be obvious that a coarse metric (respectively, uniform metric) will induce a coarse structure (respectively, uniform structure) on the underlying space. A natural question is how to characterize those coarse structures and uniform structures coming from such generalized notions of metric. In fact, it is not hard to check that any coarse space actually comes from a coarse metric. Furthermore, there is a bijective correspondence between the collection of coarse structures on a set X and the collection of coarse metrics on X that are “saturated” (Theorem 5). Through the correspondence of coarse structures and coarse metrics, one can rephrase some terminologies in coarse spaces back in metric terms, which makes them easier to understand, and hopefully easier to manipulate. For example, a coarse space is coarsely connected if and only if one (and hence all) of its defining coarse metrics does not take the value ∞. A list of other translations can be found in Propositions 8 and 9. Philosophically, the above correspondence tells us that by studying the coarse structure of a metric space, one actually “forgets” the triangle inequality and “remembers” only the growth condition (as in Definition 2(a)). On the other hand, we will show that a coarse structure has a totally ordered base if and only if it is defined by a coarse metric taking values in a totally ordered set (Corollary 12(a)). Furthermore, we investigate the relation between the Hausdorff coarse structure on the set of non-empty subsets of a coarse space (Definition 10) and the Hausdorff coarse metric on the same collection of subsets induced by the coarse metric defining the original coarse space (Proposition 11). In the case of uniform structures, the correspondence is not as perfect. We will show in Section 3 that for a uniform structure U, there is a map d : X × X → I∞ (without any further requirement on d) such that
Dα : α ∈ I \ {0I }
forms a base for U, where Dα := {(x, y) : d(x, y) ≤ α},
(1.1)
if and only if U admits a base B with B ∪ { U} being closed under arbitrary intersections. In this case, U is actually defined by a pseudo uniform metric (Theorem 17). In particular, if U admits a totally ordered base, then it is defined by a pseudo uniform metric (Corollary 18). In Section 4, we give a mild condition, under which a coarse metric will become a pseudo uniform metric (Proposition 19). We will close this short article by giving an example of a coarse and pseudo uniform metric coming from a valuation ring (Example 20). In the remainder of this section, let us set some notation. Suppose that I and I∞ are as in the second paragraph of this Introduction. • The smallest element in I, if it exists, is called the zero of I and will always be denoted by 0I . • I is called an upward directed set if for any α, β ∈ I, there exists γ ∈ I with α ≤ γ and β ≤ γ.
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
3
• If J is another partially ordered set, and Λ : I → J is a map, then we set Λ∞ : I∞ → J∞ to be the extension of Λ such that Λ∞ (∞) = ∞.
(1.2)
A partially ordered set I is said to be meet-complete if every non-empty subset S of I admits a greatest lower-bound inf S in I. Furthermore, I is called a complete lattice if it is meet-complete and each non-empty subset of I has a least upper-bound in I. For a set X, we use P0 (X) to denote the collection of all non-empty subsets of X and use ΔX to denote the diagonal in X × X; namely, ΔX := {(x, x) ∈ X × X : x ∈ X}. Moreover, for any A, B ⊆ X × X, we put A−1 := {(y, x) : (x, y) ∈ A} as well as A ◦ B := (x, z) ∈ X × X : there exists y ∈ X such that (x, y) ∈ A and (y, z) ∈ B . In this case, A−1 is called the inverse of A and A ◦ B is called the product of A and B. Furthermore, if S ⊆ X, we denote B[S] := {y ∈ X : (x, y) ∈ B, for some x ∈ S}, and B[x] := B[{x}]. Definition 1. Let I be a partially ordered set with a zero, and d : X × X → I∞ be a map. (a) We set D(z, α) := Dα [z]
(z ∈ X; α ∈ I)
(where Dα is as in (1.1)), and call D(z, α) the ball of radius α with center z. Moreover, we denote Bd := Dα : α ∈ I \ {0I } . (b) Suppose that d satisfies (D1) d(x, x) = 0I for any x ∈ X; (D2) d(x, y) = d(y, x) for any x, y ∈ X. Then d is called a semi-I-metric. 2. Coarse I-metric In this section, we define and study coarse metrics and relate them to coarse structures. Recall that if E ⊆ P0 (X × X) such that ΔX ∈ E, and E is closed under the formation of subsets, inverses, products and finite unions, then E is called a coarse structure on X. In this case, (X, E) (or simply X) is called a coarse space. A subcollection B ⊆ E is called a base for E if every element in E is contained in an element of B. Definition 2. Let I be a upward directed set with a zero, and d : X × X → I∞ be a semi-I-metric. (a) Suppose that there is a function Φ : I → I such that d(x, z) ≤ Φ(α) whenever α ∈ I and x, y, z ∈ X satisfy d(x, y) ≤ α and d(y, z) ≤ α; in other words,
4
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
Dα ◦ Dα ⊆ DΦ(α)
(α ∈ I).
Then d is called a coarse I-metric on X, and (X, I, d) is a called a coarse metric space. (b) A coarse I-metric d is said to be saturated if for each α, β ∈ I, • the inclusion Dα ⊆ Dβ implies α ≤ β; • for any subset S ⊆ Dα with S −1 = S and ΔX ⊆ S, one can find γ ∈ I such that S = Dγ . (c) Suppose that I is another upward directed set with a zero and d is a coarse I -metric on X. We say that d is coarsely dominated by d (and denote this by d d ) if there is a map Γ : I → I such that d(x, y) ≤ Γ∞ (d (x, y)) for every x, y ∈ X (see (1.2)). If d d and d d, then we say that d is coarsely equivalent to d and denote this by d ∼ d . Remark 3. Let d be a coarse I-metric. (a) If Φ is a map satisfying the requirement in Definition 2(a), then it is not hard to see that d(x, y) ≤ Φ(d(x, y)) (x, y ∈ X). Moreover, as I is upward directed, one can always find a map Φ with α ≤ Φ(α) (α ∈ I) that satisfies the requirement of Definition 2(a). (b) Suppose, in addition, that I is meet-complete. Then clearly, α∈S Dα ⊆ Dinf S for S ⊆ I. We set I(α) := {β ∈ I : Dα ◦ Dα ⊆ Dβ }
(α ∈ I),
ˆ ˆ is increasing. If x, y, z ∈ X satisfying d(x, y) ≤ α and and define Φ(α) := inf I(α). It is obvious that Φ ˆ d(y, z) ≤ α, then d(x, z) ≤ β for any β ∈ I(α), and hence d(x, z) ≤ Φ(α). Thus, in the case when I is meet-complete, we can always find an increasing map Φ satisfying the requirement in Definition 2(a). Let us begin will the following easy fact. Lemma 4. Let T be a subcollection of P0 (X × X). We set Ts := {(A ∩ A−1 ) ∪ ΔX : A ∈ T} as well as T :=
S : ∅ = S ⊆ Ts .
The collection T is closed under arbitrary intersections (i.e. the intersection of any subset of T belongs to T). Moreover, if T is totally ordered, then so is T. Proof. The first claim is obvious. Suppose now that T is totally ordered. Then Ts is also totally ordered. Consider C, D ⊆ Ts . Then either there exists C0 ∈ C such that C0 ⊆ D for every D ∈ D, or for each C ∈ C, one can find D ∈ D with D ⊆ C. In the first case, we have C ⊆ C0 ⊆ D. In the second case, we know that D ⊆ C. 2 Theorem 5. Let X be a set. (a) Suppose that I is a upward directed set with a zero and d is a coarse I-metric on X. Then Bd ∪ {D0I } (see Definition 1(a)) is a base for a coarse structure Ed on X. Moreover, if I is another upward directed set with a zero and d is a I -metric on X, then d is coarsely equivalent to d if and only if Ed = Ed . (b) Let E be a coarse structure on X. There is a unique upward directed set IE with a zero such that one can find a (necessarily unique) saturated coarse IE -metric dE with E = EdE . In this case, IE ∞ is a complete lattice. (c) Let B be a base for a coarse structure E on X, and B be as in Lemma 4. Then B is a meet-complete upward directed set, and there is a coarse B-metric dB such that E = EdB .
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
5
Proof. (a) For any α, β ∈ I, if γ ∈ I satisfying α ≤ γ and β ≤ γ, then Dα ◦ Dβ ⊆ DΦ(γ) . This gives the first statement. For the second statement, we note d d if and only if Ed ⊆ Ed . (b) As E is closed under the formation of finite unions and subsets, we see that IE := {E ∈ E : E −1 = E; ΔX ⊆ E}
(2.1)
is a meet-complete lattice (under inclusion), and it contains ΔX as its smallest element (i.e. zero). Moreover, since every subset of IE that has an upper bound in IE has a least upper bound in IE , we know that IE ∞ is a complete lattice. Set E
d (x, y) :=
ΔX ∪ {(x, y), (y, x)}
when (x, y) ∈ E for some E ∈ E
∞
otherwise.
E E E E E Obviously, dE : X × X → IE ∞ is a semi-I -metric. If we put Φ (E) := E ◦ E ∈ I for each E ∈ I , then Φ will satisfy the requirement in Definition 2(a), and dE is a coarse IE -metric. Furthermore, as
E = {(x, y) ∈ X × X : dE (x, y) ≤ E}
(E ∈ IE ),
(2.2)
we see that dE is saturated and that E = EdE . Suppose now that I is another upward directed set with a zero and d is a saturated coarse I-metric on X with E = Ed . Then the saturation assumption of d implies that α → Dα is an order isomorphism from I onto IE . Furthermore, it follows from the definitions and (2.2) that for any u, v ∈ X and β ∈ I, one has d(u, v) ≤ β
if and only if dE (u, v) ≤ Dβ .
(2.3)
For every x, y ∈ X, it is not hard to verify, through (2.3), that dE (x, y) = Dd(x,y) . In other words, d is the same as dE under the order isomorphism α → Dα . (c) The meet-completeness of B follows from Lemma 4. Suppose that S, T ∈ P0 (Bs ), where Bs is as in Lemma 4. Take any S ∈ S and T ∈ T. As Bs is a base for E, there exists B ∈ Bs with S ∪ T ⊆ B. Then B ∈ B and ( S) ∪ ( T) ⊆ B. This shows that B is upward directed. Let us define dB (x, y) := here we use the convention that
{D ∈ B : (x, y) ∈ D}
(x, y ∈ X);
(2.4)
∅ = ∞. We also set
ΦB (B) :=
{D ∈ B : B ◦ B ∈ D}
(B ∈ B).
Clearly, ΦB satisfies the requirement in Definition 2(a), and dB is a coarse B-metric. Moreover, as B = {(x, y) ∈ X × X : dB (x, y) ≤ B} for any B ∈ B, we know that E = EdB . 2 Obviously, in the case when I = {0I } and d is a coarse-I-metric, then Bd is a base for Ed . On the other E hand, although IE ∞ in part (b) above is a complete lattice, it does not mean that I ∪ {X} is closed under E E arbitrary unions. In fact, if S ⊆ I such that S ∈ / I , then the least upper bound of S in IE ∞ is ∞. Remark 6. The maps ΦE and ΦB in the proof of parts (b) and (c) of Theorem 5 are increasing and satisfy the requirement in Definition 2(a) for dE and dB , respectively. Moreover, one has E ⊆ ΦE (E) (respectively, B ⊆ ΦB (B)) for every E ∈ IE (respectively, B ∈ B).
6
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
The following example tells us that if d is a coarse I-metric and J is a upward directed set containing I, then the coarse structure induced by d when d is considered as a coarse J-metric may not be the same as the one when d is considered as a coarse I-metric. Example 7. Let (R, d1 ) be the Euclidean metric space. Clearly, d1 is a R+ -metric and the coarse structure generated by this R+ -metric is the usual one. However, if we set J = R+ ∞ , then d1 is also a J-metric, but the coarse structure generated by this J-metric is the “trivial one”, because R = D∞ is a controlled set (here ∞ is the largest element in J, but not the largest element in J∞ ). One can express many concepts concerning coarse structures in terms of metric, which seem easier to understand and handle. Let us list some of them in the following. Proposition 8. Let (X, E) and (Y, F) be two coarse spaces. Suppose that dX (respectively, dY ) is a coarse I-metric on X (respectively, coarse J-metric on Y ) that induces the underlying coarse structure. Let f, g : X → Y be two maps. (a) (X, E) is coarsely connected if and only if the largest element ∞ ∈ I∞ does not belong to dX (X × X). (b) B ⊆ X is bounded if and only if one can find (x, α) ∈ X × I with B ⊆ D(x, α). (c) f is bornologous if and only if dY ◦ (f × f ) dX . (d) f is proper if and only if there is a map Υ : Y × J → X × I with f −1 (D(y, β)) ⊆ D(Υ(y, β)), for each (y, β) ∈ Y × J. (e) f is effectively proper if and only if dX dY ◦ (f × f ). (f) f and g are close if and only if there exists β ∈ J with dY (f (x), g(x)) ≤ β, for any x ∈ X. Proof. (a) Recall that (X, E) is coarsely connected if and only if {(x, y)} ∈ E, for every x, y ∈ X. Clearly, this is equivalent to dX (x, y) ∈ I, for every x, y ∈ X. (b) Recall that B is bounded if and only if B ⊆ E[x] for some E ∈ E and x ∈ X. The equivalence in the statement is more or less trivial. (c) Recall that f is bornologous if and only if (f × f )(E) ⊆ F; equivalently, for every α ∈ I, one can find Γ(α) ∈ J such that (f × f )(Dα ) ⊆ DΓ(α) . Now, for a map Γ : I → J, the condition (f × f )(Dα ) ⊆ DΓ(α) is equivalent to the requirement as in Definition 2(c) for dY ◦ (f × f ) dX . (d) Recall that f is proper if and only if f −1 (B) is bounded for any bounded set B ⊆ Y . Thus, this part follows directly from part (b). (e) Recall that f is effectively proper if and only if (f × f )−1 (F) ⊆ E. This is the same as saying that for β ∈ J, one find Γ(β) ∈ I such that (f × f )−1 (Dβ ) ⊆ DΓ(β) , or equivalently, dX (x, y) ≤ Γ(β) whenever dY (f (x), f (y)) ≤ β. Now, a map Γ : J → I satisfies the above displayed statement if and only if it satisfies the requirement in Definition 2(c) for dY ◦ (f × f ) dX . (f) Recall that f and g are close if and only if {(f (x), g(x)) : x ∈ X} ∈ F, which is obviously the same as the requirement in the statement of part (f). 2 Notice that in Example 7, although the range of d1 take the value “∞”, it is the largest element in J = R+ ∞ , but not the largest element of J∞ . Therefore, the resulting coarse structure is connected. On the other hand, we can “reverse-engineer” some concepts in coarse structure back to metric space terms. The following is such an example. Let us recall from [12, Definition 3.9] that a coarse space (X, E) is said to have bounded geometry if one can find E ∈ E containing ΔX such that E −1 = E and
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
supx∈X max capE ((F ◦ E)[x]), capE ((F −1 ◦ E)[x]) < ∞
7
(F ∈ E),
(2.5)
where capE (S) := sup{m ∈ N : there exist y1 , ..., ym ∈ S with (yi , yj ) ∈ / E when i = j}. Clearly, capE (S) ≤ capE (S) if E ⊆ E and capE (S) ≤ capE (S ) if S ⊆ S . Moreover, one has F ⊆ F ◦E ∈ E. Consequently, one may replace (2.5) with supx∈X capE (F [x]) < ∞, for every F ∈ E with F −1 = F . Thus, in the case when E is defined by a coarse I-metric, (X, E) has bounded geometry if and only if one can find α1 ∈ I satisfying supx∈X capDα1 (D(x, α)) < ∞
(α ∈ I).
(2.6)
Proposition 9. Let (X, E) be a coarse space and dX be a coarse I-metric defining E. The following statements are equivalent. B1) (X, E) has bounded geometry. B2) There is α1 ∈ I satisfying: for any α ∈ I, there exists n1 ∈ N such that each ball of radius α contains at most n1 points with their pairwise dX -distances not dominated by α1 (i.e. dX (x, y) α1 ). B3) There is α2 ∈ I satisfying: for any α ∈ I, there exists n2 ∈ N such that for each x ∈ X, the ball D(x, α) contains at most n2 disjoint relative balls of radius α2 (here, relative balls of radius α2 are subsets of the form D(x, α) ∩ D(y, α2 ) for some y ∈ D(x, α)). B4) There is α3 ∈ I satisfying: for any α ∈ I, there exists n3 ∈ N such that each ball of radius α is contained in the union of n3 balls of radius α3 . Proof. (B1) ⇔ (B2) This equivalence is simply a matter of reformulating (2.6). (B2) ⇒ (B3) If D(x, α) contains at most n1 points with their mutual dX -distances not dominated by α1 , then clearly, it cannot contain more than n1 disjoint relative balls of radius α1 . (B3) ⇒ (B2) Suppose that D(x, α) contains at most n2 disjoint relative balls of radius α2 . Let Φ be as in Definition 2(a) and α1 := Φ(α2 ). Then D(x, α) cannot contain more than n2 points with their mutual dX -distance not dominated by α1 . (B1) ⇔ (B4) Let us recall from [12, Definition 3.1(a)] the following definition: entE (S) := inf{n ∈ N : there exist x1 , ..., xn ∈ X with S ⊆ E[x1 ] ∪ · · · ∪ E[xn ]} (where inf ∅ := ∞). It was shown in [12, Proposition 3.2(d)] that capE◦E (S) ≤ entE (S) ≤ capE (S). Therefore, one can replace “cap” by “ent” in the definition of bounded geometry; namely, supx∈X entE (F [x]) < ∞,
for every F ∈ E with F −1 = F.
In other words, (X, E) has bounded geometry if and only if there exists α3 ∈ I such that supx∈X entDα3 (D(x, α)) < ∞
(α ∈ I).
This statement is clearly equivalent to Statement (B4). 2 Using the equivalence of Statements (B1) and (B2), one sees an easy path to the fact that every bounded geometry space is coarse equivalent to a uniformly discrete space. In fact, consider B = {D(xi , α1 ) : i ∈ I} to be a maximal collection of disjoint balls of radius α1 . Since i∈I D(xi , Φ(α1 )) = X, we know that X is
8
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
coarse equivalent to its subspace {xi : i ∈ I} (as in Remark 3(a), we may assume that α1 ≤ Φ(α1 )), and the later is uniformly discrete. We end this section with a discussion of the coarse structure induced on the collection P0 (X) of non-empty subsets of a coarse space X. Definition 10. Let (X, E) be a coarse space. For any E ∈ IE (see (2.1)), we set ˇ := (R, S) ∈ P0 (X) × P0 (X) : R ⊆ E[S] and S ⊆ E[R] . E
(2.7)
ˇ on P0 (X) generated by {E ˇ : E ∈ IE } is called the Hausdorff coarse structure The coarse structure E associated with E. ˇ ˇ : B ∈ Bs } (see Lemma 4) is a base for E. Notice that if B is a base for E, then {B ˇ are defined by coarse metrics. It is natural to ask the relation between By Theorem 5(b), both E and E ˇ In the case when I is meet-complete, one natural guess is the metrics defining E and those defining E. following Hausdorff semi-metric associated with a coarse I-metric d:
ˇ D(s, α) and S ⊆ D(r, α) (R, S ∈ P0 (X)); (2.8) d(R, S) := inf α ∈ I : R ⊆ s∈S
r∈R
again, we use the convention that inf ∅ = ∞. Two natural questions are: ˇ actually a coarse I-metric? (1) is d ˇd? ˇ coincide with E (2) does the coarse structure defined by d We doubt if these two questions have positive answers in general. However, we will consider a situation when they do. Proposition 11. Let I be a meet-complete totally ordered set, and d is a coarse I-metric on a set X. ˇ is a coarse I-metric on P0 (X). (a) d ˇd. ˇ is precisely E (b) The coarse structure induced by d ˇ ˇ ˇ R) = 0I and d(R, S) = d(S, R) for R, S ∈ P0 (X). By Remark 3(b), there is Proof. (a) It is clear that d(R, an increasing map Φ : I → I satisfying the requirement in Definition 2(a). Consider α ∈ I. When α is the ˜ largest element of I (if it exists), then we set Φ(α) := α. When α is not the largest element of I, we fix an ˜ element β(α) ∈ I with α β(α) and set Φ(α) := Φ(β(α)). Let R, S, T ∈ P0 (X). Consider
I(R, S) := δ ∈ I : R ⊆
s∈S
D(s, δ) and S ⊆
r∈R
D(r, δ) .
ˇ ˇ Assume that d(R, S) ≤ α and d(S, T ) ≤ α. Since α = ∞, we know that I(R, S) = ∅ and I(S, T ) = ∅. If ˜ ˇ ˇ α is the largest element of I, then obviously, d(R, T ) ≤ α = Φ(α). Otherwise, since d(R, S) β(α), there exists γ ∈ I(R, S) with γ ≤ β(α) (because I is totally ordered). Similarly, one can find γ ∈ I(S, T ) with γ ≤ β(α). Hence, if γ := max{γ , γ }, then R ⊆ t∈T D(t, Φ(γ)) and T ⊆ r∈R D(r, Φ(γ)). This implies ˜ ˇ that d(R, T ) ≤ Φ(γ) ≤ Φ(α), because Φ is increasing. ˇ d , where D ˇ α : α ∈ I is a base for E ˇ α is defined as in (2.7). Fix α0 ∈ I. (b) As said in the above, D When α0 is the largest element in I (if it exists), one has
ˇ α = (R, S) ∈ P0 (X) × P0 (X) : R ⊆ D 0
s∈S
D(s, β) and S ⊆
ˇ = {(R, S) ∈ P0 (X) × P0 (X) : d(R, S) ≤ α0 }.
r∈R
D(r, β), for some β ∈ I
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
9
Assume that α0 is not the largest element in I. Choose any β ∈ I with α0 β. If R, S ∈ P0 (X) satisfying ˇ S) ≤ α0 , then, as in the argument of part (a), one can find γ ∈ I(R, S) with γ ≤ β. From this, we d(R, ˇ β . Conversely, if (R, S) ∈ D ˇ α , then it is clear that d(R, ˇ know that (R, S) ∈ D S) ≤ α0 . 2 0 We end this section with the following result concerning the case when E has a totally ordered base. Note that part (a) of this result comes from Lemma 4, as well as parts (a) and (c) of Theorem 5. Part (b) is a corollary of Proposition 11 and Theorem 5(c). Corollary 12. Let E be a coarse structure on a set X. (a) E has a totally ordered base B if and only if there is a meet-complete totally ordered set I and a coarse I-metric d with E = Ed . In this case, (I, d) can be chosen to be (B, dB ) (see Theorem 5(c)). ˇ B is a coarse B-metric on P0 (X), and the (b) Suppose that E admits a totally ordered base B. Then d B ˇ coarse structure induced by d is precisely the Hausdorff coarse structure associated with E. 3. Uniform I-metric In this section, we consider the uniform structure that comes from some form of metric. Let us recall that a uniform structure on a set X is a subcollection U ⊆ P0 (X × X) such that for every U, V ∈ U and S ∈ P0 (X × X) with U ⊆ S, one has U ∩ V, U −1 , S ∈ U, ΔX ⊆ U and there exists W ∈ U with W ◦ W ⊆ U . A subcollection B ⊆ U is called a base for U if for every U ∈ U, there exist V ∈ B with V ⊆ U . Definition 13. (a) If I is a partially ordered set with a zero such that I \ {0I } is non-empty and is downward directed, then we say that I is a D-index set. (b) Suppose that I is a D-index set and d is a semi-I-metric. If there is a function Ψ : I \ {0I } → I \ {0I } such that for any β ∈ I \ {0I }, one has d(x, z) ≤ β whenever x, y, z ∈ X satisfying d(x, y) ≤ Ψ(β) and d(y, z) ≤ Ψ(β); i.e., DΨ(β) ◦ DΨ(β) ⊆ Dβ , then d is called a pseudo uniform-I-metric. (c) A uniform I-metric is a pseudo uniform-I-metric d satisfying α∈I\{0I } Dα = ΔX . Remark 14. If I \ {0I } does not have a smallest element, then inf I \ {0I } exists and equals 0I . In this case, a pseudo uniform I-metric is a uniform I-metric if and only if the relation d(x, y) = 0I implies x = y. The following result is more or less trivial. Proposition 15. Let X be a set, and I be a D-index set. Suppose that d is a pseudo uniform I-metric on X. Then Bd (see Definition 1(a)) is a base for a uniform structure Ud on X. If, in addition, d is a uniform I-metric, then the topology induced by Ud is Hausdorff. We say that a uniform structure U is trivial if it is a principal filter (i.e. there exists U0 ∈ U with U = {U ⊆ X × X : U0 ⊆ U }); otherwise, U is said to be non-trivial. It is clear that U is non-trivial if and only if 0U :=
U∈ / U.
If U is a trivial uniform structure and we define d : X × X → R+ by
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
10
d(x, y) := then U = Ud (because
0
when (x, y) ∈ 0U
1
otherwise,
U = D1/2 ).
Lemma 16. Let U be a non-trivial uniform structure. Suppose that A is a base for U satisfying:
{A ∈ A : (x, y) ∈ A} ∈ U,
for every (x, y) ∈ X × X \ 0U .
If B := A \ {0U } (see Lemma 4), then B is a base for U. Proof. Clearly, the collection As as in Lemma 4 is a base for U and we have As ⊆ B (as 0U ∈ / As by the assumption of non-triviality). Therefore, it remains to show that B ⊆ U. Pick any (x, y) ∈ X × X \ 0U . As (y, x) ∈ / 0U , one has {B ∈ As : (x, y) ∈ B} = {A ∩ A−1 : A ∈ A; (x, y) ∈ A} ∩ {A ∩ A−1 : A ∈ A; (y, x) ∈ A}. The hypothesis implies that {B ∈ As : (x, y) ∈ B} ∈ U. Suppose now that D ∈ B. There exists C ⊆ As with D = C. As 0U D, there exists (x, y) ∈ C \ 0U . It then follows from C ⊆ {B ∈ As : (x, y) ∈ B} that and the above gives D ∈ U.
{B ∈ As : (x, y) ∈ B} ⊆ D
2
Theorem 17. Let U be a non-trivial uniform structure on a set X. (a) There exist a partially ordered set I with a zero and a map d : X × X → I∞ such that Bd (see Definition 1(a)) becomes a base for U if and only if U admits a base B such that B ∪ {0U } is closed under arbitrary intersections. In this case, one can choose I to be the meet-complete D-index set JB := Bs ∪ {0U } (see Lemma 4) and d to be a pseudo uniform I-metric. (b) If U admits a base B with B ∪{0U } being closed under arbitrary intersections and the topology induced by U is Hausdorff, then one can find a uniform JB -metric d with U = Ud . Proof. (a) Suppose that such a map d exists. For any (x, y) ∈ X × X \ 0U , as 0U = d(x, y) = 0I
and Dd(x,y) ⊆
Bd , we have
{B ∈ Bd : (x, y) ∈ B}.
Thus, one can apply Lemma 16 to conclude that B := Bd \ {0U } is a base for U. Moreover, we know from Lemma 4 that B ∪ {0U } = Bd is closed under arbitrary intersections. Conversely, suppose that such a base B exists. Then Bs is a base of U such that Bs ∪ {0U } is closed under arbitrary intersections. As U is non-trivial, JB \ {0U } = Bs is downward directed. Define dB (x, y) :=
{B ∈ Bs : (x, y) ∈ B} ∈ JB ∞
(x, y ∈ X)
(we again use the convention that ∅ := ∞). It is clear that dB is a semi-JB -metric (observe that Bs = 0U ). Moreover, when S ∈ JB and x, y ∈ X, one has dB (x, y) ≤ S if and only if (x, y) ∈ S. Thus, {(x, y) ∈ X × X : dB (x, y) ≤ S} = S
(S ∈ JB ).
(3.1)
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
11
Consider any S ∈ Bs ⊆ U. Pick an arbitrary B ∈ Bs with B ◦ B ⊆ S. If x, y, z ∈ X satisfying dB (x, y) ≤ B and dB (y, z) ≤ B, then (x, z) ∈ S, or equivalently, dB (x, y) ≤ S. These show that dB is a pseudo uniform JB -metric. Moreover, (3.1) tells us that U = UdB . (b) It follows from (3.1) that 0U =
B∈Bs
{(x, y) ∈ X × X : dB (x, y) ≤ B}.
From this, we see that dB is a uniform JB -metric if and only if 0U = ΔX , or equivalently, the topology induced by U is Hausdorff. 2 It follows from the proof of part (a) above that we have one more equivalent condition of U being defined by a pseudo uniform I-metric: U admits a base A satisfying the requirement in Lemma 16. Corollary 18. Let U be a non-trivial uniform structure on a set X. There exist a totally ordered set I with a zero and a pseudo uniform I-metric d on X with U = Ud if and only if there is a totally ordered base A of U. In the case, we can take I = A. Proof. Suppose that there is such a pseudo uniform I-metric d. Then, obviously, Dα : α ∈ I \ {0I } is a totally ordered base for U. Conversely, suppose that one can find a totally ordered base A for U. Let (x, y) ∈ X × X \ 0U . As 0U = A, there is A0 ∈ A with (x, y) ∈ / A0 . If B ∈ A contains (x, y), then A0 ⊆ B (as A is totally ordered), and A satisfies the hypothesis of Lemma 16. Hence, B := A \ {0U } is a base for U. Furthermore, Lemma 4 tells us that A is totally ordered and is closed under arbitrary intersections. The conclusion now follows from Theorem 17(a). 2 4. An example Before we give the example concerning valuation rings, we first consider a connection between coarse metrics and pseudo uniform metrics. We recall that a subset S of a partially ordered set I is downward cofinal if for every α ∈ I, there exists β ∈ S such that β ≤ α. The following result is more or less obvious. Proposition 19. Let I be upward directed set which is also a D-index set. Suppose that d is a coarse I-metric on X. If there exists Φ : I → I satisfying the requirement in Definition 2(a) such that Φ(I \ {0I }) is a downward cofinal subset of I \ {0I }, then d is also a pseudo uniform I-metric. Consequently, if I is a upward directed set which is also a D-index set, and d is a coarse I-metric on X such that for every α ∈ I and x, y, z ∈ X, one has d(x, z) ≤ α whenever d(x, y) ≤ α and d(y, z) ≤ α, then d is also a pseudo uniform I-metric. In this case, we called d an pseudo ultra I-metric. The following is an example of a pseudo ultra I-metric with I being a totally ordered set. Example 20. Suppose that R is a (unital) ring and Γ is a totally ordered abelian group. Let Γ0 be the ordered semi-group obtained by adjoining to Γ a new element ω, such that ω is greater than all elements in Γ and that α + β = ω when either α = ω or β = ω. As in [2, Definition VI.3.1], a map ν : R → Γ0 is called a valuation if for any x, y ∈ R, one has (V1) (V2) (V3) (V4)
ν(xy) = ν(x) + ν(y); ν(x + y) ≥ min{ν(x), ν(y)}; ν(1) = 0; ν(0) = ω.
C.-K. Ng / Topology and its Applications 260 (2019) 1–12
12
Under the reverse ordering ≤op , the set Γ0 is a upward directed set with a zero (namely, ω) and is also a D-index set (because it is totally ordered). Let us define dν : R × R → Γ0 by dν (x, y) := ν(x − y)
(x, y ∈ R).
Obviously, Condition (V4) implies Condition (D1) in Definition 1(b). On the other hand, Conditions (V1) and (V3) implies that ν(−x) = ν(x), and this verifies Condition (D2). Using Condition (V2), we know that for every x, y, z ∈ R, we have dν (x, z) = ν(x − z) ≤op max{dν (x, y), dν (y, z)}. Consequently, dν is a pseudo ultra Γ0 -metric. Suppose, furthermore, that R is a division ring. Then for every x ∈ R \ {0}, we learn from Condition (V1) that 0 = ν(xx−1 ) = ν(x) + ν(x−1 ), which implies that ν(x) = ω. Thus, in this case, (x, y) ∈ R × R : dν (x, y) = ω = ΔR . Furthermore, since a totally ordered group can never has a greatest element, we know that (Γ, ≤op ) can never has a smallest element. Therefore, if R is a division ring, then dν is a uniform Γ0 -metric (see Remark 14). Acknowledgement The author is supported by National Natural Science Foundation of China (11471168) and (11871285). References [1] F. Baudier, G. Lancien, Th. Schlumprecht, The coarse geometry of Tsirelson’s space and applications, J. Amer. Math. Soc. 31 (2018) 699–717. [2] N. Bourbaki, Commutative Algebra, Springer-Verlag, Berlin, 1989. [3] X. Chen, Q. Wang, G. Yu Guoliang, The coarse Novikov conjecture and Banach spaces with Property (H), J. Funct. Anal. 268 (2015) 2754–2786. [4] D. Dikranjan, N. Zava, Some categorical aspects of coarse spaces and balleans, Topology Appl. 225 (2017) 164–194. [5] J.R. Isbell, Uniform Spaces, Math. Surveys, vol. 12, Amer. Math. Soc., 1964. [6] N. Higson Nigel, E.K. Pedersen, J. Roe, C ∗ -algebras and controlled topology, K-Theory 11 (1997) 209–239. [7] N. Higson, J. Roe, The coarse Baum-Connes conjecture, in: Novikov Conjectures, Index Theorems and Rigidity, vol. 2, Oberwolfach, 1993, in: London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, 1995, pp. 227–254. [8] N. Higson, J. Roe, G. Yu, A coarse Mayer-Vietoris principle, Math. Proc. Cambridge Philos. Soc. 114 (1993) 85–97. [9] K. Mine, A. Yamashita, Metric compactifications and coarse structures, Canad. J. Math. 67 (2015) 1091–1108. [10] A. Nicas, D. Rosenthal, Coarse structures on groups, Topology Appl. 159 (2012) 3215–3228. [11] W. Page, Topological Uniform Structures, Dover Publ., Inc., New York, 1988. [12] J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, Amer. Math. Soc., 2003. [13] N. Wright, C0 -coarse geometry and scalar curvature, J. Funct. Anal. 197 (2003) 469–488. [14] N. Wright, Simultaneous metrizability of coarse spaces, Proc. Amer. Math. Soc. 139 (2011) 3271–3278. [15] T. Yamauchi, Straight finite decomposition complexity implies property A for coarse spaces, Topology Appl. 231 (2017) 329–336. [16] S. Zhang, Coarse quotient mappings between metric spaces, Israel J. Math. 207 (2015) 961–979.