Inductive dimensions of coarse proximity spaces

TL;DR

This paper extends the concept of asymptotic inductive dimension to coarse proximity spaces, establishing relationships with boundary dimensions and identifying conditions for their equality, with implications for spaces with specific boundary properties.

Contribution

It introduces the notion of complete traceability for boundaries, linking the asymptotic inductive dimension of coarse proximity spaces to boundary inductive dimensions.

Findings

Asymptotic inductive dimension is greater or equal to boundary inductive dimension.

Complete traceability ensures equality of dimensions.

Spaces with Z-set boundaries or metrizable compactifications have traceable boundaries.

Abstract

In this paper, we generalize Dranishnikov's asymptotic inductive dimension to the setting of coarse proximity spaces. We show that in this more general context, the asymptotic inductive dimension of a coarse proximity space is bigger or equal to the inductive dimension of its boundary, and consequently may be strictly bigger than the covering dimension of the boundary. We also give a condition, called complete traceability, on the boundary of the coarse proximity space under which the asymptotic inductive dimension of a coarse proximity space and the inductive dimension of its boundary coincide. Finally, we show that spaces whose boundaries are $Z$-sets and spaces admitting metrizable compactifications have completely traceable boundaries.

Equations

$$\overline{A}=\{x\in X\mid \{x\}\delta A\}.$$

$$\mathcal{A}{\delta }^{\ast }\mathcal{B}⟺A\delta B$$

$${\mathcal{B}}_X=\{B\cap X\mid B\in \mathcal{B}\}$$

$$A{\mathbf{b}}_{X}C⟺A\mathbf{b}C\text{ (as subsets of Y)}$$

$$A\mathbf{b}B⟺\exists ϵ>0,\forall D\in \mathcal{B},d(A\setminus D,B\setminus D)<ϵ$$

$$A\mathbf{b}B⟺c{l}_{\overline{X}}(A)\cap c{l}_{\overline{X}}(B)\cap (\overline{X}\setminus X)\ne \varnothing$$

$$A\varphi B\text{ and }C\varphi D⟹(A\cup C)\varphi (B\cup D)$$

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

$$A\delta B⟺(A\cap B\ne \varnothing \text{ or }A\mathbf{b}B).$$

$$A^{\prime }\text{mathchar}58=c{l}_{\mathfrak{X}}(A)\cap \mathcal{U}X.$$

$$\mathcal{U}f\text{mathchar}58\mathcal{U}X\to \mathcal{U}Y,$$

$$\mathcal{U}f(\sigma )=\{D\subseteq Y\mid D{\mathbf{b}}_{2}f(C)\text{ for all }C\in \sigma \}.$$

$$\sigma \in A^{\prime }⟺C{\mathbf{b}}_{1}A\text{ for all }C\in \sigma .$$

$$\mathcal{U}f(\sigma )=\{D\subseteq Y\mid D{\mathbf{b}}_{2}f(C)\text{ for all }C\in \sigma \}.$$

$$\sigma \text{mathchar}58=\mathcal{U}g({\sigma }^{\prime })=\{C\subseteq X\mid C{\mathbf{b}}_{1}g(D)\text{ for all }D\in {\sigma }^{\prime }\}.$$

$$f(C){\mathbf{b}}_{2}f(g(D))\varphi D.$$

$$Ind(C^{\prime })=Ind(\mathcal{U}C)\le asInd(C)\le n-1.\text{qed}$$

$$asInd(D)\le Ind(\mathcal{U}D)=Ind(K)\le n-1.$$

$$A\mathbf{b}B⟺c{l}_{\overline{X}}(A)\cap c{l}_{\overline{X}}(B)\cap (\overline{X}\setminus X)\ne \varnothing .$$

$$K_{\ast }=H(K,[0,1])\text{ and }D=K_{\ast }\setminus K$$

$$D^{\prime }=c{l}_{\mathfrak{X}}(D)\cap \mathcal{U}X\subseteq c{l}_{\mathfrak{X}}(K_{\ast })\cap \mathcal{U}X=K_{\ast }\cap \mathcal{U}X=K.\text{qed}$$

$${\mathcal{C}}_n=\{B(x_{1_n},1/n),\dots ,B(x_{m_n},1/n)\}$$

$$V\cap \mathcal{U}X\subseteq c{l}_{\mathfrak{X}}(V\cap X)\cap \mathcal{U}X=(V\cap X{)}^{\prime }=C^{\prime },$$

$$C^{\prime }=c{l}_{\mathfrak{X}}(V\cap X)\cap \mathcal{U}X\subseteq c{l}_{\mathfrak{X}}(V)\cap \mathcal{U}X\subseteq \mathcal{U}X\setminus c{l}_{\mathfrak{X}}(W)\subseteq (\mathcal{U}X\setminus (\mathcal{U}X\setminus U))\subseteq U.$$

$$A\ll B⟹A{\ll }_{w}B⟹A⊑B.$$

$$A⪯B⟺(A{\ll }_{w}B\text{ or }A\varphi B)⟺(A^{\prime }\subseteq int(B^{\prime })\text{ or }{A}^{\prime }=B^{\prime }).$$

$${\mathcal{N}}_K\text{mathchar}58=\{A\subseteq X\mid K\subseteq int(A^{\prime })\}$$

$$⟨U;V_1,\dots ,V_n⟩\text{mathchar}58=\{D\subseteq X\mid D\ne \varnothing ,D\text{ is closed},D\subseteq U,\forall 1\le j\le n,D\cap V_j\ne \varnothing \},$$

$$K=\bigcap _{A\in {\mathcal{N}}_K}{A}^{\prime }$$

$$\hat{K}=\bigcap _{A\in \mathcal{N}}{A}^{\prime }.$$

Full text

1. 1 Introduction
2. 2 Preliminaries
3. 3 Asymptotic inductive dimension of coarse proximity spaces
4. 4 Relationship between $asInd(X)$ and $Ind(\mathcal{U}X)$
5. 5 Filter approach to dimension of coarse proximity spaces

1. Introduction

The coarse dimensional invariant called asymptotic dimension was originally described by Gromov in [8], and was used to study infinite discrete groups. The theory was later extended more broadly to proper metric spaces, with particular interest being generated by Yu’s result relating the property of a proper metric space having finite asymptotic dimension to the Novikov conjecture (see [17]).

Asymptotic dimension can be thought of as a coarse analog of Lebesque covering dimension. It provides a large-scale notion of dimension via a “going to infinity” perspective. That is, the invariant is defined by specifying what happens at particular “scales” (represented by uniformly bounded families) within a metric space. This stands in contrast to an alternative perspective in coarse geometry which focuses on properties defined “at infinity,” typically by means of defining properties on boundary spaces associated to metric spaces such as the Higson corona of proper metric spaces or the Gromov boundary of hyperbolic metric spaces. The prototypical definition of a large-scale dimension of a proper metric space “at infinity” is the covering dimension of the Higson corona of the space. A strong relationship between asymptotic dimension of a proper metric space and the covering dimension of its Higson corona was found by Dranishnikov who proved that if a proper metric space has finite asymptotic dimension, then its asymptotic dimension and the covering dimension of the Higson corona agree (see [5]). Pursuing the dimension theory of Higson coronae in more detail, Dranishnikov went on to define the asymptotic inductive dimension and asymptotic Brouwer dimensiongrad of proper metric spaces in [4]. These dimensional invariants are meant to serve as coarse analogs of the large inductive dimension and Brouwer dimension of topological spaces studied in classical dimension theory. As the large inductive dimension and the covering dimension coincide in the class of metrizable spaces, Dranishnikov asked if the covering dimension of the Higson corona and the asymptotic inductive dimension coincide. Alongside this question, the problem of wether or not the asymptotic inductive dimension of a proper metric space coincides with the large inductive dimension of its Higson corona was posed. In this paper, we investigate both of these questions in the broader context of coarse proximity spaces.

In [10], coarse proximity spaces were introduced to axiomatize the “at infinity” perspective of coarse geometry, providing general definitions of coarse neighborhoods (whose metric space specific definition was given by Dranishnikov in [5]), asymptotic disjointness, and closeness “at infinity.” Coarse proximity structures lie between metric spaces and coarse spaces (as defined by Roe in [15]) in a way similar to how proximity spaces relate to metric spaces and uniform spaces (see [11]). In [9], the authors construct a functor from the category of coarse proximity spaces to the category of compact Hausdorff spaces that assigns to each coarse proximity space a certain “boundary space.” This functor provides a common language for speaking of boundary spaces such as the Higson corona, the Gromov boundary, and other well-known boundary spaces. In this paper, we generalize the notion of asymptotic inductive dimension to all coarse proximity spaces (whose definition agrees with Dranishnokov’s definition for proper metric spaces) and investigate both of Dranishnikov’s questions in this more general context. In section 2, we review the necessary background information surrounding proximities as well as coarse proximities and their boundaries. In section 3, we define the asymptotic inductive dimension of coarse proximity spaces and show that it is an invariant within the category of coarse proximity spaces. We also show that the answer to Dranishnikov’s first question (“Does the asymptotic inductive dimension of a proper metric space coincide with the covering dimension of its Higson corona”) generalized to this broader context is negative. In section 4, we describe two classes of coarse proximity spaces in which the answer to the second of Dranishnokov’s questions (“Does the asymptotic inductive dimension of a proper metric space coincide with the large inductive dimension of its Higson corona”) generalized to this broader context is positive. Specifically, these are locally compact Hausdorff spaces that admit metrizable compactification and spaces admitting compactifications whose boundaries are $Z$-sets. These classes include well-known boundaries such as the Gromov and visual boundaries, as well as the boundaries of the “coarse-compactification,” described in [7]. Finally, in section 5 we utilize neighborhood filters to define another inductive dimension of coarse proximity spaces. This dimension agrees with the Brouwer dimension of the boundary of the given coarse proximity space, and consequently gives an internal characterization of the Brouwer dimension “at infinity.”

2. Preliminaries

In this section, we provide the basic definitions surrounding proximity spaces and coarse proximity spaces. First, let us review basic definitions and theorems about proximity spaces from [14].

Definition 2.1**.**

Let $X$ be a set. A binary relation $\delta$ on the power set of $X$ is called a proximity on $X$ if it satisfies the following axioms for all $A,B,C\subseteq X$:

1. (1)

$A\delta B\implies B\delta A,$ 2. (2)

$A\delta B\implies A,B\neq\emptyset,$ 3. (3)

$A\cap B\neq\emptyset\implies A\delta B,$ 4. (4)

$A\delta(B\cup C)\iff A\delta B\text{ or }A\delta C,$ 5. (5)

$A\bar{\delta}B\implies\exists E\subseteq X,\,A\bar{\delta}E\text{ and }(X\setminus E)\bar{\delta}B,$

where by $A\bar{\delta}B$ we mean that the statement $"A\delta B"$ does not hold. If in addition to these axioms a proximity satisfies $\{x\}\delta\{y\}\iff x=y$ for all $x,y\in X,$ we say that the proximity $\delta$ is separated. A pair $(X,\delta)$ where $X$ is a set and $\delta$ is a proximity on $X$ is called a proximity space.

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

[TABLE]

This topology is referred to as the induced topology of $\delta$. It is always completely regular, and is Hausdorff if and only if the proximity $\delta$ is separated. Every separated proximity space admits a unique (up to $\delta$-homeomorphism) compactification, which we describe briefly below.

Definition 2.2**.**

A cluster in a separated proximity space $(X,\delta)$ is a collection $\sigma$ of nonempty subsets of $X$ satisfying the following:

1. (1)

For all $A,B\in\sigma$, $A\delta B,$ 2. (2)

If $C\delta A$ for all $A\in\sigma,$ then $C\in\sigma,$ 3. (3)

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

A cluster $\sigma$ is called a point cluster if $\{x\}\in\sigma$ for some $x\in X.$

The set of all clusters in a separated proximity space is denoted by $\mathfrak{X}$. Given a set $\mathcal{A}\subseteq\mathfrak{X}$ and a subset $C\subseteq X,$ we say that $C$ absorbs $\mathcal{A}$ if $C\in\sigma$ for every $\sigma\in\mathcal{A}$.

Theorem 2.3**.**

Let $(X,\delta)$ be a separated proximity space and $\mathfrak{X}$ the corresponding set of clusters. The relation $\delta^{*}$ on the power set of $\mathfrak{X}$ defined by

[TABLE]

for all sets $A,B\subseteq X$ that absorb $\mathcal{A}$ and $\mathcal{B}$, respectively, is a proximity on $\mathfrak{X}.$ In fact, $(\mathfrak{X},\delta^{*})$ is a compact separated proximity space into which $X$ embeds as a dense subspace (by mapping each point to its corresponding point cluster). $\square$

The compactification described above is the Smirnov compactification of the proximity space $(X,\delta)$. We call the subset $\mathfrak{X}\setminus X$ the Smirnov boundary of $X$.

Now let us recall basic definitions and theorems surrounding coarse proximity spaces, as found in [9].

Definition 2.4**.**

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 2.5**.**

Let $X$ be a set equipped with a bornology $\mathcal{B}$. A coarse proximity on a set $X$ is a relation ${\bf b}$ on the power set of $X$ satisfying the following axioms for all $A,B,C\subseteq X\mathrel{\mathop{\mathchar 58\relax}}$

1. (1)

$A{\bf b}B\implies B{\bf b}A,$ 2. (2)

$A{\bf b}B\implies A,B\notin\mathcal{B},$ 3. (3)

$A\cap B\notin\mathcal{B}\implies A{\bf b}B,$ 4. (4)

$(A\cup B){\bf b}C\iff A{\bf b}C$ or $B{\bf b}C,$ 5. (5)

$A\bar{\bf b}B\implies\exists E\subseteq X,A\bar{\bf b}E$ 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) is called the union axiom and axiom (5) is 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 2.6**.**

Let $(Y,\mathcal{B},{\bf b})$ be a coarse proximity space and $X\subseteq Y$ any subset. Then the coarse proximity structure on $X$ given by the bornology

[TABLE]

and the binary relation ${\bf b}_{X}$ defined by

[TABLE]

makes $(X,\mathcal{B}_{X},{\bf b}_{X})$ into a coarse proximity space. This coarse proximity structure is called the subspace coarse proximity structure on $X$.

The proof that the following is a coarse proximity space can be found in [10].

Example 2.7**.**

Let $(X,d)$ be a metric space and $\mathcal{B}$ the set of all metrically bounded sets in $X$. The relation ${\bf b}$ defined by

[TABLE]

makes $(X,\mathcal{B},{\bf b})$ into a coarse proximity space. This coarse proximity structure is called the metric coarse proximity structure on the space $(X,d)$.

The proof that the following is a coarse proximity space can be found in [9].

Example 2.8**.**

Let $X$ be a localy compact Hausdorff space and $\overline{X}$ a compactification thereof. Define $\mathcal{B}$ to be the set of all $K\subseteq X$ for which $cl_{X}(K)$ is compact. Then the relation ${\bf b}$ defined by

[TABLE]

makes $(X,\mathcal{B},{\bf b})$ into a coarse proximity space. This coarse proximity structure is called the coarse proximity structure induced by the compactification $\overline{X}$.

While the ${\bf b}$ relation captures “closeness at infinity” (as will be explained shortly), the following theorem introduces a relation capturing “equality at infinity.” For the proof of the following theorem, see [10].

Theorem 2.9**.**

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 an equivalence relation satisfying

[TABLE]

for any $A,B,C,D\subseteq X.$ We call this equivalence relation the weak asymptotic resemblance induced by the coarse proximity ${\bf b}$. $\square$

When $X$ is a metric space and ${\bf b}$ is the metric coarse proximity structure, then $\phi$ is equivalent to the relation of having finite Hausdorff distance (for the proof, see [10]). The $\phi$ relation is used to define morphisms between coarse proximity spaces.

Definition 2.10**.**

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 2.11**.**

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 $A\subseteq X$

[TABLE]

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

Definition 2.12**.**

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 coarse proximity isomorphic (or just isomorphic) if there exists a coarse proximity isomorphism $f\mathrel{\mathop{\mathchar 58\relax}}X\to Y.$

Coarse proximity spaces together with closeness classes of coarse proximity maps form a category of coarse proximity spaces.

Another important relation on subsets of $X$ that will be used in this paper captures “strong inclusion at infinity” and is denoted by $\ll.$ If $A$ and $B$ are subsets of a coarse proximity space satisfying $A\bar{{\bf b}}(X\setminus B),$ then we say that $B$ is a coarse neighborhood of $A$ and denote this by $A\ll B$. It was shown in [11] that a coarse proximity can be alternatively characterized using coarse neighborhoods.

As it was shown in [9], given a coarse proximity space $(X,\mathcal{B},{\bf b})$, the boundary space associated to $X$ is defined using the following proximity associated to a coarse proximity:

[TABLE]

This is a separated proximity called the discrete extension of the coarse proximity ${\bf b}$. The subset $\mathcal{U}X$ of $\mathfrak{X}$ (where $\mathfrak{X}$ denotes the Smirnov compactification associated to $\delta$) containing only those clusters that do not contain any bounded sets is called the boundary of the coarse proximity space $X$. As it was shown in [9], $\mathcal{U}X$ is always compact and Hausdorff. Given a set $A\subseteq X$, the trace of $A$ is defined to be

[TABLE]

It was shown in [9] that $\mathcal{U}X$ encodes the asymptotic behavior of subsets of $X.$ In particular, for $A,B\subseteq X,$ one has

1. (1)

$A{\bf b}B\iff A^{\prime}\cap B^{\prime}\neq\emptyset,$ 2. (2)

$A\phi B\iff A^{\prime}=B^{\prime}.$ 3. (3)

$A\ll B\implies A^{\prime}\subseteq int(B^{\prime}),$ 4. (4)

$A,B\in\sigma\in\mathcal{U}X$ $\implies$ $A{\bf b}B,$

where $int(B^{\prime})$ is the interior of $B^{\prime}$ in $\mathcal{U}X$. In particular, the above explains the intuitive notion of ${\bf b},\phi,$ and $\ll$ capturing closeness, equality, and strong inclusion at infinity, respectively.

We finish this section with two important examples of boundaries of coarse proximity spaces. For details on these examples, see [9].

Example 2.13**.**

If $(X,d)$ is a proper metric space (where proper means that closed bounded subsets of $X$ are compact) equipped with its metric coarse proximity structure as in Example 2.7, then $\mathcal{U}X$ is homeomorphic to the Higson corona $\nu X$ of $X$.

Example 2.14**.**

Let $X$ be a locally compact Hausdorff space with compactification $\overline{X}$. If $X$ is equipped with the coarse proximity structure induced by the compactification $\overline{X}$ as in Example 2.8, then $\mathcal{U}X$ is homeomorphic to $\overline{X}\setminus X$.

3. Asymptotic inductive dimension of coarse proximity spaces

In this section, we define a notion of asymptotic inductive dimension for coarse proximity spaces. Our definition is a generalization of the definition of asymptotic inductive dimension for proper metric spaces as defined by Dranishnikov in [4], whose definition is a coarse analog of the large inductive dimension of Brouwer and Poincare. Our definition will provide an invariant within the category of coarse proximity spaces.

Let us first review the definition of the large inductive dimension of Brouwer and Poincare. For a thorough treatment of the theory thereof, see [6].

Definition 3.1**.**

Given disjoint subsets $A$ and $B$ of a topological space $X$, a separator between them is a subset $C\subseteq X$ such that there are disjoint open sets $U,V\subseteq X$ such that $X\setminus C=U\cup V$, $A\subseteq U$, and $B\subseteq V$.

Notice that separators are necessarily closed.

Definition 3.2**.**

Let $X$ be a normal space. The large inductive dimension of $X,$ denoted $Ind(X)$, is defined in the following way:

• •

$Ind(X)=-1$ if and only if $X$ is empty;

• •

for $n\geq 0$, $Ind(X)\leq n$ if for every pair of disjoint closed subsets $A,B\subseteq X,$ there exists a separator $C\subseteq X$ between $A$ and $B$ such that $Ind(C)\leq n-1;$

• •

$Ind(X)=n$ if $n\geq-1$ is the smallest integer for which $Ind(X)\leq n$ holds;

• •

if $Ind(X)\leq n$ doesn’t hold for any integer, then we say that $Ind(X)=\infty$.

The following coarse analog of large inductive dimension for proper metric spaces given by Dranishnikov can be found in [4] and [1]. To understand the definition, recall that two subsets $A$ and $B$ of a metric space are called asymptotically disjoint if and only if $\lim_{r\to\infty}(A\setminus N_{r}(x_{0}),B\setminus N_{r}(x_{0}))=\infty$ for any base point $x_{0},$ where $N_{r}(x_{0})$ denotes the ball of radius $r$ with the center $x_{0}.$

Definition 3.3**.**

Given a proper metric space $(X,d)$ and two subsets $A,B\subseteq X$ that are asymptotically disjoint, a set $C\subseteq X$ is an asymptotic separator between $A$ and $B$ if the trace of $C$ in $\nu X$ (i.e., the intersection of the closure of $C$ in the Higson compactification and the Higson Corona $\nu X$) is a separator in $\nu X$ between traces of $A$ and $B$ in $\nu X.$

Definition 3.4**.**

Let $(X,d)$ be a proper metric space. The asymptotic inductive dimension of $X,$ denoted $asInd(X)$, is defined in the following way:

• •

$asInd(X)=-1$ if and only if $X$ is bounded;

• •

for $n\geq 0$, $asInd(X)\leq n$ if for every pair of asymptotically disjoint subsets $A,B\subseteq X,$ there exists an asymptotic separator $C\subseteq X$ between $A$ and $B$ such that $asInd(C)\leq n-1;$

• •

$asInd(X)=n$ if $n\geq-1$ is the smallest integer for which $asInd(X)\leq n$ holds;

• •

if $asInd(X)\leq n$ doesn’t hold for any integer, then we say that $asInd(X)=\infty$.

Now we generalize Dranishnikov’s asymptotic inductive dimension to all coarse proximity spaces.

Definition 3.5**.**

Given subsets $A$ and $B$ of a coarse proximity space $(X,\mathcal{B},{\bf b})$ such that $A\bar{\bf b}B$, a set $C\subseteq X$ is an asymptotic separator between $A$ and $B$ if $C^{\prime}$ is a separator in $\mathcal{U}X$ between $A^{\prime}$ and $B^{\prime}$.

Note that the above definition coincides with Dranishnikov’s definition of an asymptotic separator in the case of proper metric spaces.

Definition 3.6**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space. The asymptotic inductive dimension of $X,$ denoted $asInd(X)$, is defined in the following way:

• •

$asInd(X)=-1$ if and only if $X$ is bounded;

• •

for $n\geq 0$, $asInd(X)\leq n$ if for $A,B\subseteq X$ such that $A\bar{\bf b}B,$ there exists an asymptotic separator $C\subseteq X$ between $A$ and $B$ such that $asInd(C)\leq n-1$ (where $C$ is equipped with the subspace coarse proximity structure);

• •

$asInd(X)=n$ if $n\geq-1$ is the smallest integer for which $asInd(X)\leq n$ holds;

• •

if $asInd(X)\leq n$ doesn’t hold for any integer, then we say that $asInd(X)=\infty$.

Note that the above definition coincides with Dranishnikov’s definition of an asymptotic inductive dimension when the proper metric space is given the metric coarse proximity structure.

To show that the asymptotic inductive dimension is invariant in the category of coarse proximity spaces, we need the following lemma.

Lemma 3.7**.**

Let $f\mathrel{\mathop{\mathchar 58\relax}}(X,\mathcal{B}_{1},{\bf b}_{1})\rightarrow(Y,\mathcal{B}_{2},{\bf b}_{2})$ be a coarse proximity isomorphism with coarse proximity inverse $g$. Let $\phi_{1}$ and $\phi_{2}$ be weak asymptotic resemblances associated to ${\bf b}_{1}$ and ${\bf b}_{2}$, respectively. Then given $A,B\subseteq X,$ we have that:

1. (1)

$A{\bf b}_{1}B$* if and only if $f(A){\bf b}_{2}f(B),$* 2. (2)

$A\phi_{1}B$* if and only if $f(A)\phi_{2}f(B).$*

Proof.

That $A{\bf b}_{1}B$ implies $f(A){\bf b}_{2}f(B)$ is given simply by the definition of a coarse proximity map. If $f(A){\bf b}_{2}f(B)$, then $gf(A){\bf b}_{1}gf(B)$. By the definition of coarse proximity isomorphisms, we have that $gf(A)\phi_{1}A$ and $gf(B)\phi_{1}B$. By Corollary $6.18$ in [10], get that $A{\bf b}_{1}B$.

If $A\phi_{1}B,$ then $f(A)\phi_{2}f(B)$ because coarse proximity isomorphisms preserve $\phi$ relations (see Proposition 7.14 in [10]). To see the opposite direction, let $f(A)\phi_{2}f(B).$ Then, $gf(A)\phi_{1}gf(B).$ By the definition of coarse proximity isomorphisms, this shows that $A\phi_{1}gf(A)\phi_{1}gf(B)\phi_{1}B$, which implies that $A\phi_{1}B.$ ∎

Recall from [9] that given two coarse proximity spaces $(X,\mathcal{B}_{1},{\bf b}_{1})$ and $(Y,\mathcal{B}_{2},{\bf b}_{2}),$ their corresponding discrete extensions $\delta_{1}$ and $\delta_{2},$ and their corresponding boundaries $\mathcal{U}X$ and $\mathcal{U}Y,$ and given a coarse proximity map $f\mathrel{\mathop{\mathchar 58\relax}}X\to Y,$ the map $f\mathrel{\mathop{\mathchar 58\relax}}(X,\delta_{1})\rightarrow(Y,\delta_{2})$ is a proximity map. In fact, there exists a unique extension of $f$ to the Smirnov compactifications of $(X,\delta_{1})$ and $(Y,\delta_{2})$ that maps $\mathcal{U}X$ to $\mathcal{U}Y$ (see Corollary 4.12 in [9]). That unique extension (restricted to $\mathcal{U}X$) is denoted $\mathcal{U}f,$ and is defined by

[TABLE]

[TABLE]

Proposition 3.8**.**

Let $f\mathrel{\mathop{\mathchar 58\relax}}(X,\mathcal{B}_{1},{\bf b}_{1})\rightarrow(Y,\mathcal{B}_{2},{\bf b}_{2})$ be a coarse proximity map and $\mathcal{U}f\mathrel{\mathop{\mathchar 58\relax}}\mathcal{U}X\rightarrow\mathcal{U}Y$ the corresponding continuous map between boundaries. Then for an unbounded set $A\subseteq X$ with trace $A^{\prime},$ we have that $\mathcal{U}f(A^{\prime})\subseteq f(A)^{\prime}$. If $f$ is a coarse proximity isomorphism with coarse inverse $g$, then this is an equality.

Proof.

Let $f\mathrel{\mathop{\mathchar 58\relax}}(X,\mathcal{B}_{1},{\bf b}_{1})\rightarrow(Y,\mathcal{B}_{2},{\bf b}_{2})$ be given. Identify $X$ and $Y$ as the corresponding sets of point clusters given by the respective discrete extensions $\delta_{1}$ and $\delta_{2}$. Let $A\subseteq X$ be unbounded. Note that if $\sigma$ is an element of $\mathcal{U}X,$ then

[TABLE]

Also, recall that $\mathcal{U}f(\sigma)$ is given by

[TABLE]

Let $\sigma\in A^{\prime}.$ We wish to show that $\mathcal{U}f(\sigma)\in f(A)^{\prime}$. It will suffice to show that if $D\in\mathcal{U}f(\sigma)$ then $D{\bf b}_{2}f(A)$. However, this is trivial, since $A\in\sigma,$ and consequently $f(A)\in\mathcal{U}f(\sigma)$. Thus $\mathcal{U}f(A^{\prime})\subseteq f(A)^{\prime}$.

Now assume that $f$ is a coarse proximity isomorphism with a coarse inverse $g.$ Let $\sigma^{\prime}\in f(A)^{\prime}.$ Then $D{\bf b}_{2}f(A)$ for all $D\in\sigma^{\prime}.$ We wish to show that $\sigma^{\prime}\in\mathcal{U}f(A^{\prime}),$ i.e., there exists $\sigma\in A^{\prime}$ such that $\mathcal{U}f(\sigma)=\sigma^{\prime}.$ Define

[TABLE]

Let us show that $\sigma\in A^{\prime}$ and $\mathcal{U}f(\sigma)=\sigma^{\prime}.$ To see that $\sigma\in A^{\prime},$ recall that $\sigma^{\prime}\in f(A)^{\prime}$ implies that $f(A)\in\sigma^{\prime}.$ Consequently, $g(f(A))\in\sigma.$ Since clusters in the boundary are closed under the $\phi$ relation (see Lemma $4.4$ in [9]) and $g(f(A))\phi_{1}A$ by the definition of a coarse proximity isomorphism, we have that $A\in\sigma.$ To see that $\mathcal{U}f(\sigma)=\sigma^{\prime},$ let $D\in\sigma^{\prime}.$ To show that $D\in\mathcal{U}f(\sigma),$ we need to show that $D{\bf b}_{2}f(C)\text{ for all }C\in\sigma.$ Let $C\in\sigma.$ This means that $C{\bf b}_{1}g(E)$ for all $E\in\sigma^{\prime}.$ In particular, $C{\bf b}_{1}g(D).$ Consequently,

[TABLE]

Thus, $f(C){\bf b}_{2}D,$ finishing the proof that $\sigma^{\prime}\subseteq\mathcal{U}f(\sigma).$ But this implies that $\mathcal{U}f(\sigma)=\sigma^{\prime}.$ Thus, $\mathcal{U}f(A^{\prime})=f(A)^{\prime},$ as desired. ∎

Theorem 3.9**.**

Isomorphic coarse proximity spaces have the same asymptotic inductive dimension.

Proof.

As could be expected, the proof will be by induction. Let $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Y$ be a coarse proximity isomorphism with coarse inverse $g$. If $asInd(X)=-1,$ then $X$ is bounded, which implies that $Y$ is bounded as well, giving $asInd(Y)=-1$. Now assume that the result holds up to (and including) $n-1$ and assume that $asInd(X)=n$. Let $A,B\subseteq Y$ be such that $A\bar{\bf b}B$. Then by Lemma 3.7 we have that $g(A)\bar{\bf b}g(B)$. Because $asInd(X)=n$ we have that there is an asymptotic separator $C\subseteq X$ between $g(A)$ and $g(B)$ such that $asInd(C)\leq n-1$. By the definition of coarse proximity isomorphisms, we have that $fg(A)\phi A$ and $fg(B)\phi B$. Thus, by Lemma 3.7 we have that $(fg(A))^{\prime}=A^{\prime}$ and $(fg(B))^{\prime}=B^{\prime}$. By inductive hypothesis, we have that $asInd(f(C))=asInd(C)\leq n-1$. It will then suffice to show that $f(C)$ is an asymptotic separator between $A$ and $B$. However, this follows from Proposition 3.8. Therefore $asInd(Y)=n$. ∎

Dranishnikov’s question regarding the relation between $dim(\nu X)$ and $asInd(X)$ for proper metric spaces can be generalized to:

Does $dim(\mathcal{U}X)=asInd(X)$ for all coarse proximity spaces $(X,\mathcal{B},{\bf b})?$

The answer to this generalized question is negative. To see that, let us first prove a useful lemma.

Lemma 3.10**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space and let $A_{1},B_{1}\subseteq\mathcal{U}X$ be disjoint closed subsets. Then there are unbounded subsets $A_{2},B_{2}\subseteq X$ such that $A_{2}\bar{{\bf b}}B_{2}$, $A_{1}\subseteq A_{2}^{\prime}$, and $B_{1}\subseteq B_{2}^{\prime}$.

Proof.

Let $\delta$ be the discrete extension of ${\bf b}$, and $\mathfrak{X}$ the Smirnov compactification of $(X,\delta)$. Let $A_{1},B_{1}\subseteq\mathcal{U}X$ be given. Becausee $\mathcal{U}X$ is a closed and compact subset of $\mathfrak{X},$ we have that $A_{1}$ and $B_{1}$ are disjoint closed subsets of $\mathfrak{X}$. Then, using Urysohn’s lemma there is a continuous (proximity) function $f\mathrel{\mathop{\mathchar 58\relax}}\mathfrak{X}\rightarrow[0,1]$ such that $f(A_{1})=0$ and $f(B_{1})=1$. We then define $A_{2}=f^{-1}([0,1/3])\cap X$ and $B_{2}\mathrel{\mathop{\mathchar 58\relax}}=f^{-1}([2/3,1])\cap X$. These two sets are clearly disjoint. They are also nonempty and unbounded as $f^{-1}([0,1/3])$ and $f^{-1}([2/3,1])$ are compact neighborhoods of $A_{1}$ and $B_{1}$ in $\mathfrak{X},$ respectively. Now we will show that $A_{1}\subseteq A_{2}^{\prime}$. Let $\sigma\in A_{1}$. Let $U$ be an arbitrary open set in $\mathfrak{X}$ that contains $\sigma$. Then $f^{-1}([0,1/3])\cap U$ is an open set in $\mathfrak{X}$ that contains $\sigma.$ Because $X$ is dense in $\mathfrak{X}$ we have that every open neighborhood about $\sigma$ contains some point of $X.$ In particular, it contains an element of $A_{2}.$ Consequently, $\sigma$ is in the closure of $A_{2}$ in $\mathfrak{X}$, i.e., $\sigma\in A_{2}^{\prime}.$ Showing that $B_{1}\subseteq B_{2}^{\prime}$ is similar. ∎

Theorem 3.11**.**

$Ind(\mathcal{U}X)\leq asInd(X)$* for all coarse proximity spaces $(X,\mathcal{B},{\bf b})$.*

Proof.

The proof is by induction on $asInd(X)$. If $asInd(X)=-1,$ then $X$ is bounded and $\mathcal{U}X=\emptyset$ which implies that $Ind(\mathcal{U}X)=-1$. Now assume that the result holds for $asInd(X)<n$. Assume that $asInd(X)=n$ and let $A_{1},B_{1}\subseteq\mathcal{U}X$ be disjoint closed subsets. Then $A_{1}$ and $B_{1}$ are disjoint closed (compact) subsets in $\mathfrak{X}$, the Smirnov compactification of the proximity space $(X,\delta)$. By Lemma 3.10, there are unbounded subsets $A_{2},B_{2}\subseteq X$ such that $A_{2}\bar{{\bf b}}B_{2}$, $A_{1}\subseteq A_{2}^{\prime}$, and $B_{1}\subseteq B_{2}^{\prime}$. Because $asInd(X)=n$ there is an asymptotic separator $C\subseteq X$ between $A_{2}$ and $B_{2}$ such that $asInd(C)\leq n-1$. Because $A_{1}\subseteq A_{2}^{\prime}$ and $B_{1}\subseteq B_{2}^{\prime}$ we have that $C^{\prime}$ is a topological separator between $A_{1}$ and $B_{2}$. Since by Corollary $4.18$ in [9] we have that $\mathcal{U}C\cong C^{\prime}$, by inductive hypothesis we get

[TABLE]

In 1958, P. Vopenka described a class of compact Hausdorff spaces for which the large inductive dimension is strictly greater than the covering dimension (see [13]). Since every compact Hausdorff space can be realized as the boundary of a coarse proximity space (see section 6 in [9]), Theorem 3.11 answers the question of Dranishnikov generalized to coarse proximity spaces, as stated in the following corollary.

Corollary 3.12**.**

There is a coarse proximity space $X$ for which $dim(\mathcal{U}X)<asInd(X)$.

Since we know that $Ind(\mathcal{U}X)\leq asInd(X)$ for all coarse proximity spaces $(X,\mathcal{B},{\bf b}),$ the next most natural question is under what conditions do $Ind(\mathcal{U}X)$ and $asInd(X)$ coincide? We provide answers to this question in the next section.

4. Relationship between $asInd(X)$ and $Ind(\mathcal{U}X)$

In the previous section, the proof of Theorem 3.11 suggested that the gap (or lack thereof) between $Ind(\mathcal{U}X)$ and $asInd(X)$ for a coarse proximity space $X$ is tied up with which closed sets $K\subseteq\mathcal{U}X$ appear as the traces of unbounded subsets of $X$. It is an easy exercise to show that if $X$ is an unbounded proper metric space and $x$ is an element of the Higson corona $\nu X$, then there is no unbounded set whose trace is precisely $\{x\}$. Being unable to detect all closed subsets of the Higson corona in this way makes closing the gap between $asInd(X)$ (when $X$ is equipped with its metric coarse proximity structure) and $Ind(\nu X)$ by simply modifying the proof of Theorem 3.11 impossible. In general, whether or not $asInd(X)=Ind(\nu X)$ is an open question. However, in this section we will describe scenarios in which $asInd(X)=Ind(\mathcal{U}X)$ for certain classes of coarse proximity spaces.

One class of spaces in which $asInd(X)=Ind(X)$ is the obvious one suggested by the proof of Theorem 3.11. Specifically, this is the class of spaces for which every closed subset of the boundary can be realized as the trace of an unbounded set.

Definition 4.1**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space with boundary $\mathcal{U}X$. A closed subset $C\subseteq\mathcal{U}X$ is called traceable if there is some unbounded $A\subseteq X$ such that $A^{\prime}=C$. We say that $\mathcal{U}X$ is completely traceable if every nonempty closed $C\subseteq\mathcal{U}X$ is traceable.

It is an easy consequence of Proposition 3.8 that given two isomorphic coarse proximity spaces $X$ and $Y,$ $\mathcal{U}X$ is completely traceable if and only if $\mathcal{U}Y$ is.

Theorem 4.2**.**

If $(X,\mathcal{B},{\bf b})$ is a coarse proximity space whose boundary is completely traceable, then $asInd(X)=Ind(\mathcal{U}X)$.

Proof.

In light of Theorem 3.11, it will suffice to show that $asInd(X)\leq Ind(\mathcal{U}X).$ The proof will be by induction on $Ind(\mathcal{U}X)$. If $Ind(\mathcal{U}X)=-1,$ then $\mathcal{U}X=\emptyset,$ which implies $X$ is bounded and correspondingly $asInd(X)=-1$. Assume then that the result holds for $Ind(\mathcal{U}X)<n$ and assume that $Ind(\mathcal{U}X)=n$. Let $A,B\subseteq X$ be such that $A\bar{{\bf b}}B$. We may assume without loss of generality that both $A$ and $B$ are unbounded. Because $A\bar{{\bf b}}B$ we have that $A^{\prime}$ and $B^{\prime}$ are disjoint closed subsets of $\mathcal{U}X$. Because $Ind(\mathcal{U}X)=n$ there is a closed separator $K\subseteq\mathcal{U}X$ between $A^{\prime}$ and $B^{\prime}$ such that $Ind(K)\leq n-1$. Because $\mathcal{U}X$ is completely traceable there is an unbounded set $D\subseteq X$ such that $D^{\prime}=K$. Equipping $D$ with its subspace coarse proximity structure, we have that $\mathcal{U}D$ is homeomorphic to $K$, and therefore by the inductive hypothesis we have that

[TABLE]

Therefore, $asInd(X)\leq n=Ind(\mathcal{U}X)$, yielding the desired result. ∎

Which spaces have completely traceable boundaries? One such class is given by spaces whose boundaries in their compactifications are $Z$-sets.

Definition 4.3**.**

Let $X$ be a topological space and $A$ a closed subset of $X.$ Then $A$ is called a $Z$-set if there exists a homotopy $H\mathrel{\mathop{\mathchar 58\relax}}X\times[0,1]\to X$ such that $H(X,0)=id_{X}$ and $H(X,t)\subseteq(X\setminus A)$ for all $t\in(0,1].$

Many spaces have boundaries that are $Z$-sets. For example, Gromov boundary of a hyperbolic proper metric space is a $Z$-set. For more on $Z$-sets, see for example [2] or [3].

Theorem 4.4**.**

Let $X$ be a locally compact Hausdorff space and $\overline{X}$ a compactification of $X$. Let $(X,\mathcal{B},{\bf b})$ be the coarse proximity structure on $X$ induced by the compactification $\overline{X},$ i.e., $\mathcal{B}$ is the collection of all sets whose closures in $X$ are compact, and ${\bf b}$ is defined by

[TABLE]

If $\overline{X}\setminus X$ is a $Z$-set in $\overline{X},$ then $\mathcal{U}X$ (identified with $\overline{X}\setminus X$) is completely traceable.

Proof.

Denote the Smirnov compactification of $X$ given by the discrete extension of ${\bf b}$ by $\mathfrak{X}$. Then $\mathfrak{X}=X\cup\mathcal{U}X$. Let $H\mathrel{\mathop{\mathchar 58\relax}}\mathfrak{X}\times I\rightarrow\mathfrak{X}$ be a homotopy that witnesses $\mathcal{U}X$ being a $Z$-set of $\mathfrak{X}$ and let $K\subseteq\mathcal{U}X$ be a nonempty closed subset. Define

[TABLE]

Then $K\subseteq K_{*}$ and $D$ is an unbounded subset of $X$ such that $K\subseteq D^{\prime}$. This latter statement can be seen as given $x\in K,$ we have that given any sequence $(s_{n})$ in $(0,1]$ converging to $0,$ we have that $(H(x,s_{n}))$ converges to $x$. To see that $D^{\prime}$ is precisely $K,$ we simply note that by the closed map lemma $K_{*}=K\cup D$ is closed in $\mathfrak{X},$ and thus

[TABLE]

Corollary 4.5**.**

Let $X$ be a locally compact Hausdorff space and $\overline{X}$ a compactification of $X$ such that $\overline{X}\setminus X$ is a $Z$-set in $\overline{X}.$ Let $(X,\mathcal{B},{\bf b})$ be the coarse proximity structure on $X$ induced by the compactification $\overline{X}$. Then $asInd(X)=Ind(\mathcal{U}X)=Ind(\overline{X}\setminus X)$ $\square.$

Another class of spaces with completely traceable boundaries are spaces admitting metrizable compactifications. Such compactifications were described in detail using controlled products in [7].

Theorem 4.6**.**

Let $X$ be a locally compact Hausdorff space and $\overline{X}$ a metrizable compactification of $X$. Let $(X,\mathcal{B},{\bf b})$ be the coarse proximity structure on $X$ induced by the compactification $\overline{X}$. Then $\overline{X}\setminus X$, identified with $\mathcal{U}X$, is completely traceable.

Proof.

Let $K\subseteq\mathcal{U}X$ be a given nonempty closed subset and let $d$ be a metric on $\overline{X}$ that is compatible with the topology on $\overline{X}$. For each $n\in\mathbb{N},$ let

[TABLE]

be a finite open cover of $K$ where each $x_{i_{n}}\in K$. As $X$ is dense in $\overline{X},$ we have that $B(x_{i_{n}},1/n)$ intersects $X$ nontrivially for each $i$ and $n$. We then let $y_{i_{n}}$ be an element of $B(x_{i_{n}},1/n)\cap X$ for each $i$ and $n$. Define $D\subseteq X$ to be the collection of all these $y_{i_{n}}$ as $n$ rangers over $\mathbb{N}$. It is clear that $D$ is an unbounded subset. We claim that $D^{\prime}=K$. To see that $D^{\prime}\subseteq K,$ let $(y_{n})$ be an unbounded sequence in $D$ that converges to some element of $\overline{X}$. Because this sequence is unbounded we have that for each $n\in\mathbb{N}$ there is some $m\geq n$ such that $y_{m}$ is in an element of $\mathcal{C}_{n}$. Then there is a subsequence $(y_{n_{k}})$ of $(y_{n})$ such that $\lim_{k\to\infty}d(y_{n_{k}},K)=0$, which implies that $(y_{n_{k}})$ converges to a point in $K$. As each subsequence of $(y_{n})$ must converge to the same point as $(y_{n}),$ we have that $(y_{n})$ converges to a point of $K$, which gives us that $D^{\prime}\subseteq K$. Now let $x\in K$. For each $n\in\mathbb{N},$ there is some $x_{i_{n}}\in K$ such that $x\in B(x_{i_{n}},1/n)$. Choosing one such open ball for each $n\in\mathbb{N},$ we specify an unbounded sequence in $D$ that converges to $x$. Therefore, $K\subseteq D^{\prime}$ and consequently, $K=D^{\prime}$. Thus, $\mathcal{U}X$ is completely traceable. ∎

Corollary 4.7**.**

Let $X$ be a locally compact Hausdorff space and $\overline{X}$ a metrizable compactification of $X$. Let $(X,\mathcal{B},{\bf b})$ be the coarse proximity structure on $X$ induced by the compactification $\overline{X}$. Then $asInd(X)=Ind(\mathcal{U}X)=Ind(\overline{X}\setminus X)$. $\square.$

5. Filter approach to dimension of coarse proximity spaces

In [12], Isbell introduced an inductive dimension $\delta Ind$ of proximity spaces. Like the more familiar inductive dimensions $Ind$ and $ind$, the dimension $\delta Ind$ is defined by means of separating sets within a space in a certain way. As Isbell was defining a dimension that was to be relevant for proximity space theory, his notion of separation was defined in terms of the proximity relation, as in the definition below. To understand the definition, recall that given two subsets $A$ and $B$ of a proximity space $X$, $A$ is called a $\delta$-neighborhood of $B$ iff $B\bar{\delta}(X\setminus A).$

Definition 5.1**.**

Let $A$ and $B$ be subsets of a proximity space $(X,\delta)$ such that $A\bar{\delta}B$. A subset $C\subseteq X$ is said to $\delta$-separate $A$ and $B$ in $X$ (or be a $\bm{\delta}-$separator between $A$ and $B$ in $X$) if there are disjoint subsets $U,V\subseteq X$ such that $X\setminus C=U\cup V$, $A\subseteq U$, $B\subseteq V$, and $U\bar{\delta}V$. A subset $D\subseteq X$ such that $D\bar{\delta}(A\cup B)$ is said to free $A$ and $B$ (or be a freeing set for $A$ and $B$) if every $\delta$-neighborhood of $D$ that is disjoint from $A\cup B$ $\delta$-separates $A$ and $B$.

The definition of the dimension $\delta Ind(X)$ for a proximity space $(X,\delta)$ is identical to Definition 3.2 if one replaces disjoint closed sets with far sets (i.e., sets that are not close to each other) and separators with freeing sets. The notation $\delta Ind$ would suggest that under ideal conditions the dimension function is identical to $Ind$. However, in [16] it was recently shown that $\delta Ind$ agrees with the Brouwer dimension $Dg$ on compact Hausdorff spaces and differs from $Ind$ in this class. Brouwer dimension is defined by the use of cuts.

Definition 5.2**.**

Given a topological space $X$ and two disjoint closed subsets $A,B\subseteq X$, a closed subset $C\subseteq X$ that is disjoint from $(A\cup B)$ is called a cut between $A$ and $B$ if every continuum $K\subseteq X$ such that $K\cap A\neq\emptyset$ and $K\cap B\neq\emptyset$ also satisfies $K\cap C\neq\emptyset$.

By continuum we mean a compact connected Hausdorff space. The definition of $Dg$ for $T_{4}$ spaces is then identical to Definition 3.2 upon replacing the word “separator“ with “cut.” In compact Hausdorff spaces there is only one compatible proximity. In [16], the following is proven:

Theorem 5.3**.**

Let $X$ be a compact Hausdorff space and $A,B\subseteq X$ disjoint closed sets. A closed set $C\subseteq X$ that is disjoint from $A$ and $B$ is a cut between $A$ and $B$ if and only if it frees $A$ and $B$. Moreover, $Dg(X)=\delta Ind(X)$.

A cut between disjoint closed subsets of a compact Hausdorff space is then characterized by its closed neighborhoods. Said differently, the way in which a cut in a compact Hausdorff space separates disjoint closed sets is determined by a property held by its neighborhoods. Indeed, we could define a cut as a collection of subsets of a space satisfying certain properties. In this section, we define a dimension for coarse proximity spaces in this way that provides an internal characterization of the Brouwer dimension of the boundary.

Recall from the preliminaries chapter that coarse neighborhoods (defined by $A\ll B$ iff $A\bar{\bf b}(X\setminus B)$) have the property that $A\ll B\implies A^{\prime}\subseteq int(B^{\prime}).$ However, this relation turns out to be too restrictive for the purpose of “controlling” or “approximating” sets in the boundary by the means of traces of subsets of the original space. Consequently, we need to introduce slightly less restrictive relations.

Definition 5.4**.**

Let $A$ and $B$ be subsets of a coarse proximity space $(X,\mathcal{B},{\bf b})$. We define $A\sqsubseteq B$ if for all $C\subseteq X$ we have that $A{\bf b}C$ implies that $B{\bf b}C$.

Definition 5.5**.**

Let $A$ and $B$ be subsets of a coarse proximity space $(X,\mathcal{B},{\bf b})$. We define $A\ll_{w}B$ if there is a $C\subseteq X$ such that $A\ll C\sqsubseteq B$.

To see that these relations are indeed less strict than the $\ll$ relation, we will first show some useful boundary characterizations of the above two relations. To do that, we need an intuitive but technical lemma.

Lemma 5.6**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space. Let $K,U\subseteq\mathcal{U}X$ be subsets such that $K$ is closed in $\mathcal{U}X$, $U$ is open in $\mathcal{U}X$, and $K\subseteq U.$ Then there exists $C\subseteq X$ such that $K\subseteq int(C^{\prime})\subseteq C^{\prime}\subseteq U.$ Also, if $K=A^{\prime}$ for some $A\subseteq X$ and $U=int(B^{\prime})$ for some $B\subseteq X,$ then $A\ll C.$ In particular, $A^{\prime}\subseteq int(C^{\prime})\subseteq int(B^{\prime}).$

Proof.

Let $K\subseteq U\subseteq\mathcal{U}X,$ where $K$ is closed in $\mathcal{U}X$ and $U$ is open in $\mathcal{U}X.$ Then, $\mathcal{U}X\setminus U$ is closed in $\mathcal{U}X.$ Since $\mathcal{U}X$ is closed in $\mathfrak{X},$ we know that $\mathcal{U}X\setminus U$ and $K$ are closed in $\mathfrak{X}.$ Since $\mathfrak{X}$ is normal, there exist disjoint open sets $V$ and $W$ in $\mathfrak{X}$ such that $K\subseteq V,$ $(\mathcal{U}X\setminus U)\subseteq W,$ and the closures of $V$ and $W$ in $\mathfrak{X}$ are disjoint. Then $C\mathrel{\mathop{\mathchar 58\relax}}=V\cap X$ is a nonempty subset of $X$ such that $K\subseteq int(C^{\prime})\subseteq C^{\prime}\subseteq U.$ To see the first inclusion (i.e., $K\subseteq int(C^{\prime})$) notice that for any $x\in K,$ we know that $V\cap\mathcal{U}X$ is an open set in $\mathcal{U}X$ containing $x$ that is also contained in $C^{\prime},$ since

[TABLE]

where the first inclusion follows from the density of $X$ in $\mathfrak{X}.$ To see that $C^{\prime}\subseteq U,$ simply note that

[TABLE]

Now assume that $K=A^{\prime}$ for some $A\subseteq X$ and $U=int(B^{\prime})$ for some $B\subseteq X.$ To see that $A\ll C,$ notice that since $A^{\prime}\subseteq V$ and $V$ is open, $\mathfrak{X}\setminus V$ is closed in $\mathfrak{X}$ and does not intersect $A^{\prime}.$ Since $(X\setminus C)\subseteq(\mathfrak{X}\setminus V),$ it is also true that the closure of $X\setminus C$ in $\mathfrak{X}$ does not intersect $A^{\prime}.$ Thus, $(X\setminus C)^{\prime}\cap A^{\prime}=\emptyset.$ But this is equivalent to $(X\setminus C)\bar{\bf b}A,$ which in turn is equivalent to $A\ll C.$ ∎

Proposition 5.7**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space and $A,B\subseteq X$. Then:

1. (1)

$A\sqsubseteq B$* if and only if $A^{\prime}\subseteq B^{\prime}$.* 2. (2)

$A\ll_{w}B$* if and only if $A^{\prime}\subseteq int(B^{\prime})$.* 3. (3)

$A\phi B$* if and only if for all $C\subseteq X$, $A\ll_{w}C\iff B\ll_{w}C$.* 4. (4)

$A\phi B$* if and only if for all $C\subseteq X$, $A\sqsubseteq C\iff B\sqsubseteq C$.*

Proof.

The proofs of $(3)$ and $(4)$ are clear when one proves $(1)$ and $(2)$ and recalls that $A\phi B$ if and only if $A^{\prime}=B^{\prime}$. We will prove $(1)$ and $(2)$.

$(1)$ If $A\sqsubseteq B$ and $x\in A^{\prime}\cap(\mathcal{U}X\setminus B^{\prime}),$ then by normality of the boundary there is an open set $U\subseteq\mathcal{U}X$ such that $x\in U$ and $U\cap B^{\prime}=\emptyset$. By Lemma 5.6, there exists an unbounded set $C\subseteq X$ such that $x\in int(C^{\prime})\subseteq C^{\prime}\subseteq U$. However, this implies that $A{\bf b}C$, but $C\bar{{\bf b}}B$, contradicting $A\sqsubseteq B$. Conversely, assume that $A^{\prime}\subseteq B^{\prime}$ and let $C\subseteq X$ be such that $A{\bf b}C$. Then $C^{\prime}\cap A^{\prime}\neq\emptyset$. As $A^{\prime}\subseteq B^{\prime},$ we have that $C^{\prime}\cap B^{\prime}\neq\emptyset,$ which gives us that $C{\bf b}B.$ Hence, $A\sqsubseteq B$.

$(2)$ Assume that $A\ll_{w}B$ and let $C\subseteq X$ be such that $A\ll C\sqsubseteq B$. Then $A^{\prime}\subseteq int(C^{\prime})\subseteq B^{\prime}$, establishing $A^{\prime}\subseteq int(B^{\prime})$. Conversely, assume that $A^{\prime}\subseteq int(B^{\prime})$. By Lemma 5.6, there is an unbounded set $C\subseteq X$ such that $A\ll C$ and $C^{\prime}\subseteq int(B^{\prime})$. Then $A\ll C\sqsubseteq B,$ and hence $A\ll_{w}B$. ∎

Notice that the above proposition implies that for any coarse proximity space $(X,\mathcal{B},{\bf b})$, we have that

[TABLE]

The opposite implications are not true, though. To see that, equip $\mathbb{R}^{2}$ with the metric coarse proximity structure and let $A=\mathbb{R}^{2}$ and $B=\mathbb{Z}^{2}.$ Then $A\ll_{w}B$ and $A\sqsubseteq B,$ but $A\not\ll B.$ Also, to see that $A\sqsubseteq B$ does not imply $A\ll_{w}B,$ let $X\mathrel{\mathop{\mathchar 58\relax}}=[0,1]\times[0,1)$, and let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space induced by the compactification $\overline{X}\mathrel{\mathop{\mathchar 58\relax}}=[0,1]\times[0,1]$ (see Example 2.8). Then let $A=B=[0,\frac{1}{2}]\times[0,1).$ By Example 2.14, it is clear that $A\sqsubseteq B,$ but $A\not\ll_{w}B.$

As it turns out, the $\ll_{w}$ relation seems to be the most appropriate to define an internal characterization of the Brouwer dimension of the boundary of a coarse proximity space, as we will soon see.

Recall that given a set $X$, a preorder on $X$ is a binary relation $\preceq$ on $X$ that is transitive and reflexive. Every coarse proximity space $(X,\mathcal{B},{\bf b})$ admits a natural preorder on its power set defined by $A\preceq B$ if and only if $A\ll_{w}B$ or $B\phi A$. In other words:

[TABLE]

Definition 5.8**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space. Let $\preceq$ be the preorder on the power set of $X$ described above. A collection $\mathcal{N}$ of subsets of $X$ is called a neighborhood filter if it satisfies the following axioms:

1. (1)

$\mathcal{N}\neq\emptyset;$ 2. (2)

for every $A,B\in\mathcal{N}$ there is some $C\in\mathcal{N}$ such that $C\preceq A$ and $C\preceq B;$ 3. (3)

if $A\in\mathcal{N}$ and $C\subseteq X$ is such that $A\preceq C,$ then $C\in\mathcal{N};$ 4. (4)

if $A\in\mathcal{N},$ then there is a $B\in\mathcal{N}$ such that $B\ll_{w}A.$

Notice that by axiom 3 neighborhood filters in coarse proximity spaces are closed under the $\phi$ relation.

Example 5.9**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space with boundary $\mathcal{U}X$ and let $K\subseteq\mathcal{U}X$ be any closed set. Then the set

[TABLE]

is a neighborhood filter in $X$.

Proof.

Since $X\in\mathcal{N}_{K}$ and $A\preceq C\implies int(A^{\prime})\subseteq int(C^{\prime}),$ axioms $1$ and $3$ are clear. Axioms $2$ and $4$ follow from Lemma 5.6. ∎

To show that any neighborhood filter in a coarse proximity space has the form given in example 5.9, let us first prove an easy lemma.

Lemma 5.10**.**

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

1. (1)

$\mathcal{N}=\mathcal{N}_{\emptyset},$** 2. (2)

$\mathcal{N}$* contains a bounded set,* 3. (3)

$\mathcal{N}=2^{X}.$**

Proof.

if $\mathcal{N}=\mathcal{N}_{\emptyset},$ then $\emptyset\in\mathcal{N},$ since $\emptyset\subseteq int(\emptyset^{\prime})=\emptyset$. If $\mathcal{N}$ contains a bounded set, then by axiom $3$ of a neighborhood filter it has to contain all sets (since $\emptyset$ is in the interior of a trace of any set). Finally, if $\mathcal{N}=2^{X},$ then notice that $\mathcal{N}_{\emptyset}=2^{X}$ (again because the empty set is in the interior of a trace of every set), and consequently $\mathcal{N}=\mathcal{N}_{\emptyset}.$ ∎

In the next proposition, we show that any neighborhood filter is of the form given in Example 5.9. To be able to understand the proof, recall that given a topological space $X,$ the Vietoris topology on the collection of nonempty compact subsets of X is given by a basis consisting of the sets of the form

[TABLE]

for any open $U,V_{1},\ldots,V_{n}\subseteq X.$

Proposition 5.11**.**

Let $\mathcal{N}$ be a neighborhood filter in a coarse proximity space $(X,\mathcal{B},{\bf b})$. Then there is a closed set $K\subseteq\mathcal{U}X$ such that $\mathcal{N}=\mathcal{N}_{K}$.

Proof.

If $\mathcal{N}$ contains a bounded set, then by Lemma 5.10 we have that $\mathcal{N}=\mathcal{N}_{\emptyset}$. Otherwise, the set $\mathcal{N}$ is directed by $\preceq$ and may be viewed as a net in the Vietoris topology on the collection of nonempty closed subsets of $\mathcal{U}X$ by viewing each element $A\in\mathcal{N}$ as its trace $A^{\prime}$. Since $\mathcal{U}X$ is compact and Hausdorff, the Vietoris topology on the collection of nonempty closed subsets of $\mathcal{U}X$ is compact and Hausdorff. Consequently, $\mathcal{N}$ must have a convergent subnet, denoted by $\mathcal{M}$, that converges to some element $K\subseteq\mathcal{U}X$.

To see that $\mathcal{N}_{K}\subseteq\mathcal{N}$, notice that we have that if $U\subseteq\mathcal{U}X$ is an open (or closed) neighborhood of $K,$ then there is some $C\in\mathcal{N}$ such that $C\preceq U$ (it is because $\langle U,U\rangle$ is open in the Vietoris topology and consequently contains some $C\in\mathcal{N}$). This gives us that $\mathcal{N}$ contains a neighborhood basis for $K,$ and thus $\mathcal{N}_{K}\subseteq\mathcal{N}$. To show that $\mathcal{N}\subseteq\mathcal{N}_{K},$ it will suffice to show that $K\subseteq int(C^{\prime})$ for all $C\in\mathcal{M}$ (because then given $A\in\mathcal{N},$ by cofinality of $\mathcal{M}$ we get a $C\in\mathcal{M}$ such that $C\preceq A,$ and consequently $K\subseteq int(C^{\prime})\subseteq int(A^{\prime}),$ which shows that $A\in\mathcal{N}_{K}$). If $C\in\mathcal{M}$ is such that $K\not\subseteq int(C^{\prime}),$ then with the help of axiom 4 of a neighborhood filter and cofinality of $\mathcal{M}$ we can find an element $\tilde{C}\in\mathcal{M}$ such that $\tilde{C}\ll_{w}C.$ Notice that $K\not\subseteq\tilde{C}^{\prime}$. Consequently, there is an $x\in K\setminus\tilde{C}^{\prime}$. Let $\{U_{\alpha\}}$ be a neighborhood basis for $x$ in $\mathcal{U}X$ and let $\hat{\mathcal{M}}\subseteq\mathcal{M}$ be the subnet of $\mathcal{M}$ containing all $\hat{C}\in\mathcal{M}$ such that $\hat{C}\preceq\tilde{C}$. Then $\hat{\mathcal{M}}$ also converges to $K$, which implies that $\hat{\mathcal{M}}$ is eventually in $\langle X;U_{\alpha}\rangle$ for all $\alpha$. This implies that $\tilde{C}^{\prime}\cap U_{\alpha}\neq\emptyset$ for all $\alpha$, which is to say that there is a net in $\tilde{C}^{\prime}$ converging to $x$, implying that $x\in\tilde{C}^{\prime}$, a contradiction. Therefore, $K\subseteq int(C^{\prime})$ for all $C\in\mathcal{M},$ and thus $\mathcal{N}\subseteq\mathcal{N}_{K}$. ∎

Remark 5.12*.*

In the above proof, we utilized a convergent subnet $\mathcal{M}$ of the net $\mathcal{N}.$ In fact one can show that the original net $\mathcal{N}$ from the above proof also converges to $K$ in the Vietoris topology. To see this, let $\langle U;V_{1},\ldots,V_{n}\rangle$ be a basic open set containing $K$. Because $\mathcal{M}$ converges to $K$ we have that there is some $A\in\mathcal{M}$ such that for all $C\in\mathcal{M}$ such that $C\preceq A,$ we have $C^{\prime}\in\langle U;V_{1},\ldots,V_{n}\rangle$. Now let $C_{1}\in\mathcal{M}$ be any element such that $C_{1}\preceq A$ and let $D\in\mathcal{N}$ be any element such that $D\preceq C_{1}$. We claim that $D^{\prime}\in\langle U;V_{1}\ldots,V_{n}\rangle$. Since $D\preceq C_{1},$ we know that $D^{\prime}\subseteq C_{1}^{\prime}\subseteq U.$ To see that $D^{\prime}$ intersects all the $V_{j}$’s, note that the cofinality of $\mathcal{M}$ in $\mathcal{N}$ gives us that there is some $C_{2}\in\mathcal{M}$ such that $C_{2}\preceq D$. Because $C_{2}\in\langle U;V_{1},\ldots,V_{n}\rangle,$ we know that $C_{2}^{\prime}\cap V_{j}\neq\emptyset$ for all $1\leq j\leq n.$ Since $C_{2}^{\prime}\subseteq D^{\prime},$ this also shows that $D^{\prime}\cap V_{j}\neq\emptyset$ for all $1\leq j\leq n.$ Consequently, we have that $D^{\prime}\in\langle U;V_{1},\ldots,V_{n}\rangle,$ which gives us that $\mathcal{N}$ converges to $K.$

For a given neighborhood filter $\mathcal{N}$ in a coarse proximity space $X,$ we will refer to the corresponding set $K$ in Proposition 5.11 as the center of $N$. The following proposition shows that centers are unique and have an explicit form.

Proposition 5.13**.**

Let $\mathcal{N}=\mathcal{N}_{K}$ be a neighborhood filter in a coarse proximity space $(X,\mathcal{B},{\bf b})$. Then

[TABLE]

Proof.

If $\mathcal{N}$ contains a bounded set, then by Lemma 5.10 we have that $\mathcal{N}=\mathcal{N}_{\emptyset}$ and the conclusion is trivial. Otherwise, let $\mathcal{N}=\mathcal{N}_{K\neq\emptyset}$ be a neighborhood filter in a coarse proximity space $(X,\mathcal{B},{\bf b})$ as in Proposition 5.11, and define

[TABLE]

We will show that $\hat{K}=K.$ Notice that for any $A\in\mathcal{N},$ we have that $\hat{K}\subseteq A^{\prime}.$ To see that this implies $\hat{K}\subseteq K,$ assume on the contrary that there exists $x\in\hat{K}\setminus K.$ Then by normality of the Smirnov compactification $\mathfrak{X},$ there exist open sets $U$ and $V$ such that $x\in U,$ $K\subseteq V$ and the closures of $U$ and $V$ in $\mathfrak{X}$ are disjoint. Then $A\mathrel{\mathop{\mathchar 58\relax}}=V\cap X$ is an open set in $X$ such that $K\subseteq int(A^{\prime}).$ Consequently, $A\in\mathcal{N},$ but $x\notin A^{\prime}$ (since $x\in U$), a contradiction. Thus, $\hat{K}\subseteq K.$ To see that $K\subseteq\hat{K},$ notice that if $x\in K,$ then $x\in int(A^{\prime})$ for all $A\in\mathcal{N}.$ In particular, $x\in\bigcap_{A\in\mathcal{N}}A^{\prime}=\hat{K}.$ Thus, $\hat{K}=K$. ∎

To understand the behavior of centers of neighborhood filters, in the next few propositions we give internal characterizations of filters whose centers are disjoint, whose centers are such that one center is contained in the interior of the other one, and whose centers consist of the union of the centers of two other neighborhood filters.

Definition 5.14**.**

Given $\mathcal{N},\mathcal{M}\in\mathcal{N}(X)$ we define $\mathcal{N}\curlyvee\mathcal{M}$, read as “$\mathcal{N}$ and $\mathcal{M}$ diverge”, if there are $A\in\mathcal{N}$ and $B\in\mathcal{M}$ such that $A\bar{{\bf b}}B$.

Proposition 5.15**.**

If $\mathcal{N}_{K_{1}}$ and $\mathcal{N}_{K_{2}}$ are two neighborhood filters in a coarse proximity space $(X,\mathcal{B},{\bf b}),$ then $\mathcal{N}_{K_{1}}\curlyvee\mathcal{N}_{K_{2}}$ if and only if $K_{1}\cap K_{2}=\emptyset$.

Proof.

Let $\mathcal{N}_{K_{1}}$ and $\mathcal{N}_{K_{1}}$ be given. Assume $\mathcal{N}_{K_{1}}\curlyvee\mathcal{N}_{K_{2}}.$ This means that there are $A\in\mathcal{N}_{K_{1}}$ and $B\in\mathcal{N}_{K_{2}}$ such that ${A}\bar{{\bf b}}B$. As $K_{1}\subseteq A^{\prime}$ and $K_{2}\subseteq B^{\prime}$ and $A\bar{{\bf b}}B\implies(A^{\prime}\cap B^{\prime})=\emptyset,$ we have that $K_{1}\cap K_{2}=\emptyset$. Conversely, assume that $K_{1}\cap K_{2}=\emptyset.$ Because $\mathcal{U}X$ is Hausdorff that are disjoint open sets $U,V\subseteq\mathcal{U}X$ such that $K_{1}\subseteq U$ and $K_{2}\subseteq V$. Then by Lemma 5.6 there are unbounded sets $A,B\subseteq X$ such that $K_{1}\subseteq int(A^{\prime})\subseteq U$ and $K_{2}\subseteq int(B^{\prime})\subseteq V$. Then $A\in\mathcal{N}_{K_{1}}$ and $B\in\mathcal{N}_{K_{2}}$ are such that $A\bar{\bf b}B$ (since their traces are disjoint), showing that $\mathcal{N}_{K_{1}}\curlyvee\mathcal{N}_{K_{2}}$. ∎

Proposition 5.16**.**

Let $\mathcal{N}_{K_{1}},\mathcal{N}_{K_{2}}$ be neighborhood filers in a coarse proximity space $(X,\mathcal{B},{\bf b})$. Then $K_{1}\subseteq int(K_{2})$ if and only if there is an $A\in\mathcal{N}_{K_{1}}$ such that for all $B\in\mathcal{N}_{K_{2}}$ we have that $A\sqsubseteq B$.

Proof.

If $K_{1}\subseteq int(K_{2}),$ then by Lemma 5.6 there is an unbounded $A\subseteq X$ such that $K_{1}\subseteq int(A^{\prime})\subseteq K_{2}$. Then $A\in\mathcal{N}_{K_{1}},$ and if $C\in\mathcal{N}_{K_{2}},$ then we have that $A\sqsubseteq C$, as $A^{\prime}\subseteq K_{2}\subseteq C^{\prime}$. Conversely, let $A\in\mathcal{N}_{K_{1}}$ be given such that for all $B\in\mathcal{N}_{K_{2}}$ we have that $A\sqsubseteq B.$ Notice that $K_{1}\subseteq int(A^{\prime})$ by Proposition 5.11, and since $A\sqsubseteq C$ for all $C\in\mathcal{N}_{K_{2}},$ we have that

[TABLE]

Thus $K_{1}\subseteq int(A^{\prime})\subseteq int(K_{2})$.

∎

If $\mathcal{N}_{K_{1}}$ and $\mathcal{N}_{K_{2}}$ are neighborhood filters as in Proposition 5.16, we will simply say that $\mathcal{N}_{K_{1}}$** is in the interior of $\mathcal{N}_{K_{2}}$**.

Proposition 5.17**.**

Let $\mathcal{N},\mathcal{M}$ be two neighborhood filters in a coarse proximity space $(X,\mathcal{B},{\bf b})$. The the collection

[TABLE]

is a neighborhood filter.

Proof.

Axiom 1 is immediate. To see axiom 2, let $C_{1}$ and $C_{2}$ be arbitrary elements of $\mathcal{N}\vee\mathcal{M}.$ Then there exist $A_{1}\cup B_{1}$ and $A_{2}\cup B_{2}$ such that $(A_{1}\cup B_{1})\preceq C_{1}$ and $(A_{2}\cup B_{2})\preceq C_{1}.$ Since $\mathcal{N}$ and $\mathcal{M}$ are neighborhood filters, by axiom 2 and 4 there exist $A\in\mathcal{N}$ and $B\in\mathcal{M}$ such that $A^{\prime}\subseteq int(A_{1}^{\prime}),int(A_{2}^{\prime})$ and $B^{\prime}\subseteq int(B_{1}^{\prime}),int(B_{2}^{\prime}).$ Consequently,

[TABLE]

for $i=1,2.$ Thus, $(A\cup B)\preceq(A_{i}\cup B_{i})\preceq C_{i}$ for $i=1,2.$ Since $(A\cup B)\in(\mathcal{N}\vee\mathcal{M}),$ this shows axiom 2. Axiom 3 is immediate by transitivity of $\preceq.$ Finally, to see axiom 4 let $C$ be an arbitrary element of $\mathcal{N}\vee\mathcal{M}.$ Then there exist $A\in\mathcal{N}$ and $B\in\mathcal{N}$ such that $(A\cup B)\preceq C.$ Since $\mathcal{N}$ and $\mathcal{M}$ are neighborhood filters, by axiom 4 there exist $A_{1}\in\mathcal{N}$ and $B_{1}\in\mathcal{M}$ such that $A_{1}^{\prime}\subseteq int(A^{\prime})$ and $B_{1}^{\prime}\subseteq int(B^{\prime}).$ Consequently, we have that

[TABLE]

Since $A_{1}\cup B_{1}$ is an element of $\mathcal{N}\vee\mathcal{M},$ this shows axiom 4. ∎

Proposition 5.18**.**

If $\mathcal{N}$ and $\mathcal{M}$ are two neighborhood filters in a coarse proximity space $(X,\mathcal{B},{\bf b})$ with respective centers $K_{1}$ and $K_{2}$, then the center of $\mathcal{N}\vee\mathcal{M}$ is $K_{1}\cup K_{2}$.

Proof.

By Lemma 5.13, we have that the center of $\mathcal{N}\vee\mathcal{M}$, denoted by $K_{3}$, is given by

[TABLE]

∎

From now on, we are going to denote the set of all neighborhood filters on a coarse proximity space $(X,\mathcal{B},{\bf b})$ by $\mathcal{N}(X)$. This set admits a natural partial order given by

[TABLE]

for any $\mathcal{N},\mathcal{M}\in\mathcal{N}(X).$ Notice that this is equivalent to saying that $\mathcal{N}\leq\mathcal{M}$ if and only if the center of $\mathcal{N}$ is a subset of the center of $\mathcal{M}.$ We utilize this natural partial order to introduce the notion of a ${\bf b}$-separator in a coarse proximity space.

Definition 5.19**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space with boundary $\mathcal{U}X$. Let $\mathcal{W},\mathcal{N}_{1},\mathcal{N}_{2}\in\mathcal{N}(X)$ be such that $\mathcal{N}_{1},\mathcal{N}_{2}\leq\mathcal{W}$ and $\mathcal{N}_{1}\curlyvee\mathcal{N}_{2}$. Given a third neighborhood filter $\mathcal{D}\leq\mathcal{W}$ such that $\mathcal{D}\curlyvee(\mathcal{N}_{1}\vee\mathcal{N}_{2}),$ we say that $\mathcal{D}$ is a ${\bf b}$-separator between $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ in $\mathcal{W}$ if there are $\mathcal{M}_{1},\mathcal{M}_{2}\in\mathcal{N}(X)$ such that

1. (1)

$\mathcal{N}_{1}\leq\mathcal{M}_{1}$ and $\mathcal{N}_{2}\leq\mathcal{M}_{2},$ 2. (2)

$\mathcal{M}_{1}\curlyvee\mathcal{M}_{2},$ 3. (3)

$\mathcal{M}_{1}\vee\mathcal{M}_{2}\vee\mathcal{D}=\mathcal{W}.$

Next proposition shows that ${\bf b}$-separators in a coarse proximity space correspond to $\delta$-separators of its boundary. To understand the proof, recall that any compact Hausdorff space $X$ has a unique proximity (inducing that topology) given by $A\delta B\iff\bar{A}\cap\bar{B}\neq\emptyset.$

Proposition 5.20**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space and $\mathcal{N}_{1},\mathcal{N}_{2},\mathcal{D},$ and $\mathcal{W}$ be neighborhood filters in $X$ with respective centers $K_{1},K_{2},D,$ and $W$. Then $\mathcal{D}$ is a ${\bf b}$-separator between $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ in $\mathcal{W}$ if and only if $D$ is a $\delta$-separator between $K_{1}$ and $K_{2}$ in $W$.

Proof.

($\Rightarrow$) Let $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ be as in Definition 5.19 with centers $M_{1}$ and $M_{2},$ respectively. Define $U\mathrel{\mathop{\mathchar 58\relax}}=M_{1}\setminus D$ and $V\mathrel{\mathop{\mathchar 58\relax}}=M_{2}\setminus D$. Because $(\mathcal{N}_{1}\vee\mathcal{N}_{2})\curlyvee\mathcal{D}$ we have that $K_{1}\subseteq U$ and $K_{2}\subseteq V$. Because $\mathcal{M}_{1}\curlyvee\mathcal{M}_{2}$ we have that $U\bar{\delta}V,$ as $M_{1}\cap M_{2}=\emptyset$. What remains to be shown is that $W\setminus D=U\cup V$. However, because $\mathcal{W}=\mathcal{M}_{1}\vee\mathcal{M}_{2}\vee\mathcal{D}$ we have that $W=M_{1}\cup M_{2}\cup D=U\cup V\cup D$. Since $D$ does not intersect $U$ or $V,$ this shows that $W\setminus D=U\cup V.$

($\Leftarrow$) Assume that $D$ is a $\delta$-separator in $W$ between $K_{1}$ and $K_{2}$. Since $D\subseteq W,$ we know that $\mathcal{D}\leq\mathcal{W}.$ Since $D$ is a $\delta$-separator in $W$ between $K_{1}$ and $K_{2},$ then $W\setminus D=U\cup V$ where $U\bar{\delta}V$, $K_{1}\subseteq U$, and $K_{2}\subseteq V$. Consequently, $D$ is disjoint from $K_{1}$ and $K_{2},$ i.e., $\mathcal{D}\curlyvee(\mathcal{N}_{1}\vee\mathcal{N}_{2}).$ Define $\mathcal{M}_{1}=\mathcal{N}_{\overline{U}}$ and $\mathcal{M}_{2}=\mathcal{N}_{\overline{V}}$. As $\overline{U}\cap\overline{V}=\emptyset,$ we have that $\mathcal{M}_{1}\curlyvee\mathcal{M}_{2}$. Moreover, because $K_{1}\subset U$ and $K_{2}\subseteq V$ we have that $\mathcal{N}_{1}\leq\mathcal{M}_{1}$ and $\mathcal{N}_{2}\leq\mathcal{M}_{2}$. As $W=D\cup K_{1}\cup K_{2},$ we have that $\mathcal{W}=\mathcal{M}_{1}\vee\mathcal{M}_{2}\vee\mathcal{N}_{D}$. Thus $\mathcal{D}$ is a ${\bf b}$-separator in $\mathcal{W}$ between $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$. ∎

Remark 5.21*.*

Notice that in the “$\Leftarrow$” direction in the above proof, we also know that because $\overline{U}\cap int(D)=\emptyset$ and $\overline{V}\cap int(D)=\emptyset$ we have that $(\mathcal{M}_{1}\vee\mathcal{M}_{2})\curlyvee\tilde{\mathcal{D}}$ for all $\tilde{\mathcal{D}}$ in the interior of $\mathcal{D}$.

Definition 5.22**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space with boundary $\mathcal{U}X$. Let $\mathcal{W},\mathcal{N}_{1},\mathcal{N}_{2}\in\mathcal{N}(X)$ be such that $\mathcal{N}_{1},\mathcal{N}_{2}\leq\mathcal{W}$ and $\mathcal{N}_{1}\curlyvee\mathcal{N}_{2}$. Given a third neighborhood filter $\mathcal{C}\leq\mathcal{W}$ such that $\mathcal{C}\curlyvee(\mathcal{N}_{1}\vee\mathcal{N}_{2})$ we say that $\mathcal{C}$ coarsely frees $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ in $\mathcal{W}$ if for all neighborhood filters $\mathcal{D}\leq\mathcal{W}$ such that $\mathcal{C}$ is in the interior of $\mathcal{D}$ and $\mathcal{D}\curlyvee(\mathcal{N}_{1}\vee\mathcal{N}_{2})$ we have that $\mathcal{D}$ is a b-separator in $\mathcal{W}$ between $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$.

To finally define the notion of a coarse proximity Brouwer dimension, we need to show that coarsely freeing neighborhood filters correspond to freeing sets in the boundary, as the following proposition does.

Proposition 5.23**.**

Let $\mathcal{W},\mathcal{N}_{1},\mathcal{N}_{2},$ and $\mathcal{C}$ be neighborhood filters in a coarse proximity space $(X,\mathcal{B},{\bf b})$ with respective centers $W,K_{1},K_{2},$ and $C.$ Then $\mathcal{C}$ frees $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ in $\mathcal{W}$ if and only if $C$ frees $K_{1}$ and $K_{2}$ in $W$.

Proof.

First assume that $\mathcal{C}$ frees $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ in $\mathcal{W}.$ Let $D$ be an arbitrary $\delta$-neighborhood of $C$ that is far from $K_{1}\cup K_{2}$ (i.e., $D\bar{\delta}(K_{1}\cup K_{2})$). Without loss of generality we can assume that $D$ is closed in $\mathcal{U}X$ (because $D$ $\delta$-separates $K_{1}$ and $K_{2}$ if and only if its closure does). Then notice that $C\subseteq int(D)$ and $D$ is disjoint from $K_{1}\cup K_{2}.$ This implies that $\mathcal{N}_{D}$ is a neighborhood filter such that $\mathcal{N}_{D}\leq\mathcal{W},$ $\mathcal{C}$ is in the interior of $\mathcal{N}_{D}$ and $\mathcal{N}_{D}\curlyvee(\mathcal{N}_{1}\vee\mathcal{N}_{2}).$ Proposition 5.20 implies then that $D$ is a $\delta$-separator between $K_{1}$ and $K_{2}$ in $W,$ showing that $C$ frees $K_{1}$ and $K_{2}$ in $W.$

Conversely, assume that $C$ frees $K_{1}$ and $K_{2}$ in $W.$ Let $\mathcal{D}$ be an arbitrary neighborhood filter with the center $D$ such that $\mathcal{D}\leq\mathcal{W},$ $\mathcal{C}$ is in the interior of $\mathcal{D}$ and $\mathcal{D}\curlyvee(\mathcal{N}_{1}\vee\mathcal{N}_{2}).$ Then notice that $D$ is a $\delta$-neighborhood of $C$ (because $C\bar{\delta}(X\setminus D)\iff\bar{C}\subseteq int(D)$ in any coarse proximity space) that is disjoint from $K_{1}\cup K_{2}.$ Consequently, $D$ $\delta$-separates $K_{1}$ and $K_{2}.$ By Propositon 5.20, this shows that $\mathcal{D}$ is a ${\bf b}$-separator between $\mathcal{N}_{1}$ and $\mathcal{N}_{2}.$ Consequently, $\mathcal{C}$ frees $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ in $\mathcal{W}.$ ∎

Definition 5.24**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space and $\mathcal{W}$ a neighborhood filter in $X$. The coarse proximity Brouwer dimension of $\mathcal{W}$, denoted ${\bf b}Dg(\mathcal{W})$, is defined in the following way:

1. (1)

${\bf b}Dg(\mathcal{W})=-1$ if $\mathcal{W}$ contains a bounded set; 2. (2)

For $n\geq 0$, ${\bf b}Dg(\mathcal{W})\leq n$ if for all $\mathcal{N}_{1},\mathcal{N}_{2}\leq\mathcal{W}$ such that $\mathcal{N}_{1}\curlyvee\mathcal{N}_{2},$ there is a $\mathcal{C}\leq\mathcal{W}$ that coarsely frees $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ in $\mathcal{W}$, and satisfies ${\bf b}Dg(\mathcal{C})\leq n-1$; 3. (3)

We say that ${\bf b}Dg(\mathcal{W})=n$ if ${\bf b}Dg(\mathcal{W})\leq n$ and ${\bf b}Dg(\mathcal{W})\leq n-1$ does not hold; 4. (4)

We say that ${\bf b}Dg(\mathcal{W})=\infty$ in the case that ${\bf b}Dg(\mathcal{W})\leq n$ does not hold for any $n\in\mathbb{N}\cup\{0,-1\}$.

The value ${\bf b}Dg(X)$ is defined by identifying $X$ with the neighborhood filter $\mathcal{N}_{\mathcal{U}X}$.

Theorem 5.25**.**

Let $(X,\mathcal{B},{\bf b})$ be a coarse proximity space with boundary $\mathcal{U}X$. If $\mathcal{N}_{K}$ is a neighborhood filter in $X$, then

[TABLE]

In particular,

[TABLE]

Proof.

This is a consequence of Theorem 5.3 and Proposition 5.23. ∎

Corollary 5.26**.**

If $X$ and $Y$ are isomorphic coarse proximity spaces, then ${\bf b}Dg(X)={\bf b}Dg(Y)$.

Proof.

If $X$ and $Y$ are isomorphic coarse proximity spaces, then their corresponding boundaries are homeomorphic. Consequently,

[TABLE]