# Relating Asymptotic Dimension to Ponomarev’s Cofinal Dimension via Coarse
Proximities

Jeremy Siegert  Ben Gurion University of the Negev, Beer Sheva, Israel
<EMAIL_ADDRESS>
###### Abstract.

In this paper we show that the asymptotic dimension of an unbounded proper
metric space is bounded above by a coarse analog of Ponomarev’s cofinal
dimension of topological spaces, which we call the coarse cofinal dimension.
We also show that asymptotic dimension is bounded below by the cofinal
dimension of the Higson corona by existing results of Miyata, Austin, and
Virk. We do this by introducing several constructions in the theory of coarse
proximity spaces. In particular we introduce the inverse limit of coarse
proximity spaces. We end with some open problems.

###### Key words and phrases:

asymptotic dimension, cofinal dimension, Higson corona, coarse proximity space

###### 2020 Mathematics Subject Classification:

54F45 , 54F50, 54G20

###### Contents

  1. 1 Introduction
  2. 2 Basic constructions
    1. 2.1 Small-scale Proximity Spaces and the Smirnov compactification
    2. 2.2 Coarse Proximity Spaces
    3. 2.3 Cofinal Dimension
  3. 3 Asymptotic dimension of coarse proximity spaces
  4. 4 Metrizability of coarse proximity spaces
  5. 5 Inverse limits of coarse proximity spaces
  6. 6 Coarsely $n-to-1$ maps and canonical covers of coarse proximity spaces
  7. 7 Coarsely canonical covers and the coarse cofinal dimension
  8. 8 From coarsely canonical covers to an asymptotic dimension zero coarse proximity space
  9. 9 Discussion and Problems

The author was supported by Israel Science Foundation grant no. 2196/20

##  1\. Introduction

Coarse geometry studies metric spaces and abstract spaces from a large scale
perspective where phenomena occurring in sets designated as bounded are
irrelevant and one considers instead the properties that manifest as
interactions between unbounded sets and are persistent through “zooming out”.
One such large scale invariant is the asymptotic dimension defined originally
for metric spaces by Gromov in [6] as a large scale analog of classical
covering dimension. The use of this property would find application in the
study of groups when Yu showed in [19] that the Novikov conjecture is
satisfied for groups of finite asymptotic dimension. There remain open
problems directly involving asymptotic dimension that would advance progress
towards other major problems. In particular, in [3] two open problems about
the asymptotic dimension of geometrically finite groups are stated that,
together, imply the Novikov conjecture. In [5] a conjecture on the asymptotic
dimension of groups is stated that has implications for the Burghelea
conjecture. As these problems remain open, greater investigation of asymptotic
dimension is merited. A related large scale invariant is the Higson corona of
a proper metric space, which is a compact Hausdorff remainder of a
compactification that models the large scale behavior of a space. Dranishnikov
asked in [3] if the asymptotic dimension of an unbounded proper metric space
is equal to the covering dimension of its Higson corona. Dranishnikov,
Keesling, and Uspenkij would show in [3] and [4] that if the asymptotic
dimension of an unbounded proper metric space is finite, then the two
dimensions do indeed agree. The case where asymptotic dimension is infinite is
still open. A characterization of asymptotic dimension or sufficient criteria
for finite asymptotic dimension purely in terms of topological dimensions of
the Higson corona allows questions regarding asymptotic dimension to be
evaluated with the much more mature techniques of classical dimension theory.

In this paper we make the case that asymptotic dimension is more closely
related to the cofinal dimension of the Higson corona than the covering
dimension. The cofinal dimension of a topological space is defined in terms of
covers that are made up of closures of pairwise disjoint open sets. Such
covers are called canonical covers. The cofinal dimension of spaces, and
compact Hausdorff spaces in particular, can be characterized by certain
finite-to-one surjections of zero dimensional spaces. A detailed explanation
of cofinal dimenison can be found in [15]. Recently, Austin, Miyata, and Virk
proved characterizations of asymptotic dimension using large scale analogs of
surjective finite-to-one maps in [1] and [13]. This makes up the motivation
for the results of this paper. Using slight variations of the results of
Austin, Miyata, and Virk and extant results about cofinal dimension we prove
that the asymptotic dimension of unbounded proper metric spaces is bounded
above by a coarse analog of Ponomarev’s cofinal dimension that we call the
coarse cofinal dimension. To do this we make use of coarse proximity spaces,
abstract large scale objects first defined by Grzegrzolka and the author in
[7]. Coarse proximity spaces are a large scale analog of small scale proximity
spaces originally introduced by Efremovic as detailed in [14]. As is
characteristic of coarse properties defined using the language of coarse
proximity spaces, our definition of the coarse cofinal dimension is purely in
terms of unbounded sets and closeness relations between them, as opposed to
the usual characterization of asymptotic dimension using uniform covers. The
convenience of coarse proximity spaces comes from the boundary functor which
assigns to each coarse proximity space a compact Hausdorff space that captures
the geometry evident in closeness relations of the unbounded sets of the base
space. In the case of proper metric spaces, the boundary of the obvious coarse
proximity structure is homeomorphic to the Higson corona, making
characterizing coarse properties in terms of unbounded sets very useful in
recharacterizing coarse properties such as asymptotic dimension in terms of
the Higson corona.

The progression of this paper is as follows. In Section 2 we cover the basic
machinery used in later sections. Much of this material is published
elsewhere, but is reproduced here in the interest of self containment.
Included herein are the definitions of small scale proximity spaces, the
construction of the Smirnov compactification, coarse proximity spaces, the
boundary functor, and cofinal dimension. While most of the definitions and
results in this section appear elsewhere, there are some new results that are
critical for the work in later sections. In section 3 we define and very
briefly explore the asymptotic dimension of coarse proximity spaces. This is
done via uniformly bounded covers of coarse proximity spaces the definition of
which is an aesthetic variation of the same definition provided by Honari and
Kalantari in [12]. In the case where a coarse proximity structure comes from a
metric, our definition of asymptotic dimension agrees with the usual
definition. Section 4 is a short section in which we prove a necessary and
sufficient condition for the structure of a coarse proximity space to be
induced by a metric. Section 5 introduces our first major construction, the
inverse limit of coarse proximity spaces. Here we construct inverse limits in
the categories of coarse proximity spaces with coarse proximity maps. Note
that the morphisms in this category are not closeness classes, but individual
functions. We show that the inverse limit construction commutes with the
boundary functor. In section 6 we review the notions and results surrounding
coarsely $n$-to-$1$ maps as defined by Austin, Miyata, and Virk. We also prove
some generalizations to coarse proximity spaces of their results. It is also
in this section that we introduce the coarse cofinal dimension, the large
scale analog of Ponomarev’s cofinal dimension. Finally in section 8 we provide
a characterization of the coarse cofinal dimension that demonstrates that the
dimension is at least the asymptotic dimension in the case of proper metric
spaces. In the final section we conclude with some open problems.

##  2\. Basic constructions

In this first preliminary section we will review many of the basic structures
and definitions used throughout the rest of the paper. Beginning with the
basic theory of coarse proximity spaces we review the definitions of
bornologies, coarse proximity spaces, maps between coarse proximity spaces,
boundaries of coarse proximity spaces, and the basic properties of those
boundaries. This material is presented roughly as it was across [7], [8], [9],
and [17]. However there are some new results. After finishing the basic coarse
constructions and concepts we proceed to a review of, and a new
charaterization, the cofinal dimension of topological spaces as described in
[15]. The definitions in this subsection are of critical importance to later
sections.

###  2.1. Small-scale Proximity Spaces and the Smirnov compactification

The presentation of small-scale proximity spaces here is as seen in [14].

######  Definition 2.1.

A binary relation $\delta$ on the power set of a nonempty set $X$ is called a
proximity relation and the pair $(X,\delta)$ a proximity space if the relation
satisfies:

  1. (1)

For all $A\subseteq X$, $A\bar{\bf\delta}\emptyset$.

  2. (2)

$A\delta B$ implies $B\delta A$.

  3. (3)

$A\cap B\neq\emptyset$ implies $A\delta B$.

  4. (4)

$A\delta(B\cup C)$ if and only if $A\delta B$ or $A\delta C$.

  5. (5)

$A\bar{\delta}B$ implies that there is an $E\subseteq X$ such that
$A\bar{\delta}(X\setminus E)$ and $E\bar{\delta}B$.

Where $A\bar{\delta}B$ denotes the statement that “$A\delta B$ does not hold”.
If in addition the relation satisfies:

$\\{x\\}\delta\\{y\\}\iff x=y$

then we say that the proximity $\delta$ and the corresponding proximity space
$(X,\delta)$ are separated.

######  Proposition 2.2.

The induced topology of a proximity space $(X,\delta)$ is the topology on $X$
determined by the closure operator defined by

$cl(A)=\\{x\in X\mid\\{x\\}\delta A\\}$

This topology is Hausdorff if and only if the proximity is separated.

######  Example 2.3.

If $(X,d)$ is a metric space, then the relation $\delta$ defined by $A\delta
B$ if and only if $d(A,B)=0$ is a proximity relation on $X$, whose induced
topology is the topology induced by the metric $d$. We call this proximity the
metric proximity.

######  Example 2.4.

If $X$ is a compact Hausdorff space, then the relation $\delta$ defined by
$A\delta B$ if and only if their closures intersect is a proximity relation on
$X$ that induces the original topology on $X$. This proximity relation is in
fact the unique proximity relation on $X$ that induces the original topology.

######  Example 2.5.

If $(X,d)$ is a proper metric space (that is, a metric space in which all
closed and bounded sets are compact), then the relation $\delta$ defined on
$X$ by

$A\delta B\iff\left\\{\begin{array}[]{ll}d(A,B)=0&\text{ or
}\lim_{r\to\infty}d(A\setminus B(x_{0},r),B\setminus
B(x_{0},r))<\infty\end{array}\right.$

where $x_{0}$ is any point in $X$, is a proximity relation on $X$ that induces
the metric topology induced by $d$. We call this proximity the Higson
proximity.

######  Example 2.6.

If $(X,\delta)$ is a proximity space and $Y\subseteq X$ is a nonempty set,
then the relation $\delta_{Y}$ defined on subsets of $Y$ by

$A\delta_{Y}B\iff A\delta B$

is a proximity relation on $Y$ that induces the subspace topology on $Y$
inherited from the topology on $X$ induced by $\delta$. We of course call this
proximity the subspace proximity.

######  Definition 2.7.

A function $f\mathrel{\mathop{\mathchar
58\relax}}(X,\delta_{X})\rightarrow(Y,\delta_{Y})$ is called a proximity map
if for all $A,B\subseteq X$ we have

$A\delta_{X}B\implies f(A)\delta_{Y}f(B)$

We call $f$ a proximity isomorphism if $f$ is bijective and the inverse
function $g\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ is also a
proximity map.

Proximity maps are continuous with respect to the topologies induced by the
proximity relations. The collection of proximity spaces together with
proximity maps makes up the category Prox.

######  Definition 2.8.

A nonempty collection $\sigma$ of subsets of a proximity space $(X,\delta)$ is
called a cluster if it satisfies the following:

  1. (1)

For all $A,B\in\sigma$, $A\delta B$.

  2. (2)

If $(A\cup B)\in\sigma$ then $A\in\sigma$ or $B\in\sigma$.

  3. (3)

If $C\delta A$ for all $A\in\sigma$, then $C\in\sigma$.

If $\sigma$ contains $\\{x\\}$ for some $x\in X$, we call $\sigma$ a point
cluster and denote it by $\sigma_{x}$.

Note that in a separated proximity space, $\sigma_{x}=\sigma_{y}$ if and only
if $x=y$.

######  Definition 2.9.

Given a collection of clusters $\mathcal{A}$ of a proximity space $(X,\delta)$
we say that a subset $A\subseteq X$ absorbs $\mathcal{A}$ if $A\in\sigma$ for
all $\sigma\in\mathcal{A}$.

Given a separated proximity space $(X,\delta)$ the Smirnov compactification of
$(X,\delta)$ is constructed in the following way. Let $\mathfrak{X}$ be the
set of all clusters in $X$. We define a relation $\delta^{*}$ on the
collection of subsets of $\mathfrak{X}$ as follows. If
$\mathcal{A},\mathcal{B}\subseteq\mathfrak{X}$ then,
$\mathcal{A}\delta^{*}\mathcal{B}$ if and only if for all absorbing sets $A$
for $\mathcal{A}$, and $B$ for $\mathcal{B}$, we have that $A\delta B$. Then
$(\mathfrak{X},\delta^{*})$ is a compact separated proximity space. Moreover,
the map $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow\mathfrak{X}$
defined by $f(x)=\sigma_{x}$ is a proximity (and consequently topological)
embedding whose image is dense in $\mathfrak{X}$. The proximity $\delta$ on
$X$ can be recharacterized by saying that $A\delta B$ if and only if
$cl_{\mathfrak{X}}(A)\cap cl_{\mathfrak{X}}(B)\neq\emptyset$. The Smirnov
compactification is the unique compact Hausdorff space into which $(X,\delta)$
proximity embeds as a dense subspace.

######  Proposition 2.10.

Let $(X,\delta_{1})$ and $(Y,\delta_{2})$ be proximity spaces. Let
$f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ be a proximity map.
Then to each cluster $\sigma_{1}$ in $X,$ there corresponds a cluster
$\sigma_{2}$ in $Y$ given by

$\sigma_{2}\mathrel{\mathop{\mathchar 58\relax}}=\\{A\subseteq Y\mid
A\delta_{2}f(B)\text{ for all }B\in\sigma_{1}\\}.$

######  Theorem 2.11.

Let $(X,\delta_{1})$ and $(Y,\delta_{2})$ be separated proximity spaces and
$(\mathfrak{X},\delta_{1}^{*})$ and $(\mathfrak{Y},\delta_{2}^{*})$ the
respective Smirnov compactifications. Let $f\mathrel{\mathop{\mathchar
58\relax}}X\to Y$ be a proximity map. Then $f$ extends to a unique proximity
map $\bar{f}\mathrel{\mathop{\mathchar 58\relax}}\mathfrak{X}\to\mathfrak{Y}$
where $\sigma_{1}\in\mathfrak{X}$ is mapped to the associated cluster
$\sigma_{2}\in\mathfrak{Y}$ as described in Proposition 2.10.

The assignment of the Smirnov compactification to a separated proximity space
and the assignment of the extended map between Smirnov compactifications to a
proximity map makes up a functor from Prox to the category of compact
Hausdorff spaces, CompHs.

###  2.2. Coarse Proximity Spaces

The methods we will use in later sections to prove our main results are
centered around the use of coarse proximity spaces, a structure in coarse
geometry first introduced in [7]. Here we will review the important basic
definitions and results. They are as they appear in [7] and [9].

######  Definition 2.12.

A bornology $\mathcal{B}$ on a set $X$ is a family of subsets of $X$
satisfying:

  1. (1)

$\\{x\\}\in\mathcal{B}$ for all $x\in X,$

  2. (2)

$A\in\mathcal{B}$ and $B\subseteq A$ implies $B\in\mathcal{B},$

  3. (3)

If $A,B\in\mathcal{B},$ then $A\cup B\in\mathcal{B}.$

Elements of $\mathcal{B}$ are called bounded and subsets of $X$ not in
$\mathcal{B}$ are called unbounded.

######  Definition 2.13.

Let $X$ be a set equipped with a bornology $\mathcal{B}$. Let $A,B,$ and $C$
be subsets of $X$. A coarse proximity on a set $X$ is a relation ${\bf b}$ on
the power set of $X$ satisfying the following axioms:

  1. (1)

$A{\bf b}B$ implies $B{\bf b}A,$

  2. (2)

$A{\bf b}B$ implies $A\notin\mathcal{B}$ and $B\notin\mathcal{B},$

  3. (3)

$A\cap B\notin\mathcal{B}$ implies $A{\bf b}B,$

  4. (4)

$(A\cup B){\bf b}C$ if and only if $A{\bf b}C$ or $B{\bf b}C,$

  5. (5)

$A\bar{\bf b}B$ implies that there exists a subset $E$ such that $A\bar{\bf
b}E$ and $(X\setminus E)\bar{\bf b}B,$

where $A\bar{{\bf b}}B$ means “$A{\bf b}B$ is not true.” If $A{\bf b}B$, then
we say that $A$ is coarsely close to (or coarsely near) $B.$ Axiom (4) will be
called the union axiom and axiom (5) will be called the strong axiom. A triple
$(X,\mathcal{B},{\bf b})$ where $X$ is a set, $\mathcal{B}$ is a bornology on
$X$, and ${\bf b}$ is a coarse proximity relation on $X,$ is called a coarse
proximity space.

######  Example 2.14.

Let $(X,d)$ be a metric space with the bornology consisting of all the
metrically bounded sets. Define two subsets $A$ and $B$ of $X$ to be coarsely
close if and only if there exists $\epsilon<\infty$ such that for all bounded
sets $D$, there exists $a\in(A\setminus D)$ and $b\in(B\setminus D)$ such that
$d(a,b)<\epsilon.$ Then this relation is a coarse proximity, called the metric
coarse proximity.

######  Example 2.15.

Let $X$ be a Hausdorff topological space and $Z$ a compact Hausdorff space
containing $X$. Define $\mathcal{B}$ to be the collection of subsets of $X$
that are closed as subsets of $Z$. For subsets $A,C\subseteq X$ not in
$\mathcal{B}$, define $A{\bf b}C$ if and only if $cl_{Z}(A)\cap
cl_{Z}(C)\cap(Z\setminus X)\neq\emptyset$. Then $(X,\mathcal{B},{\bf b})$ is a
coarse proximity space.

######  Definition 2.16.

Given subsets $A,C$ of a coarse proximity space $(X,\mathcal{B},{\bf b})$ we
say that $A$ is a coarse neighbourhood of $C$, denoted $C\ll A$, if $C\bar{\bf
b}(X\setminus A)$.

It was shown in [8] that coarse proximity spaces can be characterized entirely
by the relation $\ll$.

######  Definition 2.17.

Let $X$ be a set and $\phi$ an equivalence relation on the power set of $X$
satisfying the following property:

$A\phi B\text{ and }C\phi D\quad\quad\Longrightarrow\quad\quad(A\cup
C)\phi(B\cup D).$

Then we call $\phi$ a weak asymptotic resemblance. If $A\phi B,$ then we say
that $A$ and $B$ are $\phi$ related.

Weak asymptotic resemblances are relaxations of the stronger asymptotic
resemblances defined by Honari and Kalantari in [12]. Any coarse proximity
naturally induces a weak asymptotic resemblance, as in the following theorem:

######  Theorem 2.18.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space. Let $\phi$ be the
relation on the power set of $X$ defined in the following way: $A\phi B$ if
and only if the following hold:

  1. (1)

for every unbounded $B^{\prime}\subseteq B$ we have $A{\bf b}B^{\prime},$

  2. (2)

for every unbounded $A^{\prime}\subseteq A$ we have $A^{\prime}{\bf b}B.$

Then $\phi$ is a weak asymptotic resemblance that we call the weak asymptotic
resemblance induced by the coarse proximity ${\bf b}$.

###### Proof.

See [7]. ∎

Notice that if $\phi$ is the relation defined in Theorem 2.18 and $A$ and $B$
are bounded, then they are always $\phi$ related. If $A$ is bounded and $B$
unbounded, then they are not $\phi$ related. If $A,B$ and $C$ are any subsets
of $X$ such that $A\phi B,$ then $A{\bf b}C$ if and only if $B{\bf b}C.$ If
$\phi$ is induced by the metric coarse proximity, then for nonempty subsets
$\phi$ is the relation of having finite Hausdorff distance. For the proofs of
the above statements, see [7].

######  Remark 2.19.

As the weak asymptotic resemblance $\phi$ induced on a coarse proximity space
$(X,\mathcal{B},{\bf b})$ is completely determined by ${\bf b}$ we will often
use it without explicitly mentioning what it is. When it is not clear which
space or which coarse proximity structure it is induced by we will index it by
the space or coarse proximity structure. That is, if the weak asymptotic
resemblance $\phi$ is induced on the coarse proximity space
$(X,\mathcal{B},{\bf b})$ we may denote this relation as $\phi_{X}$ or
$\phi_{(X,\mathcal{B},{\bf b})}$ as needed to avoid confusion. Of course, when
the coarse proximity space is understood, we will not index the relation at
all.

######  Definition 2.20.

Let $(X,\mathcal{B}_{1},{\bf b}_{1})$ and $(Y,\mathcal{B}_{2},{\bf b}_{2})$ be
coarse proximity spaces. Let $f\mathrel{\mathop{\mathchar
58\relax}}X\rightarrow Y$ be a function. Then $f$ is a coarse proximity map
provided that the following are satisfied for all $A,B\subseteq X$:

  1. (1)

$B\in\mathcal{B}_{1}$ implies $f(B)\in\mathcal{B}_{2},$

  2. (2)

$A{\bf b}_{1}B$ implies $f(A){\bf b}_{2}f(B).$

######  Definition 2.21.

Let $X$ be a set and $(Y,\mathcal{B},{\bf b})$ a coarse proximity space. Two
functions $f,g\mathrel{\mathop{\mathchar 58\relax}}X\to Y$ are close, denoted
$f\sim g$, if for all $A\subseteq X$

$f(A)\phi g(A),$

where $\phi$ is the weak asymptotic resemblance relation induced by the coarse
proximity structure ${\bf b}.$

######  Definition 2.22.

Let $(X,\mathcal{B}_{1},{\bf b}_{1})$ and $(Y,\mathcal{B}_{2},{\bf b}_{2})$ be
coarse proximity spaces. We call a coarse proximity map
$f\mathrel{\mathop{\mathchar 58\relax}}X\to Y$ a coarse proximity isomorphism
if there exists a coarse proximity map $g\mathrel{\mathop{\mathchar
58\relax}}Y\to X$ such that $g\circ f\sim id_{X}$ and $f\circ g\sim id_{Y}.$
We say that $(X,\mathcal{B}_{1},{\bf b}_{1})$ and $(Y,\mathcal{B}_{2},{\bf
b}_{2})$ are isomorphic if there exists a coarse proximity isomorphism
$f\mathrel{\mathop{\mathchar 58\relax}}X\to Y.$

The collection of coarse proximity spaces and closeness classes of coarse
proximity maps makes up the category CrsProx of coarse proximity spaces. For
details, see [7]. In [9] a functor $\mathcal{U}$ from the category ${\bf
CrsProx}$ to the category of compact Hausdorff spaces and continuous maps
called the boundary functor was constructed. For a given coarse proximity
space $(X,\mathcal{B},{\bf b})$ the compact Hausdorff space $\mathcal{U}X$ is
called the boundary of the coarse proximity space. It is constructed as
follows.

######  Proposition 2.23.

Given a coarse proximity space $(X,\mathcal{B},{\bf b})$, the relation
$\delta_{dis}$ on the collection of subsets of $X$ defined by

$A\delta_{dis}B\iff\left\\{\begin{array}[]{ll}A\cap B\neq\emptyset&or\\\ A{\bf
b}B&\end{array}\right.$

is a separated proximity on $X$.

The proximity relation described above is called the discrete extension of the
coarse proximity relation ${\bf b}$. We can then take the corresponding
Smirnov compactification $(\mathfrak{X},\delta_{dis}^{*})$.

######  Definition 2.24.

Given a coarse proximity space $(X,\mathcal{B},{\bf b})$ with corresponding
discrete extension proximity space $(X,\delta_{dis})$ and Smirnov
compactification $(\mathfrak{X},\delta_{dis}^{*})$, the boundary of the coarse
proximity space $X$, is the subset $\mathcal{U}X\subseteq\mathfrak{X}$ whose
elements are precisely those clusters $\sigma\in\mathfrak{X}$ that do no
contain any element of $\mathcal{B}$.

######  Proposition 2.25.

For every coarse proximity space $(X,\mathcal{B},{\bf b})$ the boundary
$\mathcal{U}X$ is a compact Hausdorff space.

######  Proposition 2.26.

If $f\mathrel{\mathop{\mathchar 58\relax}}(X,\mathcal{B}_{X},{\bf
b}_{X})\rightarrow(Y,\mathcal{B}_{Y},{\bf b}_{Y})$ is a coarse proximity maps,
and $\overline{f}\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{X}\rightarrow\mathfrak{Y}$ is the extension of $f$ to the
Smirnov compactifications of the discrete extensions of $X$ and $Y$, then
$\mathcal{U}f=\overline{f}|_{\mathcal{U}X}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}X\rightarrow\mathcal{U}Y$ is well defined and continuous.
Moreover, if $g,f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ are
coarse proximity maps that are close, then $\mathcal{U}f=\mathcal{U}g$.

The assignment of the boundary to a coarse proximity space and the assignment
of $\mathcal{U}f$ to a closeness class of coarse proximity maps $[f]$ makes up
a functor from the category of coarse proximity space CrsProx to CompHs.

######  Example 2.27.

If $(X,d)$ is an unbounded proper metric space, then the boundary of the
metric coarse proximity structure on $X$ is homeomorphic to the Higson corona
of $X$.

######  Definition 2.28.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space, $\delta_{dis}$ the
discrete extension of b, and $\mathfrak{X}$ the corresponding Smirnov
compactification. Given $A\subseteq X$ we define the trace of $A$ on
$\mathcal{U}X$ to be $cl_{\mathfrak{X}}(A)\cap\mathcal{U}X$.

In [9] it was shown that traces are closed subsets of the boundary that
characterize coarse proximities in the following way.

######  Proposition 2.29.

Let $A$ and $C$ be subsets of a coarse proximity space $(X,\mathcal{B},{\bf
b})$, then

  1. (1)

$tr(A)\neq\emptyset$ if and only if $A$ is unbounded

  2. (2)

$A{\bf b}C\iff tr(A)\cap tr(C)\neq\emptyset$

  3. (3)

$A\phi C\iff tr(A)=tr(C)$

  4. (4)

$A\ll C\implies tr(A)\subseteq int(tr(C))$

Another convenient property of the boundary that we use often and follows
quickly from the Urysohn lemma is the following.

######  Proposition 2.30.

If $(X,\mathcal{B},{\bf b})$ is a coarse proximity space with boundary
$\mathcal{U}X$ and $K_{1},K_{2}\subseteq\mathcal{U}X$ are disjoint closed
sets, then there are unbounded sets $A,C\subseteq X$ such that $A\bar{\bf
b}C$, $K_{1}\subseteq int_{\mathcal{U}X}(tr(A))$, and $K_{2}\subseteq
int_{\mathcal{U}X}(tr(C))$.

###### Proof.

Proof can be found in [9]. ∎

######  Lemma 2.31.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space, $C,D\subseteq X$
unbounded sets such that $C\ll D$, and $\sigma\in int(tr(C))$. Then there is
an unbounded set $K\subseteq X$ such that $\sigma\in int(tr(K))$ and $K\ll D$.

###### Proof.

As $\sigma\in int(tr(C))$ we can find an open subset $U\subseteq\mathcal{U}X$
that contains $\sigma$ and whose closure is completely contained in
$int(tr(C))$. Using Urysohn’s lemma on the Smirnov compactification
$\mathfrak{X}$ of the discrete extension of $(X,\mathcal{B},{\bf b})$ there is
a continuous function $f\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{X}\rightarrow[0,1]$ that maps $\\{\sigma\\}$ to $0$ and
$\mathcal{U}X\setminus U$ to $1$. Defining $K=f^{-1}([0,1/3))\cap X$ we have
that $\sigma\in int(tr(K))\subseteq tr(K)\subseteq U\subseteq tr(C)$. It is
then clear from Proposition 2.29 that $K\ll D$.

∎

######  Proposition 2.32.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space with nonempty
boundary $\mathcal{U}X$. If $U,V\subseteq\mathcal{U}X$ are nonempty empty sets
such that $\overline{V}\subseteq U$, then there is an unbounded set
$A\subseteq X$ such that $\overline{V}\subseteq int(tr(A))\subseteq
tr(A)\subseteq U$ and if $C\subseteq X$ is an unbounded set such that
$tr(C)\subseteq\overline{V}$, then $C\ll A$.

###### Proof.

Let $\delta$ be the discrete extension of the coarse proximity on $X$ and let
$\mathfrak{X}$ be the corresponding Smirnov compactification. As in Lemma 2.31
we use Urysohn’s lemma to get a continuous function
$f\mathrel{\mathop{\mathchar 58\relax}}\mathfrak{X}\rightarrow[0,1]$ such that
$f(\overline{V})=0$ and $f(\mathcal{U}X\setminus U)=1$. Define
$A=f^{-1}([0,1/3])\cap X$. Then $\overline{V}\subseteq int(tr(A))$ and
$tr(A)\subseteq U$. Now let $C\subseteq X$ be such that
$tr(C)\subseteq\overline{V}$. If $C{\bf b}(X\setminus A)$, then by Lemma 2.29
we would have to have that $tr(C)\cap tr(X\setminus A)\neq\emptyset$. However,
as $tr(C)\subseteq\overline{V}\subseteq int(tr(A))$, this is impossible.
Therefore $C\ll A$. ∎

######  Definition 2.33.

Given a coarse proximity space $(X,\mathcal{B},{\bf b})$ and a nonempty subset
$Y\subseteq X$, the subspace coarse proximity structure on $Y$ is
$(Y,\hat{\mathcal{B}},\hat{\bf b})$ where

$\hat{\mathcal{B}}=\\{B\cap Y\mid B\in\mathcal{B}\\}$

and for subsets $A,C\subseteq Y$ we define

$A{\bf b}_{Y}C\iff A{\bf b}_{X}C$

######  Proposition 2.34.

Let $(X,\mathcal{B}_{X},{\bf b}_{X})$ be a coarse proximity space and
$(Y,\mathcal{B}_{Y},{\bf b}_{Y})$ a subspace of $X$. Then $\mathcal{U}Y$ is
homeomorphic to $tr_{\mathcal{U}X}(Y)$ and the inclusion map
$\iota\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ induces an
embedding of $\mathcal{U}Y$ into $\mathcal{U}X$ such that
$\mathcal{U}\iota(\mathcal{U}Y)=tr_{\mathcal{U}X}(Y)$.

###### Proof.

It will suffice to show that if $\sigma\in tr_{\mathcal{U}X}(Y)$ then $\sigma$
contains an element of $\mathcal{U}Y$. Let $\sigma\in tr_{\mathcal{U}X}(Y)$ be
given. Then $Y\in\sigma$. Let

$\hat{\sigma}=\\{A\subseteq Y\mid A\in\sigma\\}$

We claim that $\hat{\sigma}$ is a cluster in $Y$. If $A,C\in\hat{\sigma}$,
then $A,C\in\sigma$ which implies $A{\bf b}_{X}C$. The coarse proximity ${\bf
b}_{Y}$ is simply a restriction of ${\bf b}_{X}$ to subsets of $Y$, so $A{\bf
b}_{Y}C$. If $(A\cup C)\in\hat{\sigma}$, then $(A\cup C)\in\sigma$, so
$A\in\sigma$ or $C\in\sigma$ which implies $A\in\hat{\sigma}$ or
$C\in\hat{\sigma}$. Finally, let $A\subseteq Y$ be such such that $A{\bf
b}_{Y}C$ for all $C\in\hat{\sigma}$. If $A\notin\sigma$ there is some
$D\subseteq X$ such that $D\in\sigma$ and $D\bar{\bf b}_{X}A$. However,
$D\in\sigma$ implies $\sigma\in tr_{\mathcal{U}X}(D)$, so we would be able to
find an unbounded subset $E_{1}\subseteq X$ such that $\sigma\in
int(tr_{\mathcal{U}X}(E_{1}))$ and $tr_{\mathcal{U}X}(E_{1})\cap
tr_{\mathcal{U}X}(A)=\emptyset$. Using the strong axiom we can find two other
such sets $E_{2},E_{3}\subseteq X$ with the additional property that $E_{1}\ll
E_{2}\ll E_{3}$. We then claim that $E_{2}\cap Y$ must be unbounded. If not,
then there is a bounded set $B\in\mathcal{B}_{X}$ such that $(Y\setminus
B)\subseteq(X\setminus E_{2})$. Then $(Y\setminus B)\phi_{X}Y$ and we must
have that $E_{1}\bar{\bf b}_{X}Y$. However, this is impossible as
$Y\in\sigma$. Therefore $E_{2}\cap Y$ is unbounded. Note that
$tr_{\mathcal{U}X}(Y)=tr_{\mathcal{U}X}(Y\cap E_{2})\cup
tr_{\mathcal{U}X}(Y\setminus E_{2})$. Because $\sigma\notin
tr_{\mathcal{U}X}(X\setminus E_{2})$ we must have that because $\sigma\in
tr_{\mathcal{U}X}(Y)$ that $\sigma\in tr_{\mathcal{U}X}(Y\cap E_{2})$.
Therefore $(Y\cap E_{2})\in\sigma$ and $(Y\cap E_{2})\in\hat{\sigma}$. Then,
by assumption $A{\bf b}_{Y}(Y\cap E_{2})$. However, by the construction of
$E_{2}$ we have that $A\bar{\bf b}_{X}(Y\cap E_{2})$, so this is impossible.
We then must have that $A$ is an element of $\sigma$. We have then established
that $\hat{\sigma}$ is a cluster in $Y$. It is then easy to see that
$\mathcal{U}\iota(\hat{\sigma})=\sigma$ meaning
$\mathcal{U}\iota(\mathcal{U}Y)=tr_{\mathcal{U}X}(Y)$.

The map $\mathcal{U}\iota$ is quickly seen to be injective as if
$\sigma_{1},\sigma_{2}\in\mathcal{U}Y$ are distinct then there are
$A\in\sigma_{1}$ and $C\in\sigma_{2}$ such that $A\bar{\bf b}_{Y}C$. Then
$A\bar{\bf b}_{X}C$, so $\mathcal{U}\iota(\sigma_{1})$ can not be equal to
$\mathcal{U}\iota(\sigma_{2})$, making $\mathcal{U}\iota$ injective. Then
$\mathcal{U}\iota$ is a continuous injection of the compact Hausdorff
$\mathcal{U}Y$ into $\mathcal{U}X$ whose image is $tr_{X}(Y)$, which makes
$\mathcal{U}\iota$ and embedding whose image is exactly $tr_{X}(Y)$. ∎

Next we will construct an analog of the disjoint union for coarse proximity
spaces. Let $\\{(X_{\alpha},\mathcal{B}_{\alpha},{\bf
b}_{\alpha})\\}_{\alpha\in S}$ be a family of coarse proximity spaces indexed
by some set $S$. Define $X$ to be the disjoint union of all of the
$X_{\alpha}$’s. That is, $X=\bigsqcup_{\alpha\in S}$. We then consider the
following collection of sets.

$\mathcal{B}=\\{B\subseteq X\mid B\cap
X_{\alpha}\in\mathcal{B}_{\alpha},\,\forall\alpha\in S\\}$

######  Proposition 2.35.

The collection $\mathcal{B}$ is a bornology on $X$.

###### Proof.

If $x\in X$, then $x\in X_{\alpha}$ for some $\alpha\in S$. Then as
$\\{x\\}\in\mathcal{B}_{\alpha}$ we have that $\\{x\\}\cap
X_{\alpha}\in\mathcal{B}_{\alpha}$ and $\\{x\\}\cap
X_{\beta}=\emptyset\in\mathcal{B}_{\beta}$ for all $\beta\neq\alpha$, so
$\\{x\\}\in\mathcal{B}$, meaning $\mathcal{B}$ covers $X$. If
$A\in\mathcal{B}$ and $B\subseteq A$, then $(B\cap X_{\alpha})\subseteq(A\cap
X_{\alpha})$ for all $\alpha\in S$. As $(A\cap
X_{\alpha})\in\mathcal{B}_{\alpha}$ for all $\alpha$ we have that $(B\cap
X_{\alpha})\in\mathcal{B}_{\alpha}$ for each $\alpha$, meaning
$B\in\mathcal{B}$. Finally if $A,B\in\mathcal{B}$, then $(A\cup B)\cap
X_{\alpha}=(A\cap X_{\alpha})\cup(B\cap X_{\alpha})$, which yields $(A\cup
B)\cap X_{\alpha}\in\mathcal{B}_{\alpha}$ for each $\alpha$ and $(A\cup
B)\in\mathcal{B}$. Therefore $\mathcal{B}$ is a bornology on $X$. ∎

######  Proposition 2.36.

The relation ${\bf b}$ defined on subsets of $(X,\mathcal{B})$ defined by

$A{\bf b}B\iff\exists\alpha\in S,\,(A\cap X_{\alpha}){\bf b}_{\alpha}(B\cap
X_{\alpha})$

is a coarse proximity on $X$.

###### Proof.

The only axioms that aren’t clear are the union axiom and the strong axiom.
Let $A,B,C\subseteq X$ be such that $(A\cup B){\bf b}C$. Then there is an
$\alpha\in S$ such that $[(A\cup B)\cap X_{\alpha}]{\bf b}_{\alpha}(C\cap
X_{\alpha})$. Then $(A\cap X_{\alpha}){\bf b}_{\alpha}(C\cap X_{\alpha})$ or
$(B\cap X_{\alpha}){\bf b}_{\alpha}C$. In either case we have that one of $A$
or $B$ is related to $C$ by ${\bf b}$. The converse is trivial.

Now let $A,C\subseteq X$ be such that $A\bar{\bf b}C$. Then for each
$\alpha\in S$ we have that $(A\cap X_{\alpha})\bar{\bf b}_{\alpha}(C\cap
X_{\alpha})$. For each $\alpha\in S$ let $E_{\alpha}\subseteq X_{\alpha}$ be
such that $(A\cap X_{\alpha})\bar{\bf b}_{\alpha}(X_{\alpha}\setminus
E_{\alpha})$ and $E_{\alpha}\bar{\bf b}_{\alpha}(C\cap X_{\alpha})$. Define
$E=\bigcup_{\alpha\in S}E_{\alpha}$. Then $A\bar{\bf b}(X\setminus E)$ and
$E\bar{\bf b}C$. Therefore ${\bf b}$ is a coarse proximity on
$(X,\mathcal{B})$. ∎

######  Proposition 2.37.

If the index set $S$ is finite in the above construction with
$S=\\{1,\ldots,n\\}$, then $\mathcal{U}X$ is homeomorphic to the disjoint
union of the $\mathcal{U}X_{i}$.

###### Proof.

We will consider each $X_{i}$ as a subset of $X$. For each $A\subseteq X$ let
$A_{i}=A\cap X_{i}$. For a given $\sigma\in\mathcal{U}X$ we claim that there
is a unique $i\in S$ such that

$\sigma_{i}=\\{A_{i}\mid A\in\sigma\\}$

is an element of $\mathcal{U}X_{i}$. First let $A\in\sigma$ be given. Say,
$A=\left(\bigcup A_{i}\right)$. Then, by the definition of a cluster we must
have that there is a $j$ for which $A_{j}\in\sigma$. In fact, there is only
one such index as if $i\neq j$ is such that $A_{j}\in\sigma$ then we would
have that $A_{i}{\bf b}A_{j}$, which is a contradiction. Therefore, for each
$A\in\sigma$ there is a unique $j$ for which $A_{j}\in\sigma$. By similar
reasoning as above we have that there is a unique index $i$ such that
$A_{i}\in\sigma$ for all $A\in\sigma$. We then have that
$\sigma_{i}\subseteq\sigma$. This itself implies that $\sigma_{i}$ is a
cluster in $\mathcal{U}X_{i}$. That there is no $j\neq i$ for which
$\sigma_{j}\in\mathcal{U}X_{j}$ is obvious.

Finally, define a function $f\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}X\rightarrow\bigsqcup_{i=1}^{n}\mathcal{U}X_{i}$ by
mapping each $\sigma\in\mathcal{U}X$ to the unique $\sigma_{i}$ such that
$\sigma_{i}\in\mathcal{U}X_{i}$. This map is a homeomorphism.

∎

The following definition is a slight variation of a definition in [17] to
characterize the covering dimension of the boundaries of coarse proximity
spaces. A similar definition is given by Hartmann in [10]. Both definitions
are based on the definition of a particular kind of cover for small scale
proximity spaces defined by Smirnov in [18] to define a dimension function for
proximity spaces. We will use the following definition in later sections when
defining a coarse analog of a fine collection of canonical covers.

######  Definition 2.38.

A b-cover (or coarse cover with respect to b) of a coarse proximity space
$(X,\mathcal{B},{\bf b})$ is a finite collection of sets
$\\{A_{1},\ldots,A_{n}\\}$ satisfying:

  1. (1)

There is a unique, potentially empty, bounded $A_{i}$.

  2. (2)

The collection covers $X$

  3. (3)

There are set $C_{1},\ldots,C_{n}$ that also cover and satisfy $C_{i}\ll
A_{i}$ for each $i$.

######  Proposition 2.39.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space with boundary
$\mathcal{U}X$. If $\\{A_{1},\ldots,A_{n}\\}$ is a ${\bf b}$-cover of $X$,
then $\\{int(tr(A_{1})),\ldots,int(tr(A_{n})\\}$ is an open cover of
$\mathcal{U}X$.

######  Proposition 2.40.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space with boundary
$\mathcal{U}X$. If $\mathcal{U}=\\{U_{1},\ldots,U_{n}\\}$ and
$\mathcal{V}=\\{V_{1},\ldots,V_{m}\\}$ are open covers of $\mathcal{U}X$ such
that $\overline{\mathcal{V}}=\\{\overline{V}_{1},\ldots,\overline{V}_{m}\\}$
refines $\mathcal{U}$, then there is a ${\bf b}$-cover
$\\{A_{1},\ldots,A_{m}\\}$ of $X$ such that $\overline{V}_{i}\subseteq
tr(A_{i})$ for each $i$ and $\overline{V}_{i}\subseteq U_{j}$ implies that
$tr(A_{i})\subseteq U_{j}$.

It is clear from the fact that coarse proximity maps copreserve the coarse
neighbourhood operator $\ll$ that the preimages of ${\bf b}$-covers under
coarse proximity maps are ${\bf b}$-covers of the domain of the map. Some
final definitions and results we will make use of are the following as can be
found in [17].

######  Definition 2.41.

Given subsets $A,C$ of a coarse proximity space $(X,\mathcal{B},{\bf b})$ we
define

  1. (1)

$A\sqsubseteq C$ if for all $D\subseteq X$ we have that $A{\bf b}D$ implies
$C{\bf b}D$.

  2. (2)

$A\ll_{w}C$ if there is a $D\subseteq X$ such that $A\ll D\sqsubseteq C$.

In case $(1)$ we say that $A$ is a coarse subset of $C$. In case $(2)$ we say
that $C$ is a weak coarse neighbourhood of $A$.

The weak coarse neighbourhood operator is in some ways nicer than the normal
coarse neighbourhood operator as can be seen in the following proposition.

######  Proposition 2.42.

Let $A,C$ be subsets of coarse proximity spaces $(X,\mathcal{B},{\bf b})$.
Then

  1. (1)

$A\sqsubseteq C$ if and only if $tr(A)\subseteq tr(C)$.

  2. (2)

$A\ll_{w}C$ if and only if $tr(A)\subseteq int(tr(C))$.

###  2.3. Cofinal Dimension

A large part of our results will be focused on defining an analog of
Ponomarev’s cofinal dimension (a topological property) for coarse proximity
spaces. For the sake of self containment we provide all of the necessary
definitions and important results about the cofinal dimension here. They are
as found in [15].

######  Definition 2.43.

A locally finite closed cover $\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$
of a topological space $X$ is called a canonical cover if there is a disjoint
collection of open sets $\\{U_{\alpha}\\}_{\alpha\in\Omega}$ such that
$\mathcal{F}_{\alpha}=cl(U_{\alpha})$ for each $\alpha\in\Omega$.

######  Lemma 2.44.

If $\\{F_{\alpha}\\}_{\alpha\in\Omega}$ is a canonical cover of a topological
space $X$, then the collection $\\{int(F_{\alpha})\\}_{\alpha\in\Omega}$ is a
disjoint family.

###### Proof.

Let $\\{F_{\alpha}\\}_{\alpha\in\Omega}$ be a canonical cover of a space $X$
with $\\{G_{\alpha}\\}_{\alpha\in\Omega}$ a disjoint collection of open sets
such that $F_{\alpha}=cl(G_{\alpha})$ for each $\alpha$. Assume towards a
contradiction that there are $\alpha\neq\beta$ and $x\in X$ such that $x\in
int(F_{\alpha})\cap int(F_{\beta})$. As $cl(G_{\alpha})=F_{\alpha}$ and
$cl(G_{\beta})=F_{\beta}$ we have that $int(F_{\alpha})\cap
G_{\beta}\neq\emptyset$ and $int(F_{\beta})\cap G_{\alpha}\neq\emptyset$. This
implies that $G_{\alpha}$ and $G_{\beta}$ must intersect, which is a
contradiction. Thus the family $\\{int(F_{\alpha})\\}_{\alpha\in\Omega}$ must
be disjoint. ∎

######  Lemma 2.45.

If $\\{F_{\alpha}\\}_{\alpha\in\Omega}$ is a canonical cover of a compact
Hausdorff space $X$, then $\Omega$ must be finite.

###### Proof.

By Lemma 2.44 we have that $\\{int(F_{\alpha})\\}_{\alpha\in\Omega}$ is a
disjoint family. For each $\alpha$ let $x_{\alpha}\in int(F_{\alpha})$ be
given and $W_{\alpha}\subseteq int(F_{\alpha})$ be an open set such that
$cl(W_{\alpha})\subseteq int(F_{\alpha})$. Then define
$U=X\setminus\left(\bigcup_{\alpha}cl(W_{\alpha})\right)$. Then the collection
$\\{U\\}\cup\\{int(F_{\alpha})\mid\alpha\in\Omega\\}$ is an open cover of $X$
which must admit a finite subcover by compactness. As each $int(F_{\alpha})$
contains some point not contained in any other element of the open cover, we
have that the finite subcover must contain each $int(F_{\alpha})$. Therefore
$\Omega$ must be finite.

∎

######  Proposition 2.46.

If $\\{U_{\alpha}\\}_{\alpha\in\Omega}$ is an open cover of a topological
space $X$ whose corresponding closed cover
$\\{cl(U_{\alpha})\\}_{\alpha\in\Omega}$ is locally finite, then there is a
canonical cover $\\{F_{\lambda}\\}_{\lambda\in\Lambda}$ of $X$ that refines
$\\{cl(U_{\alpha})\\}_{\alpha\in\Omega}$.

######  Definition 2.47.

A collection $\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ of canonical
covers of a topological space $X$ is called fine if every open cover
$\mathcal{U}$ of $X$ is refined by some $\mathcal{F}_{\alpha}$.

######  Definition 2.48.

A topological space $X$ with a basis of open and closed sets such that each
open covering $X$ has a disjoint open refinement is called a perfectly zero-
dimensional space.

######  Definition 2.49.

A collection of canonical covers
$\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ of a topological space $X$ is
called directed if $\Omega$ is a directed set and $\alpha\leq\beta$ implies
$\mathcal{F}_{\beta}$ refines $\mathcal{F}_{\alpha}$, where
$\alpha,\beta\in\Omega$.

######  Definition 2.50.

Given a topological space $X$, the cofinal dimension of $X$, denoted
$\Delta(X)$, is defined to be $-1$ if $X$ is empty. Otherwise $\Delta(X)$ is
the least nonnegative integer $n$ such that $X$ admits a fine directed
collection of canonical covers $\\{F_{\alpha}\\}_{\alpha\in\Omega}$ such that
the order of $\mathcal{F}_{\alpha}$ does not exceed $n+1$ for each
$\alpha\in\Omega$. If no such $n$ exists, we say that $\Delta(X)$ is infinite.

######  Proposition 2.51.

For a topological space $X$, the following are equivalent:

  1. (1)

$X$ is perfectly zero dimensional,

  2. (2)

$X$ is a paracompact regular space such that $dim(X)=0$,

  3. (3)

$X$ is a regular space such that $\Delta(X)=0$.

######  Definition 2.52.

A continuous surjection $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$
between topological spaces is called irreducible if $f(A)\neq Y$ for each
proper closed set $A$ of $X$.

As the following proposition from [15] shows, irreducible maps are easy to
find.

######  Proposition 2.53.

If $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ is a perfect map,
then there is a closed set $A\subseteq X$ such that $f\mid
A\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow Y$ is a perfect irreducible
mapping.

In particular, every surjection between compact Hausdorff spaces can be
restricted to a subset of its domain on which it is irreducible.

Given a surjection $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ and
subset $B\subseteq X$ define

$f^{*}(B)=\\{y\in Y\mid f^{-1}(y)\subseteq B\\}$

These sets can be used to characterize irreducible maps as well as closed
maps.

######  Lemma 2.54.

A continuous surjection $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$
is an irreducible map if and only if $f^{*}(U)$ is nonempty for each nonempty
open set $U\subseteq X$. The map $f$ is closed if and only if $f^{*}(U)$ is
open for each nonempty open set $U\subseteq X$.

######  Lemma 2.55.

If $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ is a closed
irreducible map, then for each open $U\subseteq X$ we have that

$f(cl(U))=cl(f^{*}(U))$

######  Proposition 2.56.

If $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ is a perfect
irreducible map and $\\{F_{\alpha}\\}_{\alpha\in\Omega}$ is a canonical
covering of $X$, then $\\{f(F_{\alpha})\\}_{\alpha\in\Omega}$ is a canonical
covering of $Y$.

######  Theorem 2.57.

For a regular topological space $X$, the following are equivalent:

  1. (1)

$\Delta(X)\leq n$

  2. (2)

there is a perfectly zero dimensional space $Z$ and a continuous closed
surjection $f\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow X$ such that
$f^{-1}(x)$ contains at most $n+1$ points for each $x\in X$.

The proof of the above theorem in [15] constructs a perfectly zero dimensional
space in a particular way. This construction will be mimicked in section $8$
so we will review it here. We will omit the verification of the various
details of the construction, but they can be found in [15].

Let $X$ be a regular topological space and
$\mathcal{F}=\\{\mathcal{F}^{\alpha}\\}_{\alpha\in\Omega}$ be a fine directed
set of canonical covers of $X$ such that the order of each
$\mathcal{F}^{\alpha}$ does not exceed $n+1$. Assume that for each $\alpha$ we
have that
$\mathcal{F}^{\alpha}=\\{F^{\alpha}_{\lambda}\\}_{\lambda\in\Lambda(\alpha)}$
where $\Lambda(\alpha)$ is an index set for $\mathcal{F}_{\alpha}$. The family
$\mathcal{F}$ can be strongly directed in that for $\alpha,\beta\in\Omega$
such that $\beta\geq\alpha$ there is a function
$\rho_{\beta\alpha}\mathrel{\mathop{\mathchar
58\relax}}\Lambda(\beta)\rightarrow\Lambda(\alpha)$ such that
$\\{\Lambda(\alpha),\rho_{\beta\alpha}\\}_{\alpha,\beta\in\Omega}$ is an
inverse system of sets over $\Omega$ and for each $\lambda\in\Lambda(\alpha)$
we have

$F^{\alpha}_{\lambda}=\bigcup\\{F^{\beta}_{\mu}\mid\rho_{\beta\alpha}(\mu)=\lambda\\}$

For each $\alpha\in\Omega$ a topological space $X_{\alpha}$ can be constructed
by defining
$X_{\alpha}=\bigcup_{\lambda\in\Lambda(\alpha)}F^{\alpha}_{\lambda}\times\\{\lambda\\}$,
which is the disjoint union of the elements of $\mathcal{F}^{\alpha}$ and
giving it the corresponding disjoint union topology. The obvious function
$\sigma_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}X_{\alpha}\rightarrow X$
defined by $\sigma_{\alpha}(x,\lambda)=x$ is continuous. When
$\beta,\alpha\in\Omega$ are such that $\beta\geq\alpha$ the function
$g_{\beta\alpha}\mathrel{\mathop{\mathchar 58\relax}}X_{\beta}\rightarrow
X_{\alpha}$ defined by
$g_{\beta\alpha}(x,\lambda)=(x,\rho_{\beta\alpha}(\lambda))$ is a continuous
closed surjection. These maps make
$\\{X_{\alpha},g_{\beta\alpha}\\}_{\alpha,\beta\in\Omega}$ an inverse system
of topological spaces over $\Omega$. The space $Z$ used in [15] to prove
Theorem 2.57 is the inverse limit of this system. The elements of $Z$ are of
the form $\\{(x,\lambda,\alpha)\\}_{\alpha\in\Omega}$ where
$\lambda\in\Lambda(\alpha)$. The obvious function that maps onto the first
coordinate is the irreducible closed $n+1$-to-$1$ surjection mentioned in the
theorem. Calling this function $h$ and denoting the projections from $Z$ to
$X_{\alpha}$ by $\pi_{\alpha}$ we have that
$\sigma_{\alpha}\circ\pi_{\alpha}=h$ for each $\alpha$.

We will finish this section with a minor original result. Specifically we will
show that, for compact Hausdorff spaces, the definition of canonical covers
can be weakened without changing the cofinal dimension.

######  Definition 2.58.

Let $X$ be a topological space. We say that a locally finite closed cover
$\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ is an almost canonical cover if
it satisfies the following:

  1. (1)

$int(F_{\alpha})\neq\emptyset$ for all $\alpha\in\Omega$

  2. (2)

For $\alpha\neq\beta$, $int(F_{\alpha})\cap
int(\mathcal{F}_{\beta})=\emptyset$

  3. (3)

If $x\in X$ and $U\subseteq X$ is an open subset of $X$ that contains $x$,
then there is a $y\in U$ and an $\alpha\in\Omega$ such that $y\in
int(\mathcal{F}_{\alpha})$.

######  Proposition 2.59.

Let $X$ be a compact Hausdorff space. Then the following are equivalent:

  1. (1)

$\Delta(X)\leq n$

  2. (2)

The statement “$\Delta(X)\leq n$” still holds upon replacing every instance of
“canonical cover” in the definition of $\Delta(X)$ with ”almost canonical
cover”.

###### Proof.

That $(1)$ implies $(2)$ is immediate upon observing that every canonical
cover of a compact Hausdorff space $X$ is an almost canonical cover.

To prove that $(2)$ implies $(1)$ we first show that every almost canonical
cover $\mathcal{V}=\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ induces a
canonical cover $\mathcal{U}$ that refines $\mathcal{V}$ and has order not
exceeding that of $\mathcal{V}$. If
$\mathcal{V}=\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ is an almost
canonical cover define
$\mathcal{U}=\\{\hat{\mathcal{F}}_{\alpha}=cl(int(\mathcal{F}_{\alpha}))\mid\alpha\in\Omega\\}$.
We claim that $\mathcal{U}$ is our desired cover. First, note that because
$\mathcal{V}$ is locally finite, so is $\mathcal{U}$. Secondly, let $x\in X$
be given. As $\mathcal{U}$ is locally finite, and $\mathcal{V}$ satisfies
$(2)$ of Definition 2.58 there is a finite subcollection
$\\{\hat{\mathcal{F}}_{\alpha_{1}},\ldots,\hat{\mathcal{F}}_{\alpha_{m}}\\}$
of $\mathcal{V}$ such that $x$ admits a neighbourhood basis that only
intersect the $\hat{\mathcal{F}}_{\alpha_{i}}$. By the pigeonhole principle
$x$ must be an element of one of the $\hat{\mathcal{F}}_{\alpha_{i}}$.
Therefore $\mathcal{U}$ covers $X$. Finally, as the interiors of the
$\mathcal{F}_{\alpha}$ are disjoint, so are the interiors of the
$\hat{\mathcal{F}}_{\alpha}$. We then have that $\mathcal{U}$ is a canonical
cover of $X$. It is clear from the definition of $\mathcal{U}$ that the order
of $\mathcal{U}$ does not exceed that of $\mathcal{V}$. Now, assume that we
have that $X$ admits a directed fine family
$\\{\mathcal{V}_{\alpha}\\}_{\alpha\in\Lambda}$ of almost canonical covers
none of which has order exceeding $n$. Then the corresponding collection of
canonical covers $\\{\mathcal{U}_{\alpha}\\}_{\alpha\in\Lambda}$ is a fine
directed collection of canonical covers none of which has order exceeding $n$.
Therefore $\Delta(X)\leq n$.

∎

######  Lemma 2.60.

Let $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ be a perfect
irreducible map between compact Hausdorff spaces. If
$\mathcal{F}=\\{A_{1},\ldots,A_{n}\\}$ is a almost canonical cover of $X$,
then $f(\mathcal{F})=\\{f(A_{1}),\ldots,f(A_{n})\\}$ is a almost canonical
cover of $Y$.

###### Proof.

Let $\\{A_{1},\ldots,A_{n}\\}$ be an almost canonical cover of $X$. For each
$i$ define $F_{i}=cl_{X}(int(A_{i}))$. As shown in the proof of Proposition
2.59, the set $\\{F_{1},\ldots,F_{n}\\}$ is a canonical cover of $X$. By
Proposition 2.56 the set $\\{f(F_{1}),\ldots,f(F_{n})\\}$ is a canonical cover
of $Y$. To show that $\\{f(A_{1}),\ldots,f(A_{n})\\}$ we will begin by showing
that if $y\in int(f(A_{i}))$, then $y\in int(f(F_{i}))$. Let $W\subseteq Y$ be
an open set containing $y$ such that $W\subseteq int(f(A_{i}))$. Define
$W_{i}=f^{-1}(W)\cap A_{i}$. As $W\subseteq int(f(A_{i}))$ we have that
$f(W_{i})=W$. We claim that $W_{i}\subseteq int(A_{i})$. If $x\in W_{i}$ is
not in $int(A_{i})$, then because $f^{-1}(W)$ is open in $X$ and $\mathcal{F}$
is an almost canonical cover of $X$, there is an open set $V_{j}$ contained in
$f^{-1}(W)$ that is completely contained in $int(A_{j})$ for some $j\neq i$.
Then $f(V_{j})\subseteq W$. Note that, by Lemma 2.55 we have that
$f(F_{j})=f(cl(int(A_{j})))=cl(f^{*}(int(A_{j})))$. As $V_{j}\subseteq
int(A_{j})$ we have that $f^{*}(V_{j})$ is a nonempty open subset of
$f(F_{j})$ by Lemma 2.55. However this is impossible as $f(V_{j})\subseteq
W\subseteq int(F_{i})$ which is disjoint from $int(F_{j})$. Therefore
$W_{i}\subseteq int(A_{i})$. Consequently $W_{i}$ is open as well. Now,
$f(cl(W_{i}))\subseteq f(cl(int(A_{i})))=f(F_{i})$ which gives us that $y\in
f(F_{i})$. Finally, as $f(W_{i})=W$ which is open in $Y$ we have that $y\in
int(f(F_{i}))$ and $int(f(A_{i}))=int(f(F_{i}))$ which gives us that the
interiors of the various $f(A_{i})$ are disjoint.

Finally, we need to show that if $y\in Y$ is any point and $W\subseteq Y$ is
an open subset of $Y$ containing $y$, then there is an $i$ and an $x\in W$
such that $x\in W\cap int(f(A_{i}))$. If $y$ and $W$ are given we may consider
$f^{-1}(y)$ and $f^{-1}(W)$. As the collection of $A_{j}$’s is almost
canonical there must be some $i$ for which $f^{-1}(W)\cap int(A_{i})$ is
nontrivial. Call this intersection $W_{i}$. By Proposition 2.54 we have that
$f^{*}(W_{i})$ is open in $Y$. This set is necessarily contained in $W$. Then
if $z\in f^{*}(W_{i})$ we have that $f^{-1}(z)\subseteq
f^{-1}(f^{*}(W_{i}))\subseteq int(A_{i})$. Then $f(f^{-1}(f^{*}(W)))$ is an
open set containing $z$, contained in $W$, that is also contained in
$f(A_{i})$.

∎

##  3\. Asymptotic dimension of coarse proximity spaces

In this section we will define and prove some very basic properties of the
asymptotic dimension of coarse proximity spaces. The results and definitions
are largely rephrasings of definitions and results for metric spaces and
coarse spaces as one might find in [2] and [16]. We also provide some results
about uniformly bounded covers in coarse proximity spaces that will be used in
later sections. The definition of uniformly bounded covers that we present
here is identical in form to the definition of uniformly bounded covers for
asymptotic resemblance spaces as in [12].

######  Definition 3.1.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space. A collection,
$\mathcal{U}$, of subsets of $X$ is called a uniformly bounded family if

  1. (1)

$B\in\mathcal{U}$ implies $B\in\mathcal{B}$

  2. (2)

For all $A,C\subseteq X$, $A\subseteq st(C,\mathcal{U})$ and $C\subseteq
st(C,\mathcal{U})$ implies $A\phi C$

If in addition we have that for all $x\in X$ there is a $B\in\mathcal{U}$ such
that $x\in B$ then we call $\mathcal{U}$ a uniformly bounded cover of $X$.

######  Proposition 3.2.

Let $(X,d)$ be a metric space and $(X,\mathcal{B},{\bf b})$ the corresponding
metric coarse proximity structure. Then a collection $\mathcal{U}$ of subsets
of $X$ is uniformly bounded in metric $d$ if and only if it is a uniformly
bounded family with respect to the coarse proximity structure. Consequently
$\mathcal{U}$ is a uniformly bounded cover of $X$ in the metric sense if and
only if it is so in the coarse proximity sense.

###### Proof.

Let $\mathcal{U}$ be a collection of subsets of $X$ that is uniformly bounded
with respect to the metric $d$. Then there is an $M<\infty$ such that for all
$B\in\mathcal{U}$ we have that $diam(B)\leq M$. Thus $B\in\mathcal{B}$ for all
$B\in\mathcal{U}$. If $A,C\subseteq X$ are such that $A\subseteq
st(C,\mathcal{U})$ and $C\subseteq st(A,\mathcal{U})$ then the Hausdorff
distance between $A$ and $C$ is at most $M$, which implies $A\phi C$.
Therefore $\mathcal{U}$ is a uniformly bounded family with respect to the
coarse proximity structure.

Now assume that $\mathcal{U}$ is a uniformly bounded family of $X$ with
respect to the coarse proximity structure. The second condition in the
definition of a uniformly bounded family says that if $A,C\subseteq X$ are
such that $A\subseteq st(C,\mathcal{U})$ and $C\subseteq st(A,\mathcal{U})$
then $A\phi C$ where $\phi$ is the relation defined by having finite Hausdorff
distance. If $B\in\mathcal{U}$ is unbounded in the metric sense then for any
$x\in B$ we have that $B\subseteq st(x,\mathcal{U})$ and $\\{x\\}\subseteq
st(B,\mathcal{U})$, but $\\{x\\}\bar{\phi}B$. Therefore every element of
$\mathcal{U}$ must be bounded in the metric sense. If $\mathcal{U}$ is not
uniformly bounded in the metric sense, then for every $n\in\mathcal{N}$ we can
find a $B_{n}\in\mathcal{U}$ such that $diam(B_{n})\geq n$. Fix a point
$x_{0}\in X$.

######  Claim 3.3.

For every $n\in\mathbb{N}$ there is an element $C_{n}\in\mathcal{U}$ such that
$diam(C_{n}\setminus B(x_{0},n^{2}))>n$.

###### Proof.

Assume otherwise. Then there is an $n\in\mathbb{N}$ such that if
$B\in\mathcal{U}$, then $diam(C\setminus B(x_{0},n^{2}))\leq n$. Then by the
triangle inequality we have that $diam(B)\leq n+2n^{2}$ for all
$B\in\mathcal{U}$, contradicting our assumption that $\mathcal{U}$ is not
uniformly bounded in the metric sense. Thus the claim holds. ∎

Now, let $B_{1}\in\mathcal{U}$ be such that $diam(B_{1})>1$ and let
$x_{1},y_{1}\in B_{1}$. Choose $m$ large enough so that
$B(\\{x_{1},y_{1}\\},n^{2})\subseteq B(x_{0},m^{2})$ and let
$B_{2}\in\mathcal{U}$ be such that $diam(B_{2}\setminus B(x_{0},m^{2}))>m$ as
given by the claim. Let $x_{2},y_{2}\in B_{2}\setminus B(x_{0},m^{2})$ be such
that $d(x_{2},y_{2})>m$. Proceeding in like fashion we may by induction find
sequences $\\{x_{n}\\}_{n\in\mathbb{N}}$ and $\\{y_{n}\\}_{n\in\mathbb{N}}$
with the following properties:

  1. (1)

$d(x_{j+1},\\{x_{1},\ldots,x_{j}\\})>j^{2}$ for $1\leq j\leq n-1$

  2. (2)

$d(y_{j+1},\\{y_{1},\ldots,y_{j}\\})>j^{2}$ for $1\leq j\leq n-1$

  3. (3)

$d(x_{j},y_{j})>j$ for $1\leq j\leq n$

  4. (4)

$d(y_{j+1},\\{x_{1},\ldots,x_{j}\\})>j^{2}$ for $1\leq j\leq n-1$

  5. (5)

$d(x_{j+1},\\{y_{1},\ldots,y_{j}\\})>j^{2}$ for $1\leq j\leq n-1$

  6. (6)

For $1\leq j\leq n$ there is a $B_{j}\in\mathcal{U}$ such that $x_{j},y_{j}\in
B_{j}$.

Define $A$ to be the collection of points $x_{n}$ and $C$ to be the collection
of points $y_{n}$ it is clear from construction that $A\subseteq
st(C,\mathcal{U})$ and $C\subseteq st(A,\mathcal{U})$ but $A$ and $C$ do not
have finite Hausdorff distance to each other. This contradicts $\mathcal{U}$
being uniformly bounded with respect to the metric coarse proximity structure.
Therefore we must have that $\mathcal{U}$ is uniformly bounded in the metric
sense. The statement about uniformly bounded covers then follows trivially. ∎

######  Definition 3.4.

Given a coarse proximity space $(X,\mathcal{B},{\bf b})$, the asymptotic
dimension of $X$, denoted $asdim(X)$, is the least nonnegative integer $n$
such that for every uniformly bounded cover $\mathcal{U}$ of $X$, there is a
uniformly bounded cover $\mathcal{V}$ of order at most $n+1$ such that
$\mathcal{U}<\mathcal{V}$.

This notion of asymptotic dimension agrees with the usual definition on the
metric coarse proximity structure. Arguably the greatest extant result
connecting the asymptotic dimension of metric spaces to dimensional invariants
of the Higson corona is the following result prove in [3] and [4].

######  Theorem 3.5.

If $(X,d)$ is an unbounded proper metric space with $asdim(X)<\infty$, then
$asdim(X)=dim(\nu X)$.

######  Lemma 3.6.

If $\mathcal{U}$ and $\mathcal{V}$ are uniformly bounded covers of a coarse
proximity space $(X,\mathcal{B},{\bf b})$ then $st(\mathcal{U},\mathcal{V})$
is a uniformly bounded cover of $X$.

###### Proof.

If $W\in st(\mathcal{U},\mathcal{V})$ then $W=st(U,\mathcal{V})$ for some
$U\in\mathcal{U}$. Elements of $\mathcal{U}$ are bounded and $U\phi
st(U,\mathcal{V})$, so $W$ must also be bounded. Recall that if $\mathcal{W}$
is a uniformly bounded cover of $X$, then $st(A,\mathcal{W})\phi A$ for all
$A\subseteq X$. Then note that

$st(A,st(\mathcal{U},\mathcal{V}))=st(st(st(A,\mathcal{V}),\mathcal{U}),\mathcal{V})\cup
st(st(A,\mathcal{U}),\mathcal{V})$

Which yields $A\phi st(A,st(\mathcal{U},\mathcal{V}))$. Now assume
$A,C\subseteq X$ are such that $A\subseteq st(C,st(\mathcal{U},\mathcal{V}))$
and $C\subseteq st(A,st(\mathcal{U},\mathcal{V}))$ then if $\hat{C}\subseteq
C$ is an unbounded set then $\hat{C}\subseteq
st(A,st(\mathcal{U},\mathcal{V}))$ and because
$st(A,st(\mathcal{U},\mathcal{V}))\phi A$ we have that $A{\bf b}\hat{C}$.
Similarly if $\hat{A}\subseteq A$ is unbounded we have $\hat{A}{\bf b}C$.
Therefore $A\phi C$ and $st(\mathcal{U},\mathcal{V})$ is a uniformly bounded
cover. ∎

######  Example 3.7.

Let $\mathbb{N}$ be equipped the metric bornology and the coarse proximity
${\bf b}$ induced by the one point compactification of $\mathbb{N}$. That is,
sets are considered bounded if and only if they are metrically bounded and two
sets $A,C\subseteq\mathbb{N}$ are coarsely close if and only if they are
unbounded. Then the boundary $\mathcal{U}\mathbb{N}$ is a single point. Define
$\mathcal{U}=\\{[n,n+1]\mid n\in\mathbb{N}\\}$. Then $\mathcal{U}$ is a
uniformly bounded cover of $\mathbb{N}$. If $\mathcal{V}$ is any uniformly
bounded cover of $\mathbb{N}$ that is refined by $\mathcal{U}$ then
$\mathcal{V}$ must have order at least $2$, which implies
$asdim(\mathbb{N})\geq 1$ with this coarse proximity structure.

######  Definition 3.8.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space, $\mathcal{U}$ a
collection of subsets of $X$, and $\mathcal{V}$ a uniformly bounded cover of
$X$. The $\mathcal{V}$-multiplicity of $\mathcal{U}$ is the greatest integer
$n$ such that there is a $V\in\mathcal{V}$ and
$U_{1},\ldots,U_{n}\in\mathcal{U}$ such that $U_{i}\cap V\neq\emptyset$ for
$1\leq i\leq n$.

######  Proposition 3.9.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space. Then the following
are equivalent:

  1. (1)

$asdim(X)=0$

  2. (2)

For every uniformly bounded cover $\mathcal{V}$ there is a uniformly bounded
cover $\mathcal{U}$ such that the $\mathcal{V}$-multiplicity of $\mathcal{U}$
is at most $1$.

  3. (3)

For every uniformly bounded cover $\mathcal{U}$, the set of
$\mathcal{U}$-components in $X$ is uniformly bounded.

###### Proof.

$(1)\implies(2)$ Assume that $asdim(X)=0$ and let $\mathcal{V}$ be a uniformly
bounded cover. Then there is a uniformly bounded cover $\mathcal{U}$ of order
at most $1$ that is refined by $st(st(\mathcal{V},\mathcal{V}),\mathcal{V})$.
Then the $\mathcal{V}$-multiplicity of $\mathcal{U}$ is at most $1$.

$(2)\implies(1)$ Assume $(2)$ and let $\mathcal{V}$ be a uniformly bounded
cover. Then there is a uniformly bounded cover $\mathcal{U}$ of
$\mathcal{V}$-multiplicity at most $1$. Then $\mathcal{V}$ must refine
$\mathcal{U}$, so $asdim(X)=0$.

$(1)\implies(3)$ Assume that $asdim(X)=0$, that $\mathcal{U}$ is a uniformly
bounded cover, and that $\mathcal{V}$ is a uniformly bounded cover with
$\mathcal{U}$-multiplicity $1$ (as given by $(2)$). Then $\mathcal{U}$ refines
$\mathcal{V}$ and the set of $\mathcal{U}$-components refines $\mathcal{V}$
and is therefore uniformly bounded.

$(3)\implies(1)$ Let $\mathcal{U}$ be a uniformly bounded cover of $X$. Then
the collection $\mathcal{V}$ of $\mathcal{U}$-components is uniformly bounded
and is refined by $\mathcal{U}$. The order of $\mathcal{V}$ is $1$, so
$asdim(X)=0$. ∎

######  Definition 3.10.

Given a coarse proximity space $(X,\mathcal{B},{\bf b})$ we say that the
bornology $\mathcal{B}$ is generated by a collection
$\\{B_{\alpha}\\}_{\alpha\in\Omega}\subseteq\mathcal{B}$ if for all
$B\in\mathcal{B}$ there is a $B_{\alpha}\in\\{B_{\alpha}\\}_{\alpha\in\Omega}$
such that $B\subseteq B_{\alpha}$. Similarly, if
$\\{\mathcal{U}_{\alpha}\\}_{\alpha\in\Omega}$ is a collection of uniformly
bounded covers of $X$ we say that the collection generates the uniformly
bounded covers of $X$ if for every uniformly bounded cover $\mathcal{U}$ there
is some $\mathcal{U}_{\alpha}$ such that $\mathcal{U}$ refines
$\mathcal{U}_{\alpha}$.

######  Definition 3.11.

A coarse proximity space $(X,\mathcal{B},{\bf b})$ is said to be witnessed by
uniform covers if for all $A,C\subseteq X$, $A{\bf b}C$ implies that there is
a uniformly bounded cover $\mathcal{U}$ of $X$ such that
$st(A,\mathcal{U})\cap C$ is unbounded. In general if for a particular pair of
sets $A,C\subseteq X$ we have that $A{\bf b}C$ and there is a uniformly
bounded cover $\mathcal{U}$ such that $st(A,\mathcal{U})\cap C$ is unbounded,
then we say that $\mathcal{U}$ witnesses $A$ and $B$ being close. If
$\mathcal{C}$ is a family of uniformly bounded covers of $X$ such that $A{\bf
b}C$ implies that there is some $\mathcal{V}\in\mathcal{C}$ such that $A\cap
st(C,\mathcal{V})$ is unbounded then we say that the coarse proximity is
witnessed by the family $\mathcal{C}$.

######  Remark 3.12.

An equivalent condition for a coarse proximity to be witnessed by uniform
covers if that $A{\bf b}C$ implies that there is a uniformly bounded cover
$\mathcal{U}$ for which $st(A,\mathcal{U})\cap st(C,\mathcal{U})$ is
unbounded.

######  Lemma 3.13.

Let $(X,\mathcal{B},{\bf b})$ be an infinite coarse proximity space in which
$\mathcal{B}$ is countably generated and let
$\mathcal{C}=\\{\mathcal{V}_{n}\\}_{n\in\mathbb{N}}$ be a family of uniformly
bounded covers that witnesses the coarse proximity structure on $X$. Then the
family of uniformly bounded covers of $X$ is countably generated.

###### Proof.

Say $\mathcal{B}$ is generated by the family $\\{B_{n}\\}_{n\in\mathbb{N}}$.
We then define $\mathcal{W}=\\{\mathcal{V}_{n}\cup\\{B_{m}\\}\mid
n,m\in\mathbb{N}\\}$. Then $\mathcal{W}$ is a countable set. Enumerate
$\mathcal{W}$ by denoting the elements as
$\mathcal{U}_{1},\mathcal{U}_{2},\ldots$. We then define
$\hat{\mathcal{U}}_{1}=\mathcal{U}_{1}$ and for general $n\geq 2$ we define
$\hat{\mathcal{U}}_{n}=st(\hat{\mathcal{U}}_{n-1},\mathcal{U}_{n})$. Let
$\hat{\mathcal{W}}$ be the set of all $\hat{\mathcal{U}}_{n}$. We claim that
every uniformly bounded cover of $X$ refines an element of
$\hat{\mathcal{W}}$. Define a function $d\mathrel{\mathop{\mathchar
58\relax}}X\times X\rightarrow[0,\infty)$ by setting $\hat{\mathcal{U}}_{0}$
to be the set of singleton subsets of $X$ and setting $d(x,y)$ to be

$d(x,y)=\min\\{n\mid\exists B\in\hat{\mathcal{U}}_{n},\,\\{x,y\\}\subseteq
B\\}$

As each bounded subset of $X$ is a subset of some element of some
$\hat{\mathcal{U}}_{n}$ we have that $d$ is well defined. In fact, it is a
metric on $X$ as seen in [16]. With this metric we have that the elements of
$\hat{\mathcal{U}}_{n}$ have diameter less than or equal to $n$ with respect
to this metric. If $\mathcal{V}$ is a uniformly bounded family of $X$ that
does not refine any $\hat{\mathcal{U}}_{n}$ then there is a sequence of pairs
$(x_{n},y_{n})$ in $X$ such that $\\{\\{x_{n},y_{n}\\}\\}_{n\in\mathbb{N}}$
refines $\mathcal{V}$ and $d(x_{n},y_{n})>n$ for each $n\in\mathbb{N}$. As we
saw in the proof of Proposition 3.2 we may assume that the sequence
$(d(\\{x_{n},y_{n}\\},\\{x_{n+1},y_{n+1}\\}))_{n\in\mathbb{N}}$ diverges to
infinity. Defining $A$ to be the collection of $x_{n}$’s and $C$ to be the
collection of $y_{n}$’s we have that $A$ and $C$ must be unbounded as if they
were both bounded then there would be a $\hat{\mathcal{U}}_{n}$ that contains
the union of both of them. Similarly because $A\subseteq st(C,\mathcal{V})$
and $C\subseteq st(A,\mathcal{V})$ we have that $A\phi C$, so both must either
unbounded or bounded. Because $A\phi C$ we have that $A{\bf b}C$. Because the
coarse proximity on $X$ is witnessed by the family
$\\{\hat{\mathcal{U}}_{n}\\}_{n\in\mathbb{N}}$ we must have that there is some
$n$ for which $A\cap st(C,\hat{\mathcal{U}}_{n})$ is unbounded. However, there
are only finitely many elements $(x,y)\in A\times C$ for which $d(x,y)\leq n$,
so this is impossible. This is a contradiction. Therefore $\mathcal{V}$ must
refine some $\hat{\mathcal{U}}_{n}$ meaning that the uniformly bounded covers
of $X$ are countably generated. ∎

######  Lemma 3.14.

If $(X,\mathcal{B},{\bf b})$ is a coarse proximity space whose bornology is
countably generated and whose coarse proximity is witnessed by the countably
family $\mathcal{C}=\\{\mathcal{V}_{n}\\}_{n\in\mathbb{N}}$, then there is a
countably family of uniformly bounded covers of $X$,
$\hat{\mathcal{C}}=\\{\hat{\mathcal{V}}_{n}\\}_{n\in\mathbb{N}}$ that is
linearly ordered by refinement, is such that if $\mathcal{U}$ is a uniformly
bounded cover of $X$ then $\mathcal{U}$ refines some $\hat{\mathcal{V}}_{n}$,
and satisfies the property that if $A,C\subseteq X$ are such that $A\phi C$
then there is a $\hat{\mathcal{V}}_{n}$ such that $A\subseteq
st(C,\hat{\mathcal{V}})$ and $C\subseteq st(A,\hat{\mathcal{V}})$.

###### Proof.

If $\mathcal{C}=\\{\mathcal{V}_{n}\\}_{n\in\mathbb{N}}$ is a countable family
of uniformly bounded covers of a coarse proximity space $X$, then by the proof
of Lemma 3.13 we have that the linearly ordered family
$\hat{\mathcal{C}}=\\{\hat{\mathcal{V}}_{n}\\}_{n\in\mathbb{N}}$ as described
in the statement of the Lemma, exists. Now let $A,C\subseteq X$ be unbounded
sets such that $A\phi C$. Let $d$ be the metric on $X$ constructed from the
family $\hat{\mathcal{C}}$ as in the proof of Lemma 3.13. Assume towards a
contradiction that there is no $n\in\mathbb{N}$ such that $A\subseteq
st(C,\hat{\mathcal{V}}_{n})$ and $C\subseteq st(A,\hat{\mathcal{V}}_{n})$.
This is equivalent to the Hausdorff distance between $A$ and $C$, with respect
to the metric $d$, being infinite. By the pigeonhole principle we then have
that, without loss of generality, there is an unbounded subset
$\hat{A}\subseteq A$ such that for every $n$, $\hat{A}\cap
st(C,\hat{\mathcal{U}}_{n})$ is finite. However, because $A\phi C$ we must
have that $\hat{A}{\bf b}C$ and because the coarse proximity on $X$ it
witnessed by the family $\hat{\mathcal{C}}$ there must be some
$n\in\mathbb{N}$ for which $\hat{A}\cap st(C,\hat{\mathcal{U}}_{n})$ is
unbounded and therefore infinite. This is a contradiction. Therefore there
must be some $n\in\mathbb{N}$ for which $A\subseteq
st(C,\hat{\mathcal{U}}_{n})$ and $C\subseteq st(A,\hat{\mathcal{U}}_{n})$. ∎

######  Proposition 3.15.

If $f\mathrel{\mathop{\mathchar 58\relax}}(X,\mathcal{B}_{X},{\bf
b}_{X})\rightarrow(Y,\mathcal{B}_{Y},{\bf b}_{Y})$ is an injective coarse
proximity embedding, $A,C\subseteq X$ are such that $A{\bf b}_{X}C$, and there
is a uniformly bounded cover $\mathcal{U}$ in $Y$ that witnesses $f(A)$ being
close to $f(C)$, then $f^{-1}(\mathcal{U})=\\{f^{-1}(B)\mid
B\in\mathcal{U}\\}$ is a uniformly bounded cover of $X$ that witnesses $A$
being close to $C$.

###### Proof.

Let $A,C,$ and $\mathcal{U}$ be given. First we will show that
$f^{-1}(\mathcal{U})$ is a uniformly bounded cover of $X$. As $\mathcal{U}$
covers $Y$ it is clear that $f^{-1}(\mathcal{U})$ covers $X$. Moreover, as the
preimage of bounded sets via a coarse proximity map are bounded we have that
all elements of $f^{-1}(\mathcal{U})$ are bounded. Now let
$A_{1},A_{2}\subseteq X$ be unbounded sets such that $A_{1}\subseteq
st(A_{2},f^{-1}(\mathcal{U}))$ and $A_{2}\subseteq
st(A_{1},f^{-1}(\mathcal{U}))$. Then $f(A_{1})\subseteq
st(f(A_{2}),\mathcal{U})$ and $f(A_{2})\subseteq st(f(A_{1}),\mathcal{U})$. As
$\mathcal{U}$ is a uniformly bounded cover of $Y$ we have that
$f(A_{1})\phi_{Y}f(A_{2})$ which implies that $A_{1}\phi_{X}A_{2}$ because $f$
is a coarse proximity embedding. Therefore $f^{-1}(\mathcal{U})$ is a
uniformly bounded cover of $X$.

Now, let $D$ be an unbounded subset of $st(f(A),\mathcal{U})\cap f(C)$.
Because $f$ is injective we have that $f^{-1}(D)$ is an unbounded subset of
$st(A,f^{-1}(\mathcal{U}))\cap C$, establishing the result.

∎

######  Definition 3.16.

A subspace $Y$ of a coarse proximity space $(X,\mathcal{B},{\bf b})$ is called
coarsely dense in $X$ if $Y\phi X$.

######  Proposition 3.17.

Let $(X,d)$ be a metric space equipped with its metric coarse proximity
structure, $(Y,\mathcal{B},{\bf b})$ a coarse proximity space, and
$f\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ a coarse proximity map
such that $f(Y)$ is coarsely dense in $X$. Then there are subspaces
$\hat{Y}\subset Y$ and $\hat{X}\subseteq X$ such that $f\mid_{\hat{Y}}$ is a
bijective coarse map between $\hat{Y}$ and $\hat{X}$, and $\hat{X}$ is
coarsely equivalent to $X$.

###### Proof.

Let $X$, $Y$, and $f$ be given. Define $\hat{X}=f(Y)$. As $f(Y)$ is coarsely
dense in $X$ we immediately have that $\hat{X}$ is coarsely equivalent to $X$.
For each $z\in\hat{X}$ let $y_{z}\in Y$ be an element such that $f(y_{z})=z$
and define $\hat{Y}$ to be the set of such elements. The result then follows.
∎

##  4\. Metrizability of coarse proximity spaces

In this short section we provide a characterization of coarse proximity spaces
that are equivalent to some metric space. One may compare the result of this
section to the corresponding result about the metrizability of coarse spaces
in [16].

######  Definition 4.1.

A coarse proximity space $(X,\mathcal{B},{\bf b})$ is called metrizable if
there is a metric $d$ on $X$ such that $\mathcal{B}$ is the bornology of sets
that are bounded via the metric $d$ and ${\bf b}$ is the metric coarse
proximity induced by $d$.

######  Theorem 4.2.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space. Then the following
are equivalent:

  1. (1)

$(X,\mathcal{B},{\bf b})$ is metrizable.

  2. (2)

$\mathcal{B}$ is countably generated and ${\bf b}$ is witnessed by a countable
family of uniformly bounded covers.

###### Proof.

($(1)\Rightarrow(2)$) This direction is trivial.

($(2)\Leftarrow(1)$) This will follow from Lemma 3.13 and Lemma 3.14. If
$\mathcal{V}=\\{\mathcal{V}_{n}\\}_{n\in\mathbb{N}}$ is a countable family of
uniformly bounded covers that witnesses the coarse proximity ${\bf b}$ and
$\mathcal{C}=\\{B_{n}\\}_{n\in\mathbb{N}}$ is a countable subset of
$\mathcal{B}$ that generates the bornology. By Lemma 3.14 we may assume that
$\mathcal{V}$ is linearly ordered with respect to refinement. We may also
assume that every $B_{n}$ is contained in an element of some
$\mathcal{V}_{n}$. Defining a metric $d$ as in the proof of Lemma 3.13 we
quickly have that the metric induces the original coarse proximity structure
on $X$. ∎

##  5\. Inverse limits of coarse proximity spaces

Let $\Omega$ be a directed set. That is, $\Omega$ is a nonempty set with an
order $\leq$ that is reflexive, antisymmetric, transitive, and is such that
every two elements have a common upper bound. Let
$\\{(X_{\alpha},\mathcal{B}_{\alpha},{\bf
b}_{\alpha}),f_{\alpha\beta}\\}_{\alpha,\beta\in\Omega}$ be an inverse system
of coarse proximity spaces over $\Omega$. That is, for each $\alpha\in\Omega$
there is a coarse proximity space $(X_{\alpha},\mathcal{B}_{\alpha},{\bf
b}_{\alpha})$, for each pair of distinct elements $\alpha,\beta\in\Omega$ such
that $\alpha<\beta$ there is a coarse proximity map
$f_{\beta\alpha}\mathrel{\mathop{\mathchar 58\relax}}X_{\beta}\rightarrow
X_{\alpha}$, and the collection of maps $f_{\beta\alpha}$ is such that if
$\alpha<\beta<\gamma$ then $f_{\beta\alpha}\circ
f_{\gamma\beta}=f_{\gamma\alpha}$.

Consider the product $\prod_{\alpha\in\Omega}X_{\alpha}$. Denote each element
of this set by $(x_{\alpha})_{\alpha\in\Omega}$ where $x_{\alpha}\in
X_{\alpha}$. Let $Z\subseteq\prod_{\alpha\in\Omega}X_{\alpha}$ be the set of
all $(x_{\alpha})_{\alpha\in\Omega}$ such that
$f_{\beta\alpha}(x_{\beta})=x_{\alpha}$ for all $\alpha,\beta\in\Omega$ with
$\alpha<\beta$. For each $\alpha\in\Omega$ define
$\pi_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow X_{\alpha}$ by
$\pi_{\alpha}((x_{\alpha})_{\alpha\in\Omega})=x_{\alpha}$.

######  Lemma 5.1.

With $Z$ defined as above, define $\mathcal{B}$ to the collection of all
subsets $B\subseteq Z$ with the property that there is an $\alpha\in\Omega$
for which $\pi_{\alpha}(B)\in\mathcal{B}_{\alpha}$. Then $\mathcal{B}$ is a
bornology on $\mathcal{B}$.

###### Proof.

Given any point $(x_{\alpha})_{\alpha\in\Omega}\in Z$ and any
$\alpha\in\Omega$ we have that
$\pi_{\alpha}((x_{\alpha})_{\alpha\in\Omega})=x_{\alpha}$ and the singleton
$\\{x_{\alpha}\\}$ is an element of $\mathcal{B}_{\alpha}$. Therefore, all
singleton sets in $Z$ are in $\mathcal{B}$, making $\mathcal{B}$ a cover of
$Z$. Similarly, all finite subsets of $Z$ are in $\mathcal{B}$. As images of
functions preserve the subset relation we have that $\mathcal{B}$ is closed
under subsets. Similarly as functions commute with set theoretic unions we
have that $\mathcal{B}$ is closed under finite unions. We then have that
$\mathcal{B}$ is a bornology on $Z$. ∎

######  Proposition 5.2.

Define $Z$ as above and equip it with the bornology described in Lemma 5.1.
Define a binary relation ${\bf b}$ on subsets of $(Z,\mathcal{B})$ by setting
$A{\bf b}C$ if and only if $\pi_{\alpha}(A){\bf b}_{\alpha}\pi_{\alpha}(C)$
for all $\alpha\in\Omega$. If $Z\neq\emptyset$ then the triple
$(Z,\mathcal{B},{\bf b})$ is a coarse proximity space.

###### Proof.

Symmetry and the union axiom are clear. Moreover, if $A{\bf b}C$ then it is
clear that neither $A$ nor $C$ are in $\mathcal{B}$ as otherwise there is a
$\gamma\in\Omega$ such that $\pi_{\gamma}(A)$ of $\pi_{\gamma}(C)$ is in
$\mathcal{B}_{\gamma}$ which would imply that the image of $A$ or $C$ under
$\pi_{\gamma}$ is coarsely close to a bounded set, which is impossible. As
images of unbounded subsets of $Z$ under every $\pi_{\alpha}$ are unbounded we
have that if $A,C\subseteq Z$ intersect along an unbounded set, then $A{\bf
b}C$. Next, let $A,C\subseteq Z$ be such that $A\bar{\bf b}C$. Then there is a
$\gamma\in\Omega$ such that $\pi_{\gamma}(A)\bar{\bf
b}_{\gamma}\pi_{\gamma}(C)$. As ${\bf b}_{\gamma}$ is a coarse proximity space
there is a $E\subseteq X_{\gamma}$ such that $\pi_{\gamma}(A)\bar{\bf
b}_{\gamma}E$ and $(X_{\gamma}\setminus E)\bar{\bf
b}_{\gamma}\pi_{\gamma}(C)$. Define $\hat{E}$ to be $\pi_{\gamma}^{-1}(E)$. We
then must have that $A\bar{\bf b}\hat{E}$ and
$Z\setminus\hat{E}=\pi_{\gamma}^{-1}(X_{\gamma}\setminus E)\bar{\bf b}C$.
Finally, let $A,C,D\subseteq Z$ be unbounded sets. It is clear that if $A{\bf
b}C$ or $A{\bf b}D$ then $A{\bf b}(C\cup D)$. Conversely, assume that $A{\bf
b}(C\cup D)$. If $A\bar{\bf b}C$ and $A\bar{\bf b}D$ then there is a
$\gamma\in D$ such that $\pi_{\gamma}(A)\bar{\bf b}_{\gamma}\pi_{\gamma}(C)$
and $\pi_{\gamma}(A)\bar{\bf b}_{\gamma}\pi_{\gamma}(D)$. As $X_{\gamma}$ is a
coarse proximity space, this implies that $\pi_{\gamma}(A)\bar{\bf
b}_{\gamma}(\pi_{\gamma}(C)\cup\pi_{\gamma}(D))$. Consequently
$\pi_{\gamma}(A)\bar{\bf b}_{\gamma}\pi_{\gamma}(C\cup D)$, which is a
contradiction. We then have that ${\bf b}$ satisfies the union axiom and
$(Z,\mathcal{B},{\bf b})$ is a coarse proximity space. ∎

As is the case with general inverse systems of sets it is very possible for
the inverse limit of coarse proximity spaces to be empty, even when dealing
with an inverse system indexed by $\mathbb{N}$ and whose connection morphisms
are coarse proximity isomorphisms. For example, let $(X,d)$ be any unbounded
metric space with fixed point $x_{0}$. For each $n\in\mathbb{N}$ let
$X_{n}=X_{n}\setminus B(x_{0},n)$ equipped with the metric coarse proximity
structure and let connection morphisms simply be inclusion. Then each
connection morphism is a coarse proximity isomorphism, but the set theoretic
inverse limit is empty. In the following theorems we will restrict ourselves
to the case where the set theoretic inverse limit exists and has surjective
projections. This may seem excessively restrictive, but will end up being
critical in later sections.

######  Theorem 5.3.

Let $\mathfrak{C}$ be the category of coarse proximity spaces and coarse
proximity maps. Let $\\{(X_{\alpha},\mathcal{B}_{\alpha},{\bf
b}_{\alpha}),f_{\beta\alpha}\\}_{\alpha,\beta\in\Omega}$ be an inverse system
in $\mathcal{C}$ whose set theoretic inverse limit $Z$ exists and is such that
the projections $\pi_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow
X_{\alpha}$ are surjective. Then the space $(Z,\mathcal{B},{\bf b})$ as
described above is the inverse limit of the inverse system
$\\{(X_{\alpha}),\mathcal{B}_{\alpha},{\bf
b}_{\alpha}),f_{\alpha\beta}\\}_{\alpha,\beta\in\Omega}$.

###### Proof.

We begin by noting that by the construction of $Z$ for all
$\alpha,\beta\in\Omega$ with $\alpha<\beta$ we have that
$f_{\beta\alpha}\circ\pi_{\beta}=\pi_{\alpha}$. Let $(Y,\mathcal{B}_{Y},{\bf
b}_{Y})$ be a coarse proximity space, and for each $\alpha\in\Omega$ let
$g_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X_{\alpha}$ be a
coarse proximity map satisfying such that the collection of $g_{\alpha}$
satisfies the same commutativity with the $f_{\alpha\beta}$’s as the
$\pi_{\alpha}$’s. Then for a given $y\in Y$ define
$g\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow Z$ by
$g(y)=(g_{\alpha}(y))_{\alpha\in\Omega}$. It is clear that this map is well
defined by the commutativity condition on the $g_{\alpha}$’s. Likewise, it is
trivial that $\pi_{\alpha}\circ g=g_{\alpha}$ for every $\alpha$. We must show
that $g$ is a coarse proximity map, and is the unique coarse proximity such
that $\pi_{\alpha}\circ g_{\alpha}=g$ for each $\alpha$. Uniqueness is clear.
We then begin by letting $B\subseteq Y$ be bounded. Then for any given
$\alpha$ we have that $g_{\alpha}(B)$ is bounded. As $\pi_{\alpha}\circ
g(B)=g_{\alpha}(B)$ we have that the image of $g(B)$ under $\pi_{\alpha}$ is
bounded in $X_{\alpha}$. Therefore $g(B)$ is bounded in $Z$. Next let
$A,C\subseteq Y$ be unbounded such that $A{\bf b}_{Y}C$. Then
$g_{\alpha}(A){\bf b}_{\alpha}g_{\alpha}(C)$ for every $\alpha$. Then
$\pi_{\alpha}(g(A)){\bf b}_{\alpha}\pi_{\alpha}(g(C))$ for every $\alpha$,
which gives us that $g(A){\bf b}g(C)$. Thus, $g$ is a coarse proximity map. We
have then shown that $Z$ is indeed the inverse limit of the system. ∎

######  Theorem 5.4.

Let $\\{(X_{\alpha},\mathcal{B}_{\alpha},{\bf
b}_{\alpha}),f_{\beta\alpha}\\}_{\alpha,\beta\in\Omega}$ be an inverse system
in $\mathcal{C}$ whose set theoretic inverse limit $Z$ exists and is such that
the projections $\pi_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow
X_{\alpha}$ are surjective. Equipping $Z$ with the inverse limit coarse
proximity structure $(Z,\mathcal{B},{\bf b})$ we have that $\mathcal{U}Z$ is
the inverse limit of the inverse system
$\\{\mathcal{U}X_{\alpha},\mathcal{U}f_{\alpha\beta}\\}_{\alpha,\beta\in\Omega}$.

###### Proof.

That
$\\{\mathcal{U}X_{\alpha},\mathcal{U}f_{\alpha\beta}\\}_{\alpha,\beta\in\Omega}$
is an inverse system is given by the functorality of the boundary. Similarly
we have the requisite commutativity of triangles involving
$\mathcal{U}\pi_{\alpha}$’s and the $\mathcal{U}f_{\alpha\beta}$’s. Then,
letting $Y$ be the inverse limit of the inverse system
$\\{\mathcal{U}X_{\alpha},\mathcal{U}f_{\alpha\beta}\\}_{\alpha,\beta\in\Omega}$
with projections $g_{\alpha}$, we have that there is a unique continuous
function $f\mathrel{\mathop{\mathchar 58\relax}}\mathcal{U}Z\rightarrow Y$
such that all $g_{\alpha}\circ f=\mathcal{U}\pi_{\alpha}$ for each $\alpha$.
We will show that $f$ is a homeomorphism. Let
$\sigma_{1},\sigma_{2}\in\mathcal{U}Z$ be given and assume that
$f(\sigma_{1})=f(\sigma_{2})$. Then for each $\alpha$ we have that
$g_{\alpha}(f(\sigma_{1}))=g_{\alpha}(f(\sigma_{2}))$, which by the
commutativity gives us that
$\mathcal{U}\pi_{\alpha}(\sigma_{1})=\mathcal{U}\pi_{\alpha}(\sigma_{2})$ for
each $\alpha$. If $\sigma_{1}\neq\sigma_{2}$ then there are $A\in\sigma_{1}$
and $C\in\sigma_{2}$ such that $A\bar{\bf b}C$ in $Z$. This implies that there
is some $\gamma$ for which $\pi_{\gamma}(A)\bar{\bf
b}_{\gamma}\pi_{\gamma}(C)$. We then recall that
$\mathcal{U}\pi_{\gamma}(\sigma_{1})=\\{D\subseteq X_{\gamma}\mid D{\bf
b}_{\gamma}\pi_{\gamma}(E),\,\forall E\in\sigma_{1}\\}$. Clearly
$\pi_{\gamma}(C)$ will not be in this set, which implies that
$\mathcal{U}\pi_{\gamma}(\sigma_{1})$ would not be equal to
$\mathcal{U}\pi_{\gamma}(\sigma_{2})$. We then must have that
$\sigma_{1}=\sigma_{2}$. We then have that $f$ is injective.

As $\mathcal{U}Z$ and $Y$ are compact and Hausdorff we have that
$f(\mathcal{U}Z)$ is closed in $Y$. Because $g_{\alpha}\circ
f=\mathcal{U}\pi_{\alpha}$ for each $\alpha$ and each
$\mathcal{U}\pi_{\alpha}$ is surjective we have that
$g_{\alpha}(f(\mathcal{U}Z))=\mathcal{U}X_{\alpha}$ for each $\alpha$. Then,
by general properties of inverse limits we have that
$f(\mathcal{U}Z)=\varprojlim\mathcal{U}X_{\alpha}=Y$. We have then shown that
$f$ is a continuous bijection between compact Hausdorff spaces and is
therefore a homeomorphism.

∎

##  6\. Coarsely $n-to-1$ maps and canonical covers of coarse proximity spaces

This section is dedicated to translating results of Austin and Virk from [1]
to the language of coarse proximity spaces. The results pertain to coarse
analogs of $n$-to-$1$ maps between topological spaces. In particular, using
the results of Austin and Virk, we can provide conditions under which coarse
proximity maps induce $n$-to-$1$ maps between boundaries of the coarse
proximity spaces involved. The following three definitions are as they appear
in [1].

######  Definition 6.1.

Suppose $\mathcal{A}=\\{A_{1},\ldots,A_{k}\\}$ is a collection of disjoint
subsets of a metric space $X$. The collection $\mathcal{A}$ is called
gradually disjoint if for each $R>0$ there is a bounded set $B\subseteq X$
such that the sets $B(A_{i},R)\setminus B$ are disjoint.

######  Definition 6.2.

Suppose $\mathcal{A}=\\{A_{1},\ldots,A_{k}\\}$ is a collection of subsets of a
metric space $X$. We say that the collection $\mathcal{A}$ diverges or is
divergent if there is a bounded set $B\subseteq X$ such that

$\bigcap_{i=1}^{k}B(A_{i},R)\setminus B=\emptyset$

In a discrete proper metric space $X$, a collection
$\mathcal{A}=\\{A_{1},\ldots,A_{k}\\}$ is gradually disjoint if and only if
the traces of the $A_{i}$ on the Higson corona are pairwise disjoint. The
collection is divergent if and only if the intersection of the traces of the
$A_{i}$ is empty.

######  Definition 6.3.

A coarse map $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ between
discrete proper metric spaces is called coarsely $n-to-1$ if for every $R>0$
there is an $S>0$ such that if $B\subseteq Y$ satisfies $diam(B)\leq R$ then
there are at most $n$ sets $K_{1},\ldots,K_{n}\subseteq X$ such that
$diam(K_{i})\leq S$ for each $i$ and
$f^{-1}(B)\subseteq\bigcup_{i=1}^{n}K_{i}$.

The following theorems appears in [1].

######  Theorem 6.4.

Let $X$ and $Y$ be discrete proper metric spaces. Suppose
$f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ is a coarse map. Then
the following are equivalent:

  1. (1)

$f$ is coarsely $n-to-1$.

  2. (2)

$\nu f$ is $n-to-1$.

  3. (3)

If $\mathcal{A}=\\{A_{1},\ldots,A_{k}\\}$ is a gradually disjoint collection
of subsets of $X$ for which $f(\mathcal{A})=\\{f(A_{1}),\ldots,f(A_{k})\\}$ is
not divergent, then $k\leq n$.

######  Theorem 6.5.

Let $(X,d)$ be an unbounded proper metric space such that there is some $r>0$
for which $d(x,y)>r$ for all $x\neq y$. Then the following are equivalent.

  1. (1)

$asdim(X)\leq n$

  2. (2)

There is a proper metric space $Z$ of asymptotic dimension zero and a coarsely
$n+1$-to-$1$ and coarsely surjective map $f\mathrel{\mathop{\mathchar
58\relax}}Z\rightarrow X$.

Theorem 6.5 was actually first proved by Miyata and Virk in [13]. A
consequence of Theorem 6.5 not stated in [13] or [1] is the following.

######  Theorem 6.6.

If $(X,d)$ is an unbounded proper metric space, then $asdim(X)\geq\Delta(\nu
X)$.

###### Proof.

If $asdim(X)=n$, then by Theorem 6.5 there is a metric space $Z$ such that
$asdim(Z)=0$ and a coarsely $n+1$-to-$1$ coarsely surjective map
$f\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow X$. Then, by Theorem 6.4
the continous function $\mathcal{U}f\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}Z\rightarrow\nu X$ is $n+1$-to-$1$. By Proposition 2.53
there is a closed subset $A\subseteq Z$ such that $\mathcal{U}f$ restricted to
$A$ is irreducible and still $n+1$-to-$1$. By Theorem 2.57 we then have that
$\Delta(\nu X)\leq n$. Therefore $\Delta(\nu X)\leq asdim(X)$. ∎

Now we will generalize the above definitions and Theorem 6.4 (in part) to
coarse proximity spaces.

######  Definition 6.7.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space and
$\mathcal{A}=\\{A_{1},\ldots,A_{n}\\}$ a collection of subsets of $X$. We say
that $\mathcal{A}$ is:

  1. (1)

gradually disjoint if for $i\neq j$ we have $A_{i}\bar{\bf b}A_{j}$,

  2. (2)

divergent if $\bigcup_{i=1}^{n}tr(X\setminus A_{i})=\mathcal{U}X$ and there
are sets $D_{1},\ldots,D_{n}$ such that $D_{i}\ll(X\setminus A_{i})$ and
$\bigcup_{i=1}^{n}tr(D_{i})=\mathcal{U}X$. (i.e. if the set of $X\setminus
A_{i}$ is a ${\bf b}$-cover.

The reader is encourage to compare the definition of divergent families above
to the definition of _vanishing families_ in the study of uniform spaces as
seen in [11].

######  Proposition 6.8.

Let $\mathcal{A}=\\{A_{1},\ldots,A_{n}\\}$ be a family of subsets of a coarse
proximity space $(X,\mathcal{B},{\bf b})$ then:

  1. (1)

$\mathcal{A}$ is gradually disjoint if and only if the traces $tr(A_{i})$ on
$\mathcal{U}X$ are disjoint,

  2. (2)

$\mathcal{A}$ is divergent if and only if
$\bigcap_{i=1}^{n}tr(A_{i})=\emptyset$.

###### Proof.

$(1)$ This equivalence is clear upon recalling that $tr(A_{i})\cap
tr(A_{j})\neq\emptyset$ if and only if $A_{i}{\bf b}A_{j}$.

$(2)$ Assume that $\mathcal{A}=\\{A_{1},\ldots,A_{n}\\}$ is divergent and
assume towards a contradiction that $\sigma\in\bigcap_{i=1}^{n}tr(A_{i})$. Let
$D_{1},\ldots,D_{n}$ be such that $D_{i}\ll(X\setminus A_{i})$ for each $i$
and $\bigcup_{i=1}^{n}tr(D_{i})=\mathcal{U}X$. Because the traces of the
$D_{i}$ cover $\mathcal{U}X$ we must have that there is some $i$ such that
$\sigma\in tr(D_{i})$. Without loss of generality lets say that $\sigma\in
tr(D_{1})$. Because $D_{1}\ll(X\setminus A_{1})$ we have that
$tr(D_{1})\subseteq int(tr(X\setminus A_{1}))$. Therefore $\sigma\in
int(tr(X\setminus A_{1}))$ so there is an open set $U\subseteq\mathcal{U}X$
with $\sigma\in U$ and $cl(U)\subseteq int(tr(X\setminus A_{1}))$. Then
$\sigma\notin tr(A_{1})$ which is a contradiction. Therefore
$\bigcap_{i=1}^{n}tr(A_{i})=\emptyset$.

Conversely assume that $\bigcap_{i=1}^{n}tr(A_{i})=\emptyset$ and define
$C_{i}=(X\setminus A_{i})$ for each $i$. Let $U_{i}=\mathcal{U}X\setminus
tr(A_{i})$ for each $i$. The collection $\mathcal{U}=\\{U_{1},\ldots,U_{n}\\}$
is an open cover of $\mathcal{U}X$. There is then a finite open cover
$\mathcal{V}$ of $\mathcal{U}X$ such that $\\{\overline{V}\mid
V\in\mathcal{V}\\}$ refines $\mathcal{U}$. For each $V\in\mathcal{V}$ with
$\overline{V}\subseteq U_{i}$ we can find an unbounded set $A_{V}\subseteq X$
such that $V\subseteq tr(A_{V})\subseteq U_{i}$. Then, for each
$V\in\mathcal{V}$ let $\alpha(V)\in\\{1,\ldots,n\\}$ be such that
$\overline{V}\subseteq U_{\alpha(V)}$ and let $A_{V}$ be such that $V\subseteq
tr(A_{V})\subseteq U_{\alpha(V)}$. Then define
$D_{i}=\bigcup_{\alpha(V)=i}A_{V}$. Then because $\mathcal{V}$ is finite we
have that $tr(D_{i})\ll C_{i}$ for each $i$. Similarly
$\mathcal{U}X=\bigcup_{i=1}^{n}tr(D_{i})$. Therefore
$\\{A_{1},\ldots,A_{n}\\}$ is divergent. ∎

######  Theorem 6.9.

Let $f\mathrel{\mathop{\mathchar 58\relax}}(X,\mathcal{B}_{X},{\bf
b}_{X})\rightarrow(Y,\mathcal{B}_{Y},{\bf b}_{Y})$ be a coarse proximity map.
Then the following are equivalent:

  1. (1)

$\mathcal{U}f$ is $n$-to-$1$

  2. (2)

If $\mathcal{A}=\\{A_{1},\ldots,A_{k}\\}$ is a gradually disjoint collection
of subsets of $X$ such that $f(\mathcal{A})=\\{f(A_{1}),\ldots,f(A_{k})\\}$ is
not divergent, then $k\leq n$.

###### Proof.

$(1)\implies(2)$ Assume that $\mathcal{U}f$ is $n$-to-$1$ and let
$\mathcal{A}=\\{A_{1},\ldots,A_{k}\\}$ be a gradually disjoint collection of
subsets of $X$ such that $f(\mathcal{A})=\\{f(A_{1}),\ldots,f(A_{k})\\}$ is
not divergent. Assume towards a contradiction that $k>n$. Without loss of
generality let us say that $k=n+1$. By Proposition 6.8 we have that there is
some point $\sigma\in\bigcap_{i=1}^{n+1}tr(f(A_{i}))$. Then
$\mathcal{U}f^{-1}(\sigma)\cap tr(A_{i})\neq\emptyset$ for each $i$. However,
as $\mathcal{A}$ is gradually disjoint and has $n+1$ elements we have that
$\mathcal{U}f^{-1}(\sigma)$ has at least $n+1$ elements, contradicting
$\mathcal{U}f$ being $n$-to-$1$. Therefore $k\leq n$.

$(2)\implies(1)$. Assume $(2)$ and assume further towards a contradiction that
$\sigma\in\mathcal{U}Y$ is such that $\mathcal{U}f^{-1}(\sigma)$ contains at
least $n+1$ elements. As $\mathcal{U}X$ is compact and Hausdorff we can then
separate $n+1$ elements of $\mathcal{U}f^{-1}(\sigma)$ with disjoint open
neighbourhoods. We can then find a gradually disjoint family of unbounded sets
$A_{1},\ldots,A_{n+1}$ in $X$ such that
$tr(A_{i})\cap\mathcal{U}f^{-1}(\sigma)\neq\emptyset$ for each $i$. However,
then $\\{f(A_{1}),\ldots,f(A_{n+1})\\}$ is not divergent. This contradicts
$(2)$. Therefore $\mathcal{U}f$ must be $n$-to-$1$. ∎

##  7\. Coarsely canonical covers and the coarse cofinal dimension

In this section we will introduce a coarse analog of cofinal dimension for
coarse proximity spaces. The definition itself makes use of a coarse analog of
almost canonical covers of topological spaces.

######  Definition 7.1.

A collection $\mathcal{A}=\\{A_{1},\ldots,A_{n}\\}$ of disjoint subsets of a
coarse proximity space $(X,\mathcal{B},{\bf b})$ is a coarsely canonical cover
of $X$ if:

  1. (1)

There is a unique, possibly empty, bounded element of $\mathcal{A}$.

  2. (2)

The union of the $A_{i}$ is equal to $X$.

  3. (3)

For each unbounded element $A_{i}\in\mathcal{A}$ there is an unbounded
$K\subseteq X$ such that $K\ll_{w}A_{i}$.

  4. (4)

For $i\neq j$ there is no unbounded $K$ such that $K\ll_{w}A_{i}$ and
$K\ll_{w}A_{j}$.

  5. (5)

If $K\subseteq X$ is an unbounded set and $K\ll D$, then there is an unbounded
$C\subseteq X$ and some $i$ for which $C\ll D$ and $C\ll_{w}A_{i}$.

One may take issue with $(2)$ in the above definition as it doesn’t seem
particularly “coarse”, however requiring that these collections cover our
space will be very useful in the next section.

######  Example 7.2.

If $(X,d)$ is an unbounded metric space equipped with its metric coarse
proximity structure, then $\\{X\\}$ is a coarsely canonical cover.

######  Example 7.3.

Let $X=\mathbb{R}^{2}$ be equipped with its metric coarse proximity structure.
Define $A_{1}$ to the be left half-plane including the $y$-axis, and $A_{2}$
the right half-plane containing the $y$-axis. Then $\\{A_{1},A_{2}\\}$ is a
coarsely canonical cover of $\mathbb{R}^{2}$.

######  Proposition 7.4.

Let $\mathcal{U}=\\{A_{1},\ldots,A_{n}\\}$ be a coarsely canonical cover of a
proper metric space $(X,d)$. Then the collection
$tr(\mathcal{U})=\\{tr(A_{1}),\ldots,tr(A_{n})\\}$ is a almost canonical cover
of $\nu X$. Conversely, if $\mathcal{U}=\\{A_{1},\ldots,A_{n}\\}$ is a cover
of $X$ by disjoint sets such that $\\{tr(A_{1}),\ldots,tr(A_{n})\\}$ is a
almost canonical cover of $\nu X$, then $\\{A_{1},\ldots,A_{n}\\}$ is a
coarsely canonical cover of $X$ up to the possible inclusion of a bounded set.

###### Proof.

We begin by proving the result for coarsely canonical covers. Because
$\mathcal{U}$ covers $X$ and is finite we have that $tr(\mathcal{U})$ covers
$\nu X$. If for $i\neq j$ we have that there is some $\sigma\in
int(tr(A_{i}))\cap int(tr(A_{j}))$ there is there an unbounded set $D\subseteq
X$ such that $\sigma\in tr(D)\subseteq int(tr(A_{i}))\cap int(tr(A_{j}))$
which implies $D\ll_{w}A_{i}$ and $D\ll_{w}A_{j}$ contradicting item $(2)$ of
Definition 7.1. Finally, let $\sigma\in\nu X$ be any element and
$U\subseteq\nu X$ some open set containing $\sigma$. If $U$ does not intersect
the interior of any $tr(A_{i})$ then we can find unbounded sets $K,D\subseteq
X$ such that $K\ll D$ and there is no unbounded $C\subseteq X$ and $i$ for
which $C\ll D$ and $C\ll_{w}A_{i}$, contradicting item $(3)$ of Definition
7.1. Therefore $U$ must intersect the interior of some $tr(A_{i})$. We then
have that $tr(\mathcal{U})$ is an almost canonical cover of $\nu X$.

Conversely, assume that $\\{A_{1},\ldots,A_{n}\\}$ is a cover of $X$ by
disjoint sets such that $\\{tr(A_{1}),\ldots,tr(A_{n})\\}$ is an almost
canonical cover of $\nu X$. If $K\subseteq X$ is such that there are $i,j$ for
which $K\ll_{w}A_{i}$ and $K\ll_{w}A_{j}$, then $tr(K)\subseteq
int(tr(A_{i}))\cap int(tr(A_{j}))$. As the traces of the $A_{k}$ are an almost
canonical cover of $\nu X$ they have disjoint interiors, therefore $i=j$. If
$K,D\subseteq X$ are such that $K\ll D$, then $tr(K)\subseteq int(tr(D))$.
Because the traces of the $A_{i}$ are an almost canonical cover there must be
some $\sigma\in int(tr(D))$ and some $i$ such that $\sigma\in int(tr(A_{i}))$.
We can then find some unbounded subset $C\subseteq X$ such that $\sigma\in
tr(C)\subseteq int(tr(A_{i}))\cap int(tr(D))$ and $C\ll D$ by Lemma 2.31.
Because $tr(C)\subseteq int(A_{i})$ we also have that $C\ll_{w}A_{i}$.
Finally, as the $A_{i}$ cover $X$, we have that $\\{A_{1},\ldots,A_{n}\\}$ is
a coarsely canonical cover of $X$ up to possibly the inclusion of bounded set.

∎

######  Definition 7.5.

Let $\mathcal{U}=\\{A_{1},\ldots,A_{n}\\}$ be a coarsely canonical cover of a
coarse proximity space $(X,\mathcal{B},{\bf b})$. The order of $\mathcal{U}$,
denoted $ord(\mathcal{U})$, is the least natural number $m$ that if
$A_{i_{1}},\ldots,A_{i_{m+1}}$ is any collection of $m+1$ elements of
$\mathcal{U}$, the the $A_{i_{j}}$ diverge.

######  Proposition 7.6.

If $\mathcal{U}=\\{A_{1},\ldots,A_{n}\\}$ is a coarsely canonical cover of a
metric space $X$, then $ord(\mathcal{U})=m$ if and only if the order of the
collection $tr(\mathcal{U})=\\{tr(A_{1}),\ldots,tr(A_{n})\\}$ is $m$.

###### Proof.

Assume that $ord(\mathcal{U})\leq m$ and let
$tr(A_{i_{1}}),\ldots,tr(A_{i_{m+1}})$ be $m+1$ elements of $tr(\mathcal{U})$.
If $\bigcap_{j=1}^{m+1}tr(A_{i_{j}})\neq\emptyset$, then by Proposition 6.8 we
have that $\\{A_{i_{1}},\ldots,A_{i_{m+1}}\\}$ is not a divergent family which
is a contradiction. Therefore every collection of $m+1$ elements of
$tr(\mathcal{U})$ must have nonempty intersection, so
$ord(tr(\mathcal{U}))\leq m$. Conversely assume that $ord(tr(\mathcal{U})\leq
m$ and let $A_{i_{1}},\ldots,A_{i_{m+1}}$ be $m+1$ elements of $\mathcal{U}$.
If these sets are not divergent, then by Proposition 6.8
$\bigcap_{j=1}^{m+1}tr(A_{i_{j}})$ is nonempty, which contradicts
$ord(tr(\mathcal{U}))\leq m$. ∎

######  Definition 7.7.

Given two collections of subsets $\mathcal{U}=\\{A_{1},\ldots,A_{n}\\}$ and
$\mathcal{V}=\\{C_{1},\ldots,C_{m}\\}$ of a coarse proximity space
$(X,\mathcal{B},{\bf b})$ we say that $\mathcal{V}$ coarsely refines
$\mathcal{U}$ if for each unbounded $C_{i}\in\mathcal{V}$ there is an
unbounded $A_{j}\in\mathcal{U}$ and a bounded set $B\subseteq X$ such that
$(C_{i}\setminus B)\subseteq A_{j}$.

It is possible that two distinct coarsely canonical covers may coarsely refine
each other. Fortunately, with respect to coarse refinement, this equivalence
does not matter as can be seen in the following easy proposition.

######  Proposition 7.8.

Let $\mathcal{U}$ and $\mathcal{V}$ be coarsely canonical covers of a coarse
proximity space $(X,\mathcal{B},{\bf b})$ that coarsely refine each other. For
a third coarsely canonical cover $\mathcal{W}$ the following are equivalent:

  1. (1)

$\mathcal{W}$ coarsely refines $\mathcal{U}$

  2. (2)

$\mathcal{W}$ coarsely refines $\mathcal{V}$

######  Proposition 7.9.

Let $\mathcal{U}=\\{A_{1},\ldots,A_{n}\\}$ be a ${\bf b}$-cover of a coarse
proximity space $(X,\mathcal{B},{\bf b})$ that covers $X$. Then there is a
coarsely canonical cover $\mathcal{V}=\\{C_{1},\ldots,C_{m}\\}$ that coarsely
refines $\mathcal{U}$.

###### Proof.

By Proposition 2.39 we have that
$int(tr(\mathcal{U}))=\\{int(tr(A_{1})),\ldots,int(tr(A_{n}))\\}$ is an open
cover $\mathcal{U}X$. Reduce this open cover to an open cover with the
property that every element of the cover contains an element of $\mathcal{U}X$
not contained in any of the other elements of the cover. Call this cover
$\mathcal{W}$. Say $\mathcal{W}=\\{U_{1},\ldots,U_{m}\\}$. Because
$\mathcal{U}X$ is a compact Hausdorff space it is normal, so $\mathcal{W}$
admits a shrinking $\mathcal{V}=\\{V_{1},\ldots,V_{m}\\}$. That is,
$\mathcal{V}$ is an open cover of $\mathcal{U}X$ such that
$\overline{V}_{i}\subseteq U_{i}$. As each $U_{i}$ contained an element not
contained in any other $U_{j}$ we have that every $V_{j}$ contains a
nontrivial open set that does not intersect any other $V_{j}$. Note that
$\mathcal{V}$ trivially refines $int(tr(\mathcal{U}))$. By Lemma 2.46 we have
that there is a canonical cover $\mathcal{F}=\\{F_{1},\ldots,F_{k}\\}$ of
$\mathcal{U}X$ that refines $\\{\overline{V}_{1},\ldots,\overline{V}_{m}\\}$,
and consequently $\mathcal{W}$ and $int(tr(\mathcal{U}))$.

By Proposition 2.32 can find an unbounded set $D_{1}\subseteq X$ such that
$F_{1}\subseteq int(tr(D_{1}))$, $tr(D_{1})\cap F_{i}=\emptyset$ for all
$F_{i}$ such that $F_{i}\cap F_{1}=\emptyset$, is such that $tr(D_{1})$ is
contained in some $U_{i}\in\mathcal{W}$, and is such that if $K\subseteq X$ is
an unbounded set with $tr(K)\subseteq F_{1}$, then $K\ll D_{1}$. Assume that
unbounded sets $D_{1},\ldots,D_{k}$ have been constructed such that for $i\leq
k$:

  1. (1)

$F_{i}\subseteq int(tr(D_{i}))$

  2. (2)

$D_{i}{\bf b}D_{j}$ if and only if $F_{i}\cap F_{j}\neq\emptyset$.

  3. (3)

$tr(D_{i})\cap F_{j}\neq\emptyset$ implies $F_{i}\cap F_{j}\neq\emptyset$

  4. (4)

$tr(D_{i})$ is contained in some $U_{j}\in\mathcal{W}$.

  5. (5)

If $K\subseteq X$ is such that $tr(K)\subseteq F_{i}$, then $K\ll D_{i}$

Consider $F_{k+1}$. As $F_{k+1}$ is disjoint from all $D_{i}$ for which
$F_{k+1}\cap F_{i}=\emptyset$ we can find an unbounded set $D_{k+1}$ such that
$F_{k+1}\subseteq int(tr(D_{k+1}))$, $D_{k+1}{\bf b}D_{i}$ if and only if
$F_{k+1}\cap D_{i}\neq\emptyset$, and $D_{k+1}\cap F_{j}\neq\emptyset$ implies
$F_{k+1}\cap F_{j}\neq\emptyset$. By induction we then have a collection of
unbounded sets $\mathcal{D}=\\{D_{1},\ldots,D_{m_{2}}\\}$ satisfying items
$(1)$, $(2)$, and $(4)$ as above. Without loss of generality we may assume
that for every $i$ there is some open set $V\subseteq tr(D_{i})$ that does not
intersect any other $tr(D_{j})$. We may also assume that the collection of
$D_{i}$’s covers $X$. Let $S=\\{1,\ldots,m_{2}\\}$. Because $\mathcal{F}$ is
finite it has finite order, say $k+1$. We then consider all nonempty unbounded
intersections of the form $\bigcap_{i\in F\subseteq S,\,|F|=k+1}D_{i}$. Let
$\mathcal{C}_{k+1}$ be the collection of these unbounded sets. We define
$\hat{\mathcal{C}}_{k}$ be the similarly defined collection of unbounded sets
and define $\mathcal{C}_{k}$ to be the collection of sets of the forum
$C\setminus\left(\bigcup_{F\in\mathcal{C}_{k+1}}F\right)$ where
$C\in\hat{\mathcal{C}}_{k}$. We continue in like fashion constructing
collections of unbounded sets $\mathcal{C}_{i}$ for all $i\in S$ with the
property that every every of $\mathcal{C}_{i}$ is disjoint from every element
of $\bigcup_{j>i}\mathcal{C}_{j}$. We finally let
$\hat{\mathcal{F}}=\bigcup_{i=1}^{k+1}\mathcal{C}_{i}$. This is a finite set
of unbounded sets whose corresponding set of traces refines $\mathcal{U}$. We
claim that $\hat{\mathcal{F}}$ is the desired coarsely canonical cover. It is
clear from construction that for distinct $C_{1},C_{2}\in\hat{\mathcal{F}}$
there is no unbounded set $K\subseteq X$ such that $K\ll_{w}C_{1}$ and
$K\ll_{w}C_{2}$. It is also clear that $\hat{\mathcal{F}}$ covers $X$. Let
$K,D\subseteq X$ be unbounded sets with $K\ll D$. Consider $int(tr(D))$. As
$\\{int(tr(D_{1})),\ldots,int(tr(D_{m_{2}}))\\}$ is an open cover of $\nu X$
there is a largest natural number of $k_{\sigma}$ such that there is a point
$\sigma\in\nu X$ for which $\sigma$ is the in the intersection of $int(tr(D))$
and exactly $k_{\sigma}-1$ open sets of the form $int(tr(D_{i}))$. There is an
element $C$ of $\hat{\mathcal{F}}$ that contains this intersection. We may
then find an unbounded set $E\subseteq X$ such that $E\ll D$ and $E\ll_{w}C$.
Therefore $\hat{\mathcal{F}}$ is a coarsely canonical cover. What remains to
be shown is that $\hat{\mathcal{F}}$ coarsely refines $\mathcal{U}$. However,
this is trivial as $tr(D_{i})\subseteq U_{j}=int(tr(A_{j}))$ for some
$F_{j}\in\mathcal{U}$ which implies $D_{i}\ll A_{j}$ which implies that
$D_{i}$ is contained in $A_{j}$ up to a bounded set.

∎

######  Definition 7.10.

We say that a pre-directed system of coarsely canonical covers
$\\{F_{\alpha}\\}_{\alpha\in\Omega}$ is fine if for every ${\bf b}$-cover
$\mathcal{U}$ that covers $X$ there is some $\alpha\in\Omega$ such that
$\mathcal{F}_{\alpha}$ coarsely refines $\mathcal{U}$.

######  Definition 7.11.

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space. We define the
coarse cofinal dimension of $X$, denoted $\Delta_{c}(X)$, in the following
way. If $X$ is bounded then $\Delta_{c}(X)=-1$. Otherwise $\Delta_{c}(X)$ is
the smallest nonnegative integer $n$ such that $X$ admits a fine directed
system of coarsely canonical covers
$\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ such that the order of
$\mathcal{F}_{\alpha}$ is at most $n+1$ for each $\alpha\in\Omega$. If there
is no such $n$ we say that $\Delta_{c}(X)=\infty$.

Definition 7.11, Proposition 7.4, and Proposition 7.6 together with the
observation that taking traces of coarsely canonical covers commutes with the
refinement relation yield the following.

######  Theorem 7.12.

If $X$ is a proper metric space equipped with its metric coarse proximity
structure, then $\Delta_{c}(X)\geq\Delta(\nu X)$ where $\nu X$ is the Higson
corona of $X$.

Next we will work towards proving a basic equality. Namely, that for unbounded
discrete proper metric spaces, asymptotic dimension and coarse cofinal
dimension agree in dimension $0$. We will make use of the fact that when a
proper metric space has asymptotic dimension $0$, its Higson corona has
covering dimension $0$, as given to us by Theorem 3.5.

######  Proposition 7.13.

Let $(X,d)$ be an unbounded $1$-discrete proper metric space and let
$\mathcal{U}=\\{U_{1},\ldots,U_{m}\\}$ be a disjoint open cover of $\nu X$.
Then there is a coarsely canonical cover $\\{A_{1},\ldots,A_{m},A_{m+1}\\}$ of
$X$, where $tr(A_{i})=U_{i}$ for $i\leq m$ and $A_{m+1}$ is a bounded set.

###### Proof.

Because the elements of the finite open cover $\mathcal{U}$ are disjoint, they
are all also closed. By Proposition 2.32 there is an unbounded set
$A_{1}\subseteq X$ such that $tr(A_{1})$ contains $U_{1}$ and is disjoint from
$U_{j}$ for $j>1$. Then $tr(A_{1})=U_{1}$. Assume that $A_{i}$ has been
constructed for $i\leq n<m$ in such way that the $A_{i}$ are disjoint and are
such that if $K\subseteq X$ satisfies $tr(K)\subseteq U_{i}$, then $K\ll
A_{i}$. We can again, by Proposition 2.32 find an unbounded set
$\hat{A}_{n+1}\subseteq X$ such that $tr(\hat{A}_{n+1})=U_{n+1}$. As the
$U_{i}$ are all disjoint we have that $\left(\bigcup_{i\leq
n}A_{i}\right)\bar{\bf b}\hat{A}_{n+1}$ where ${\bf b}$ is the metric coarse
proximity on $X$. Then, $\bigcup_{i\leq n}A_{i}$ intersects $\hat{A}_{n+1}$ in
at most a a bounded set. Subtract this bounded set from $\hat{A}_{n+1}$ to get
$A_{n+1}$. Proposition 2.32 gives us that if $K\subseteq X$ is such that
$tr(K)\subseteq U_{n+1}$, then $K\ll A_{n+1}$. By induction we then have our
unbounded sets $A_{1},\ldots,A_{m}$. By construction we have that
$X\setminus\left(\bigcup_{i=1}^{m}A_{i}\right)$ is bounded. We then set
$A_{m+1}$ to be this bounded set. This completes the proof. ∎

######  Proposition 7.14.

Let $(X,d)$ be an unbounded $1$-discrete proper metric space of asymptotic
dimension $0$, and let $\mathcal{U}=\\{A_{1},\ldots,A_{m_{1}}\\}$ and
$\mathcal{V}=\\{C_{1},\ldots,C_{m_{2}}\\}$ be coarsely canonical covers of $X$
of order $1$. Then there is a coarsely cofinal cover $\mathcal{W}$ of $X$ that
coarsely refines both $\mathcal{U}$ and $\mathcal{V}$.

###### Proof.

We begin by noting that by Proposition 7.6 we have that
$tr(\mathcal{U})=\\{tr(A_{1}),\ldots,tr(A_{m_{1}})\\}$ and
$tr(\mathcal{V})=\\{tr(C_{1}),\ldots,tr(C_{m_{2}})\\}$ are almost canonical
covers of $\nu X$ of order $1$ and are therefore disjoint open covers. Then,
the set of nonempty intersections $\mathcal{C}=\\{tr(A_{i})\cap tr(C_{j})\mid
1\leq i\leq m_{1},\,1\leq j\leq m_{2}\\}$ is another disjoint open cover of
$\nu X$. By Proposition 7.13 there is a coarsely canonical cover
$\hat{\mathcal{W}}=\\{\hat{D}_{1},\ldots,\hat{D}_{m_{3}}\\}$ whose
corresponding set of traces is equal to $\mathcal{C}$. If
$tr(D_{k})=tr(A_{i})\cap tr(C_{j})$ then $D_{k}\ll A_{i}$ and $D_{k}\ll
C_{j}$. Therefore, $D_{k}\ll(A_{i}\cap C_{j})$ which implies that $D_{k}$ is
contained in $A_{i}\cap C_{j}$ up to a bounded set. Define
$D_{k}=\hat{D}_{k}\cap(A_{i}\cap C_{j})$ and define $\mathcal{W}$ to be the
collection of $D_{k}$’s with $D_{m_{3}}$ being equal to
$X\setminus\left(\bigcup_{i<m_{3}}D_{i}\right)$. ∎

######  Theorem 7.15.

Let $(X,d)$ be an unbounded $1$-discrete proper metric space of asymptotic
dimension $0$. Then $\Delta_{c}(X)=0$.

###### Proof.

Let $X$ be given. Then by Theorem 3.5 we have that $dim(\nu X)=0$. Denote by
$Cov(\nu X)$ the collection of finite disjoint open covers of $\nu X$, ordered
by refinement. Denote by $CCCov(X,1)$ the collection of coarsely canonical
covers of $X$ or order $1$, ordered by coarse refinement. As the order defined
by coarse refinement is not antisymmetric we let $\mathcal{L}\subseteq
CCCov(X,1)$ be such that no two elements of $\mathcal{L}$ coarsely refine each
other and every element of $CCCov(X,1)$ is coarsely refined some element of
$\mathcal{L}$. Proposition 7.8 assures us that this is possible. By
Proposition 7.13 we have that for every $\mathcal{U}\in Cov(\nu X)$ there is a
$\mathcal{V}\in\mathcal{L}$ such that the collection of traces of elements of
$\mathcal{V}$ is exactly $\mathcal{U}$. If
$\mathcal{U},\mathcal{V}\in\mathcal{L}$ are given coarsely canonical covers of
$X$, then by Proposition 7.14 there is a coarsely canonical cover
$\mathcal{W}$ of $X$ coarsely refines both $\mathcal{U}$ and $\mathcal{V}$.
Moreover, this coarsely canonical cover has order $1$. We then have that
$\mathcal{L}$ is a fine directed system of coarsely canonical covers of $X$
none of which has order greater than $1$. Therefore $\Delta_{c}(X)\leq 0$ and
as $X$ is unbounded we have that $\Delta_{c}(X)=0$.

∎

##  8\. From coarsely canonical covers to an asymptotic dimension zero coarse
proximity space

This section is focused on proving our main result, that $\Delta_{c}(X)\geq
asdim(X)$ for every unbounded proper metric space $X$. We do this by providing
a coarse version of a result in classic dimension theory. Specifically we
“coarsify” the result that says that if a space $X$ satisfies $\Delta(X)\leq
n$, then there is a perfectly zero dimensional space $Y$ and a closed
$n+1$-to-$1$ map $f\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$. Our
coarse version of the result is made by following the methods for the
classical result as thoroughly explained in Chapter $6$ of [15]. What we show
in this section specifically is the following, if $X$ is an unbounded proper
metric space satisfying $\Delta_{c}(X)\leq n$, then there is another unbounded
proper metric space $Z$ of asymptotic dimension zero (and satisfying
$\Delta_{c}(Z)=0$) and a coarsely surjective and coarsely $n+1$-to-$1$ map
$f\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow X$. This will give us that
$asdim(X)\leq n$ by Theorem 6.5. This in conjunction with Theorem 6.6 will
yield that for every unbounded proper metric space $X$, the inequality
$\Delta(\nu X)\leq asdim(X)\leq\Delta_{c}(X)$ holds.

Let $\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ be a fine directed system
of coarsely canonical covers of a countable $1$-discrete proper metric space
$X$ where $\beta\geq\alpha$ implies that $\mathcal{F}_{\alpha}$ is coarsely
refined by $\mathcal{F}_{\beta}$. Say
$\mathcal{F}_{\alpha}=\\{A_{1},\ldots,A_{n}\\}$ and
$\mathcal{F}_{\beta}=\\{C_{1},\ldots,C_{m}\\}$. For a given $\alpha\in\Omega$
we will denote by $\Omega(\alpha)$ the index set of the coarsely canonical
cover $\mathcal{F}_{\alpha}$. We will assume that every $\Omega(\alpha)$ is a
set of natural numbers of the form $\\{1,\ldots,n\\}$ for some natural number
$n$. Condition $(1)$ of Definition 7.1 makes it so that if
$\mathcal{F}_{\alpha}$ refines $\mathcal{F}_{\beta}$ then for each
$A_{i}\in\mathcal{F}_{\alpha}$ there is a unique $C_{j}\in\mathcal{F}_{\beta}$
such that $A_{i}\subseteq C_{j}$. There is then a unique function from
$\Omega(\beta)=\\{1,\ldots,m\\}$ to $\Omega(\alpha)=\\{1,\ldots,n\\}$ denoted
by $\rho_{\alpha\beta}$ such that $C_{i}\subseteq A_{\rho_{\alpha\beta}(i)}$
for each $i$. For a given $\alpha$ we then define a coarse proximity space
$(X_{\alpha},\mathcal{B}_{\alpha},{\bf b}_{\alpha})$ by define
$X_{\alpha}=\bigcup_{i=1}^{n}(A_{i}\times\\{i\\})$. That is, $X_{\alpha}$ is
the disjoint union of the elements of $\mathcal{F}_{\alpha}$. Recall that
coarsely canonical covers of $X$ are made up of disjoint sets that cover $X$.
The function $\delta_{\alpha}\mathrel{\mathop{\mathchar
58\relax}}X_{\alpha}\rightarrow X$ defined by $(x,i)\mapsto x$ is therefore
bijective . We say that a subset $B\subseteq X_{\alpha}$ if
$\delta_{\alpha}(B)$ is bounded in $X$. It is easily seen that this is a
bornology on $X_{\alpha}$. For unbounded sets $A,C\subseteq X_{\alpha}$ we say
that $A{\bf b}_{\alpha}C$ if and only if $(\delta_{\alpha}(A)\cap A_{i}){\bf
b}(\delta_{\alpha}(C)\cap A_{i})$ for some $i$. It is again easily seen that
this relation makes $(X_{\alpha},\mathcal{B}_{\alpha},{\bf b}_{\alpha})$ into
a coarse proximity space. With this coarse proximity structure
$\delta_{\alpha}$ is a bijective coarse proximity map.

######  Lemma 8.1.

A collection $\mathcal{W}$ of bounded subsets of $X_{\alpha}$ is uniformly
bounded if and only if the collection

$\\{\delta_{\alpha}(B)\cap A_{i}\mid B\in\mathcal{W},\,1\leq i\leq n\\}$

is uniformly bounded in $X$ and the collection

$\\{B\in\mathcal{W}\mid\exists i\neq
j,\,B\cap(A_{i}\times{i})\neq\emptyset,\,B\cap(A_{j}\times\\{j\\})\neq\emptyset\\}$

is finite.

###### Proof.

Assume $\mathcal{W}$ is a collection of subsets of $X_{\alpha}$ that is
uniformly bounded. Because $\delta_{\alpha}$ is a coarse proximity map we have
that $\partial_{\alpha}(\mathcal{U})$ is uniformly bounded. As
$\\{\delta_{\alpha}(B)\cap A^{\alpha}_{i}\mid B\in\mathcal{W},\,1\leq i\leq
n\\}$ refines $\delta_{\alpha}(\mathcal{W})$ it too must be uniformly bounded.
If $\\{B\in\mathcal{W}\mid\exists i\neq
j,\,B\cap(A_{i}\times{i})\neq\emptyset,\,B\cap(A_{j}\cap\\{j\\})\neq\emptyset\\}$
is infinite then by the pigeon hole principle there are $i\neq j$ and an
infinite collection of subsets $\hat{\mathcal{W}}\subseteq\mathcal{W}$ such
that $B$ contains elements of both $A^{\alpha}_{i}\times\\{i\\}$ and
$A^{\alpha}_{j}\times\\{j\\}$ for all $B\in\hat{\mathcal{W}}$. We can then
find an unbounded subset $K_{1}\subseteq A^{\alpha}_{i}\times\\{i\\}$ and
$K_{2}\subseteq A^{\alpha}_{j}\times\\{j\\}$ such that $K_{1}\subseteq
st(K_{2},\hat{\mathcal{W}})$ and $K_{2}\subseteq st(K_{1},\hat{\mathcal{W}})$.
This implies that $K_{1}{\bf b}_{\alpha}K_{2}$. However, this is impossible as
$K_{1}\overline{\bf b}_{\alpha}K_{2}$ by the definition of ${\bf b}_{\alpha}$.

Conversely, suppose $\mathcal{W}$ is a collection of sets satisfying the two
conditions stated in the Lemma. Because $\delta_{\alpha}$ is a coarse
proximity map we have that each element of $\mathcal{W}$ is bounded. Let
$K_{1},K_{2}\subseteq X_{\alpha}$ be unbounded sets such that $K_{1}\subseteq
st(K_{2},\mathcal{W})$ and $K_{2}\subseteq st(K_{1},\mathcal{W})$. We wish to
show that $K_{1}\phi_{\alpha}K_{2}$ where $\phi_{\alpha}$ is the weak
asymptotic resemblance induced by ${\bf b}_{\alpha}$. Assume towards a
contradiction that this is not true. Then, without loss of generality, there
is an unbounded $\hat{K}_{1}\subseteq K_{1}$ such that $\hat{K}_{1}\bar{\bf
b}_{\alpha}K_{2}$. We may also assume that $\hat{K}_{1}$ does not intersect
any of the elements of $\mathcal{W}$ that intersect more than one
$A_{i}\times\\{i\\}$. Let $\tilde{\mathcal{W}}$ be the collection of elements
of $\mathcal{W}$ that do not intersect more than one $A_{i}\times\\{i\\}$ we
have that $\hat{K}_{1}\subseteq st(K_{2},\tilde{\mathcal{W}})$. Using the
pigeonhole principle we may assume that $\hat{K}_{2}$ is entirely contained in
one $A_{i}\times\\{i\\}$. Again, without loss of generality, this is
$A_{1}\times\\{1\\}$. Then there is an unbounded set $\hat{K}_{2}$ contained
entirely in $A_{1}\times\\{1\\}$ such that $\hat{K}_{1}\subseteq
st(\hat{K}_{2},\tilde{\mathcal{W}})$. However, this implies that
$(\delta_{\alpha}(\hat{K}_{1})\cap A_{1}){\bf
b}(\delta_{\alpha}(\hat{K}_{2})\cap A_{1})$ which implies $\hat{K}_{1}{\bf
b}_{\alpha}\hat{K}_{2}$ and consequently that $\hat{K}_{1}{\bf
b}_{\alpha}K_{2}$, a contradiction. Therefore $K_{1}\phi_{\alpha}K_{2}$ and
$\mathcal{W}$ is uniformly bounded in $X_{\alpha}$. ∎

Returning to our directed set $\\{F_{\alpha}\\}_{\alpha\in\Omega}$ of coarsely
canonical covers, let $\beta\geq\alpha$ and consider the corresponding coarse
proximity spaces $X_{\beta}$ and $X_{\alpha}$. The function
$\rho_{\alpha\beta}$ induces a function
$\pi_{\alpha\beta}\mathrel{\mathop{\mathchar 58\relax}}X_{\beta}\rightarrow
X_{\alpha}$ in the following way. If $(x,i)\in X_{\beta}$ then
$\pi_{\alpha\beta}(x,i)=(x,\rho_{\alpha\beta}(i))$ is well defined and
bijective. Moreover, it is easily seen that $\pi_{\alpha\beta}$ is a coarse
proximity map from $X_{\beta}$ to $X_{\alpha}$ such that the collection
$\\{(X_{\alpha},\mathcal{B}_{\alpha},{\bf
b}_{\alpha}),\pi_{\alpha\beta}\\}_{\alpha\beta\in\Omega}$ is an inverse system
of coarse proximity spaces. Let $(X_{\infty},\mathcal{B}_{\infty},{\bf
b}_{\infty})$ be the corresponding inverse limit coarse proximity space.
Because each $\rho_{\beta\alpha}$ is bijective and
$(\delta_{\alpha}\circ\pi_{\beta\alpha})=\delta_{\beta}$ for all
$\beta\geq\alpha$ we have that elements of $X_{\infty}$ can be expressed
uniquely as $(x,i,\alpha)_{\alpha\in\Omega}$ where $x\in X$, and
$i\in\Omega(\alpha)$ is such that $x\in A^{\alpha}_{i}$. We can then define a
map $\delta\mathrel{\mathop{\mathchar 58\relax}}X_{\infty}\rightarrow X$ by
$(x,i,\alpha)_{\alpha\in\Omega}\mapsto x$. It is easily seen that $\delta$ is
a bijective coarse proximity map.

Note that for each $\alpha\in\Omega$ we have that $\mathcal{U}X_{\alpha}$ is
homeomorphic to the disjoint union of the boundaries of the elements of
$\mathcal{F}_{\alpha}$ by Proposition 2.37. By Theorem 5.4 we have that
$\mathcal{U}X_{\infty}$ is homeomorphic to the inverse limit of these disjoint
unions.

######  Lemma 8.2.

Given $A,C\subseteq X_{\infty}$ we have that $A\phi_{\infty}C$ if and only if
$\pi_{\alpha}(A)\phi_{\alpha}\pi_{\alpha}(C)$ for all $\alpha$.

###### Proof.

Weak asymptotic resemblances are preserved by coarse proximity maps, so one
direction is clear. Assume that $\pi_{\alpha}(C)\phi_{\alpha}\pi_{\alpha}(A)$
for all $\alpha$. If $\hat{C}\subseteq C$ is an unbounded subset of
$X_{\infty}$ then $\pi_{\alpha}(\hat{C}){\bf b}_{\alpha}\pi_{\alpha}(A)$ for
all $\alpha$ because $\pi_{\alpha}(C)\phi_{\alpha}\pi_{\alpha}(A)$ and
$\pi_{\alpha}(\hat{C})\subseteq\pi_{\alpha}(C)$. Then $\hat{C}{\bf
b}_{\infty}A$. Similarly one shows that if $\hat{A}\subseteq A$ is unbounded,
then $C{\bf b}_{\infty}\hat{A}$, so $C\phi_{\infty}A$. ∎

######  Lemma 8.3.

A collection $\mathcal{U}$ of bounded subsets of $X_{\infty}$ is uniformly
bounded if and only if $\pi_{\alpha}(\mathcal{U})$ is a uniformly bounded
collection of subsets of $X_{\alpha}$ for every $\alpha\in\Omega$.

###### Proof.

Uniformly bounded collections are preserved by coarse proximity maps, so one
direction is clear. Assume that $\pi_{\alpha}(\mathcal{U})$ is uniformly
bounded in $X_{\alpha}$ for each $\alpha$. Then, for every $B\in\mathcal{U}$
we must have that $\pi_{\alpha}(B)$ is bounded in $X_{\alpha}$, so $B$ is
bounded in $X_{\infty}$. Next, if $A,C\subseteq X_{\infty}$ are such that
$A\subseteq st(C,\mathcal{U})$ and $C\subseteq st(A,\mathcal{U})$ then
$\pi_{\alpha}(A)\subseteq st(\pi_{\alpha}(C),\pi_{\alpha}(\mathcal{U}))$ and
$\pi_{\alpha}(C)\subseteq st(\pi_{\alpha}(A),\pi_{\alpha}(\mathcal{U}))$. As
$\pi_{\alpha}(\mathcal{U})$ is uniformly bounded for each $\alpha$ we have
that $\pi_{\alpha}(A)\phi_{\alpha}\pi_{\alpha}(C)$ for each $\alpha$, which by
Lemma 8.2 implies that $A\phi_{\infty}C$. Therefore, $\mathcal{U}$ is
uniformly bounded in $X_{\infty}$. ∎

######  Proposition 8.4.

Let $\mathcal{V}$ be a uniformly bounded cover of $X_{\infty}$, then the
collection of $\mathcal{V}$-components of $X_{\infty}$, denoted
$\mathcal{C}(\mathcal{V})$, is uniformly bounded.

###### Proof.

Let $\mathcal{V}$ be a uniformly bounded cover of $X_{\infty}$ and assume
towards a contradiction that $\mathcal{C}(\mathcal{V})$ is not uniformly
bounded. Then one of two things must be true. Either
$\mathcal{C}(\mathcal{V})$ contains an unbounded set, or there are unbounded
sets $A,C\subseteq X_{\infty}$ with $A\subseteq
st(C,\mathcal{C}(\mathcal{V}))$ and $C\subseteq
st(A,\mathcal{C}(\mathcal{V}))$, but $A\overline{\phi_{\infty}}C$. We will
consider these cases separately. Note that for a uniformly bounded cover
$\mathcal{U}$ of a space $Y$, by a $\mathcal{U}$-path of length $m$ in $Y$ we
mean a sequence of points $y_{0},y_{1},\ldots,y_{m}\in Y$ such that
$y_{i-1},y_{i+1}\in st(y_{i},\mathcal{U})$ for $1\leq i\leq m-1$ and is such
that $y_{m}\notin st^{m-1}(y_{0},\mathcal{U})$.

Case 1: ($\mathcal{C}(\mathcal{V})$ contains an unbounded set) In this case
there is an unbounded $\mathcal{V}$-component in $X$. Let us denote this
component by $K$. Then $K$ is necessarily infinite and $f(K)$ is an unbounded
$f(\mathcal{V})$-component in $X$. We can then find an $x_{0}\in f(K)$ and an
unbounded $A\subseteq f(K)$ that contains $x_{0}$ and is such that for every
$m\in\mathbb{N}$ we can find a $\mathcal{V}$-path of length $m$ in $K$ outside
$B(x_{0},n)$. There are then disjoint unbounded sets
$f(C_{1}),f(C_{2})\subseteq A$ such that $f(C_{1})\overline{\bf b}f(C_{2})$
and for every $m,n\in\mathbb{N}$ there is an $f(\mathcal{V})$-path of length
$m$ outside of $B(x_{0},n)$ between $f(C_{1})$ and $f(C_{2})$. Then
$C_{1}\overline{\bf b}_{\infty}C_{2}$ and for every $m,n\in\mathbb{N}$ there
is a $\mathcal{V}$-path of length $m$ outside of $f^{-1}(B(x_{0},n))$ of
length $m$. As $f(C_{1})\bar{\bf b}f(C_{2})$ there is a ${\bf b}$-cover
$\\{D_{1},D_{2},D_{3}\\}$ of $X$ such that $f(C_{1})\ll D_{1}$, $f(C_{2})\ll
D_{2}$, $f(C_{1})\bar{\bf b}D_{2}$, $f(C_{2})\bar{\bf b}D_{1}$, and
$(f(C_{1})\cup f(C_{2}))\bar{\bf b}D_{3}$. Such a cover can be constructed by
using the strong axiom for the coarse proximity on $X$. As
$\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ is a fine collection of coarse
canonical covers, there is an $\alpha\in\Omega$ such that
$\mathcal{F}_{\alpha}=\\{A_{1},\ldots,A_{n}\\}$ refines this ${\bf b}$-cover.
Then $\pi_{\alpha}(C_{1})$ and $\pi_{\alpha}(C_{2})$ are contained in
different $A_{i}\in\mathcal{F}_{\alpha}$. As there are $\mathcal{V}$-paths of
length $m$ between $C_{1}$ and $C_{2}$ outside of $f^{-1}(B(x_{0},k)$ for
every $k$ and $m$ in $\mathbb{N}$ and $f^{-1}(B(x_{0},k))$ exhausts the
bounded sets of $X_{\infty}$ we have that a similar condition holds for
$\pi_{\alpha}(C_{1})$ and $\pi_{\alpha}(C_{2})$ in $X_{\alpha}$. However, this
would imply that the number of elements of $\pi_{\alpha}(\mathcal{V})$ that
intersect multiple $A_{i}\in\mathcal{F}_{\alpha}$. However, $\mathcal{V}$ is a
uniformly bounded cover of $X_{\infty}$, so this contradicts Lemma 8.1.
Therefore $\mathcal{C}(\mathcal{V})$ can not contain an unbounded set.

Case 2: (There are unbounded $A,C\subseteq X_{\infty}$ with
$A\overline{\phi}_{\infty}C$, $A\subseteq st(C,\mathcal{C}(\mathcal{V}))$, and
$C\subseteq st(A,\mathcal{C}(\mathcal{V}))$) By the previous case we have that
$\mathcal{C}(\mathcal{V})$ does not contained unbounded sets. Therefore
$X_{\infty}$ can decomposed into countably many $\mathcal{V}$-components
$\\{X_{\infty,i}\\}_{i\in\mathbb{N}}$. Similarly, each $X_{\alpha}$ can be
decomposed into countably many $\pi_{\alpha}(\mathcal{V})$-components
$\\{X_{\alpha,i}\\}_{i\in\mathbb{N}}$. We may assume that
$\pi_{\alpha}(X_{\infty,i})=X_{\alpha,i}$ for each $\alpha$ and $i$. Now,
turning our attention to $A$ and $C$ we have that there is some
$\alpha\in\Omega$ such that
$\pi_{\alpha}(A)\overline{\phi}_{\alpha}\pi_{\alpha}(C)$. Then, without loss
of generality there is an unbounded $\hat{A}\subseteq\pi_{\alpha}(A)$ such
that $\hat{A}\overline{\bf b}_{\alpha}\pi_{\alpha}(C)$. By assumption, for
every $x\in\hat{A}$ there is a $y_{x}\in\pi_{\alpha}(C)$ such that $x$ is
connected to $y_{x}$ by a $\pi_{\alpha}(\mathcal{V})$-path. By case 1 we have
that neither $\hat{A}$ nor $\pi_{\alpha}(C)$ can be
$\pi_{\alpha}(\mathcal{V})$-connected. We may then assume that $\hat{A}$ can
be decomposed into countably many finite $\mathcal{V}$-connected components.
Say $\hat{A}=\\{K_{i}\\}_{i\in\mathbb{N}}$ and
$\mathcal{F}_{\alpha}=\\{A_{1},\ldots,A_{n}\\}$. Similarly $\pi_{\alpha}(C)$
can be decomposed into $\pi_{\alpha}(C)=\\{C_{i}\\}_{i\in\mathbb{N}}$. Up to
taking subsets we may assume that no $X_{\alpha,i}$ intersects distinct
$K_{j}$’s. Then there is an injective function $\psi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{N}\rightarrow\mathbb{N}$ such that $K_{i}$ is joined to
$C_{\psi(i)}$ via a finite $\pi_{\alpha}(\mathcal{V})$-chain. As the
$\pi_{\alpha}(\mathcal{V})$-chains witnessing the connections between
$\pi(\mathcal{V})$ components of $\hat{A}$ and $\pi_{\alpha}(C)$ must be
disjoint and bounded subsets of $X_{\alpha}$ are finite we must have that for
a given $x_{0}\in X_{\alpha}$ and for every $m\in\mathbb{N}$ there is a
$\pi_{\alpha}(\mathcal{V})$-chain in $X_{\alpha}$ from some $K_{i}$ to some
$C_{\psi(i)}$ lying completely outside of $B(x_{0},m)$. As a uniformly bounded
cover of $X_{\alpha}$ can only have finite many elements that intersect
multiple $A_{i}$ we must have that there is some $N\in\mathbb{N}$ and $A_{k}$
such that if $m\geq N$ then $K_{m},C_{m}\subseteq A_{k}$. We may then assume
that $\hat{A}$ is made up of those $K_{m}$ contained in $A_{k}$. Then,
$f(\hat{A})\overline{\bf b}f(C)$. Then, using an argument similar to that in
case 1 we can find some ${\bf b}$-cover $\\{D_{1},D_{2},D_{3}\\}$ with
$f(\hat{A})\ll D_{1}$, $f(C)\ll D_{2}$, $f(\hat{A})\bar{\bf b}D_{2}$,
$f(C)\bar{\bf b}D_{1}$, and $(f(\hat{A})\cup f(C))\bar{\bf b}D_{3}$. Then, by
the fineness of our collection of coarsely canonical covers we can find an
$X_{\beta}$ for which $\pi_{\beta}(\hat{A}),\pi_{\beta}(C)$ lie in different
$A_{i}\in\mathcal{F}_{\beta}$. The $\pi_{\beta}(\mathcal{V})$ with have
infinitely many elements that intersects multiple elements of
$\mathcal{F}_{\beta}$, which is a contradiction. Therefore we must have that
$A{\bf b}_{\infty}C$. ∎

Using Proposition 3.9 we then have:

######  Corollary 8.5.

If $\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ is a directed fine
collection of coarsely canonical covers of a proper discrete metric space $X$,
then $asdim(X_{\infty})=0$.

Now, we wish to show that $\Delta_{c}(X)\leq n$, then $asdim(X)\leq n$. To
this end we consider our coarse proximity space $X_{\infty}$ constructed via
coarse canonical covers of order not exceeding $n+1$ as above. To do this we
will proceed to show that $X_{\infty}$ is is a metrizable coarse proximity
space. Then we will show that $\delta$ is coarsely $n+1$-to-$1$ and employ
Theorem 6.5. Proving that $X_{\infty}$ is metrizable will require that instead
of working with the original inverse system used to define $X_{\infty}$ we
should work instead with all possible “thickenings” of the inverse system.
More specifically, let $\\{\mathcal{F}_{\alpha}\\}_{\alpha\in\Omega}$ be the
inverse system of disjoint coarse canonical covers of $X$ that is used to
define $X_{\infty}$. Say
$\mathcal{F}_{\alpha}=\\{A_{1}^{\alpha},\ldots,A_{\phi(\alpha)}^{\alpha}\\}$.
For a given $n\in\mathbb{N}$ we then define
$\mathcal{F}_{\alpha,n}=\\{A_{1,n}^{\alpha},\ldots,A_{\phi(\alpha),n}^{\alpha}\\}$