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Article

# Varieties of Coarse Spaces

Faculty of Computer Science and Cybernetics, Kyiv University, Academic Glushkov pr. 4d, 03680 Kyiv, Ukraine
Axioms 2018, 7(2), 32; https://doi.org/10.3390/axioms7020032
Received: 27 March 2018 / Revised: 22 April 2018 / Accepted: 10 May 2018 / Published: 14 May 2018

## Abstract

:
A class $M$ of coarse spaces is called a variety if $M$ is closed under the formation of subspaces, coarse images, and products. We classify the varieties of coarse spaces and, in particular, show that if a variety $M$ contains an unbounded metric space then $M$ is the variety of all coarse spaces.
MSC:
54E35; 08B85

## 1. Introduction

Following , we say that a family $E$ of subsets of $X × X$ is a coarse structure on a set X if:
• Each $ε ∈ E$ contains the diagonal $▵ X$, $▵ X = { ( x , x ) : x ∈ X }$;
• If $ε , δ ∈ E$ then $ε ∘ δ ∈ E$ and $ε − 1 ∈ E$, where $ε ∘ δ = { ( x , y ) : ∃ z ( ( x , z ) ∈ ε , ( z , y ) ∈ δ ) } ,$ and $ε − 1 = { ( y , x ) : ( x , y ) ∈ ε }$;
• And if $ε ∈ E$ and $△ X ⊆ ε ′ ⊆ ε$ then $ε ′ ∈ E$.
Each $ε ∈ E$ is called an entourage of the diagonal. A subset $E ′ ⊆ E$ is called a base for $E$ if, for every $ε ∈ E$ there exists $ε ′ ∈ E ′$ such that $ε ⊆ ε ′$.
The pair $( X , E )$ is called a coarse space. For $x ∈ X$ and $ε ∈ E$, we denote $B ( x , ε ) = { y ∈ X : ( x , y ) ∈ ε }$ and say that $B ( x , ε )$ is a ball of radius ε around x. We note that a coarse space can be considered as an asymptotic counterpart of a uniform topological space and can be defined in terms of balls, see [2,3]. In this case a coarse space is called a ballean.
A coarse space $( X , E )$ is called connected if, for any $x , y ∈ X$, there exists $ε ∈ E$ such that $y ∈ B ( x , ε )$. A subset Y of X is called bounded if there exist $x ∈ X$ and $ε ∈ E$ such that $Y ⊆ B ( x , ε )$. The coarse structure $E = { ε ∈ X × X : △ X ⊆ ε }$ is the unique coarse structure such that $( X , E )$ is connected and bounded. In what follows, all coarse spaces under consideration are assumed to be connected.
Given a coarse space $( X , E )$, each subset $Y ⊆ X$ has the natural coarse structure $E | Y = { ε ∩ ( Y × Y ) : ε ∈ E }$, where $( Y , E | Y )$ is called a subspace of $( X , E )$. A subset Y of X is called large (or coarsely dense) if there exists an $ε ∈ E$ such that $X = B ( Y , ε )$ where $B ( Y , ε ) = ∪ y ∈ Y B ( Y , ε )$.
Let $( X , E )$, and $( X ′ , E ′ )$ be coarse spaces. A mapping $f : X ⟶ X ′$ is called coarse (or bornologous in the terminology of ) if, for every $ε ∈ E$ there exists an $ε ′ ∈ E ′$ such that, for every $x ∈ X$, we have $f ( B ( x , ε ) ) ⊆ ( B ( f ( x ) , ε ′ ) )$. If f is surjective and coarse then $( X ′ , E ′ )$ is called a coarse image of $( X , E )$. If f is a bijection, such that f and $f − 1$ are coarse mappings, then f is called an asymorphism. The coarse spaces $( X , E )$, $( X ′ , E ′ )$ are called coarsely equivalent if there exist large subsets $Y ⊆ X$, and $Y ′ ⊆ X$ such that $( Y , E | Y )$ and $( Y ′ , E ′ | Y ′ )$ are asymorphic.
To conclude the coarse vocabulary, we take a family ${ ( X α , E α ) : α < κ }$ of coarse spaces and define the product $P α < κ ( X α , E α )$ as the set $P α < κ X α$ endowed with the coarse structure with the base set $P α < κ E α$. If $ε α ∈ E α$, for $α < κ$ and $x , y ∈ P α < κ X α$, where $x = ( x α ) α < κ$, and $y = ( y α ) α < κ$ then $( x , y ) ∈ ( ε α ) α < κ$ if and only if $( x α , y α ) ∈ ε α$ for every $α < κ$.
Let $M$ be a class of coarse spaces closed under asymorphisms. We say that $M$ is a variety if $M$ is closed under the formation of subspaces $( S M ⊆ M )$, coarse images $( Q M ⊆ M )$, and products $( P M ⊆ M )$.
For an infinite cardinal $κ$, we say that a coarse space $( X , E )$ is κ-bounded if every subset $Y ⊆ X$, such that $| Y | < κ$, is bounded. Additionally, we denote $M κ$ as the variety of all $κ$-bounded coarse spaces. We denote by $M s i n g l e$ and $M b o u n d$ the variety of singletons and the variety of all bounded coarse spaces, respectively. Thus we have the chain of varieties:
$M s i n g l e ⊂ M b o u n d ⊂ … ⊂ M κ ⊂ … ⊂ M ω .$
In Section 2, we prove that every variety of coarse spaces lies in this chain and, in Section 3, we discuss some extensions of this result to coarse spaces endowed with additional algebraic structures.

## 2. Results

We recall that a family $I$ of subsets of a set X is an ideal in the Boolean algebra $P X$ of all subsets of X, if $I$ is closed under finite unions and subsets. Every ideal $I$ defines a coarse structure with the base ${ E A : A ∈ I }$ where $E A = ( A × A ) ∪ △ X$. Therefore, $B ( x , E A ) = A$ if $x ∈ A$ and $B ( x , E A ) = { x }$ if $x ∈ X \ A$. We denote the obtained coarse space by $( X , I )$. For a cardinal $κ$, $[ X ] < κ$ denotes the ideal ${ Y ⊆ X : | Y | < κ }$. If $( X , E )$ is a coarse space, the family $I$ of all bounded subsets of X is an ideal. The coarse space $( X , I )$ is called the companion of $( X , E )$.
Let $K$ be a class of coarse spaces. We say that a coarse space $( X , E )$ is free with respect to $K$ if, for every $( X ′ , E ′ ) ∈ K$ every mapping $f : ( X , E ) ⟶ ( X ′ , E ′ )$ is coarse. For example, $( X , [ X ] < κ )$ is free with respect to the variety $M κ$. Since $( κ , [ κ ] < κ ) ∈ M κ$ but $( κ , [ κ ] < κ ) ∉ M κ ′$ for each $κ ′ > κ$; the inclusion $M κ ′ ⊂ M κ$ is strict.
Lemma 1.
If a coarse space $( X , E )$ is free with respect to a class $K$ then $( X , E )$ is free with respect to $S K$, $Q K$, $P K$.
Proof.
We verify only the second statement. Let $( X ′ , E ′ ) ∈ K$, $( X ′ ′ , E ′ ′ ) ∈ Q K$, and $h : ( X ′ , E ′ ) ⟶ ( X ″ , E ″ )$ be a coarse surjective mapping. We take an arbitrary $f : X ⟶ X ″$ and choose $h ′ : X ⟶ X ′$ such that $f = h h ′$. Since $( X , E )$ is free with respect to $K$, $h ′ : ( X , E ) ⟶ ( X ′ , E ′ )$ is coarse so f is coarse as the composition of the coarse mappings h, and $h ′$. ☐
Lemma 2.
Let X be a set and let $K$ be a class of coarse spaces, $K ≠ M s i n g l e$. Then there exists a coarse structure $E$ on X such that $( X , E ) ∈ SP K$ and $( X , E )$ is free with respect to $K$.
Proof.
We take a set S of all pairwise non-asymorphic coarse spaces $( X ′ , E ′ ) ∈ K$, such that $| X ′ | ≤ | X |$, and enumerate all possible triplets ${ ( X α , E α , f α ) : α < λ }$, such that $( X α , E α ) ∈ S$ and $f α : X ⟶ X α$. Then we consider the product $P α < λ ( X α , E α )$ and define $f : X ⟶ P α < λ X α$ by $f ( x ) = ( f α ( x ) ) α < λ$. Since $K ≠ M s i n g l e$, f is injective and so we can identify X with $f ( X )$ and consider the subspace $( X , E )$ of $P α < λ ( X α , E α )$ . Clearly, $( X , E ) ∈ SP K$.
To see that $( X , E )$ is free with respect to $K$, it suffices to verify that, for each $( X ′ , E ′ ) ∈ S$, every mapping $h : ( X , E ) ⟶ ( X ′ , E ′ )$ is coarse. We take $β < λ$ such that $( X ′ , E ′ ) = ( X β , E β )$ and $h = f β$. Then $f β$ is the restriction to X of the projection $p r β : P α < λ ( X α , E α ) ⟶ ( X β , E β )$. Hence, $f β$ is coarse. ☐
Theorem 1.
For every class $K$ of coarse spaces, the smallest variety, Var $K$, containing $K$ is $QSP K$.
Proof.
The inclusion $QSP K ⊆ K$ is evident. To prove the inverse inclusion, we suppose that $K ≠ M s i n g l e$ (this case is evident) and take an arbitrary $( X ′ , E ′ ) ∈ V a r ( K )$. Then $( X ′ , E ′ )$ can be obtained from $K$ by means of some finite sequence of operations $S , P , Q$. We use Lemma 2 to choose a coarse space $( X , E ) ∈ SP K$, with $| X | = | X ′ |$, which is free with respect to $K$. By Lemma 1, any bijection $f : ( X , E ) ⟶ ( X ′ , E ′ )$ is coarse so $( X ′ , E ′ ) ∈ QSP K$. ☐
Theorem 2.
Let $M$ be a variety of coarse spaces such that $M ≠ M s i n g l e$, and $M ≠ M b o u n d$. Then there exists a cardinal κ such that $M = M κ$.
Proof.
Since $M ≠ M b o u n d$ and $M ≠ M s i n g l e$, there exists a minimal cardinal $κ$ such that $M$ contains an unbounded space of cardinality $κ$; so $M ⊆ M κ$.
To verify the inclusion $M κ ⊆ M$, we take a coarse space $( X , E ) ∈ M$, which is free with respect to $M$, and show that $( X , E )$ is free with respect to $M κ$. We prove that $( X , E ) = ( X , [ X ] < κ )$. If $| X | < κ$ then $( X , E )$ is bounded and the statement is evident. Assume that $| X | ≥ κ$ but $( X , E ) ≠ ( X , [ X ] < κ )$. Assume that, for every $ε$, $ε = ε − 1$, the set $S ε = { x ∈ X : | B ( x , ε ) | > 1 }$ is bounded in $( X , E )$. By the choice of $κ$, $| S ε | < κ$ and $| B ( x , ε ) | = 1$ for all $x ∈ X \ S ε$. It follows that $( X , E ) = ( X , [ X ] < κ )$. Then there exists an $ε ∈ E$ such that the set $S ε$ is unbounded in $( X , E )$.
We choose a maximal, by inclusion, subset $Y ⊂ X$ such that $B ( y , ε ) ∩ B ( y ′ , ε ) = ∅$ for all distinct $y , y ′ ∈ Y$. We observe that Y is unbounded so $| Y | ≥ κ$. We take an arbitrary $x 0 ∈ X$ and choose a mapping $f : X ⟶ X$ such that $f ( y ) = x 0$ for each $y ∈ Y$ and f is injective on $X \ Y$. Since $( X , E )$ is free with respect to $M$, the mapping $f : ( X , E ) ⟶ ( X , E )$ must be coarse. Hence, there exists an $ε ′ ∈ E$ such that $f ( B ( x , ε ) ) ⊆ B ( f ( x ) , ε ′ )$ for each $x ∈ X$. It follows that $f ( ∪ y ∈ Y B ( y , ε ) )$ is bounded in $( X , E )$. We note that $| f ( ∪ y ∈ Y B ( y , ε ) ) | ≥ κ$ so $( X , E )$ contains a bounded subset Z such that $| Z | = κ$. Since $( X , E )$ is free with respect to $M$, every $( X ′ , ε ′ ) ∈ M$ is $κ +$-bounded and we get a contradiction with the choice of $κ$. To conclude the proof, we take an arbitrary $( X , E ′ ) ∈ M κ$ and note that the identity mapping $i d : ( X , [ X ] < κ ) ⟶ ( X , E ′ )$ is coarse so $( X , E ′ ) ∈ M$. ☐
Remark 1.
We note that $M s i n g l e$ is not closed under coarse equivalence because each bounded coarse space is coarsely equivalent to a singleton. Clearly, $M b o u n d$ is closed under coarse equivalence. We show that the same is true for every variety $M κ$. Let $( X , E )$ be a coarse space, Y be a large subset of $( X , E )$. We assume that $( Y , E | Y ) ∈ M κ$ but $( X , E ) ∉ M κ$. Then X contains an unbounded subset Z such that $| Z | < κ$. We choose $ε ∈ E$ such that $ε = ε − 1$ and $X = B ( Y , E )$. For each $z ∈ Z$, we pick $y z ∈ Y$ such that $z ∈ B ( y z , E )$. We let $Y ′ = { y z ∈ Z }$. Since $| Y ′ | < κ$, $Y ′$ is bounded in $( Y , E | Y )$. It follows that Z is bounded in $( X , E )$, a contradiction with the choice of Z.
We note also that every variety of coarse spaces is closed under formations of companions. For $M s i n g l e$ and $M b o u n d$, this is evident. Let $( X , E ) ∈ M κ$ and $I$ be the ideal of all bounded subsets of $( X , E )$. Since $( X , [ X ] < κ )$ is free with respect to $M κ$, the identity mapping $i d : ( X , [ X ] < κ ) ⟶ ( X , E )$ is coarse. Hence, $[ X ] < κ ⊆ I$ and $( X , E ) ∈ M κ$.
Remark 2.
Every metric d on a set X defines a coarse structure $E d$ on X with the base ${ ( x , y ) : d ( x , y ) ≤ n }$, $n ∈ ω$. A coarse structure $E$ on X is called metrizable if there exists a metric d on X such that $E = E d$. By (, Theorem 2.1.1), $E$ is metrizable if and only if $E$ has a countable base. From the coarse point of view, metric spaces are important in Asymptotic Topology, see .
We assume that a variety $M$ of a coarse space contains an unbounded metric space $( X , d )$ and show that $M = M ω$. We choose a countable unbounded subset Y of X and note that $( Y , d ) ∉ M κ$ for $κ > ω$ so $( Y , d ) ∈ M ω \ M κ$, and the variety generated by $( X , d )$ is $M ω$.

1. Let G be a group with the identity e. An ideal $I$ in $P G$ is called a group ideal if $[ G ] < ω ⊆ I$ and $A B − 1 ∈ I$ for all $A , B ∈ I$.
Let X be a G-space with the action $G × X ⟶ X$, and $( g , x ) ⟼ g x$. We assume that G acts on X transitively, take a group ideal $I$ on G, and consider the coarse structure $E ( G , I , X )$ on X with the bases ${ ε A : A ∈ I , e ∈ A }$, where $ε A = { ( x , g x ) : x ∈ X , g ∈ A }$. Then $B ( x , ε A ) = A x$, where $A x = { g x : g ∈ A }$.
By (, Theorem 1), for every coarse structure $E$ on X, there exist a group G of permutations of X and a group ideal $I$ in $P G$ such that $E = E ( G , I , X )$. Now let $X = G$ such that G acts on X by left shifts, $x ⟼ g x$ for $g ∈ G$. We denote $( G , E ( G , I , G ) )$ by $( G , I )$ and say that $( G , I )$ is a right coarse group. If $I = [ G ] < ω$ then $( G , I )$ is called a finitary right coarse group. In the metric form, these structures on finitely generated groups play an important role in geometric group theory, see (, Chapter 4).
A group G endowed with a coarse structure $E$ is a right coarse group if and only if, for every $ε ∈ E$, there exists $ε ′ ∈ E$ such that $( B ( x , ε ) ) g ⊆ B ( x g , ε ′ )$ for all $x , g ∈ G$. For group ideals and coarse structures on groups see (, Chapter 6) and .
2. A class $M$ of right coarse groups is called a variety if $M$ is closed under formation of subgroups, coarse homomorphic images, and products.
Let $K$ be a class of right coarse groups, and G be a group generated by a subset $X ⊂ G$. We say that a right coarse group $( G , I )$ is free with respect to $K$ if, for every $( G ′ , I ′ ) ∈ K$, any mapping $X ⟶ G ′$ extends to the coarse homomorphism $( G , I ) ⟶ ( G ′ , I ′ )$. Then Lemmas 1 and 2 and Theorem 1 hold for the right coarse groups in place of coarse spaces.
Let $M$ be a variety of right coarse groups. We take an arbitrary $( G , I ) ∈ M$, delete the coarse structure on G and the class $M ♭$ of the groups. If $( G , I ) ∈ M$ then $( G , P G ) ∈ M$. It follows that $M ♭$ is a variety of groups.
Now let $G$ be a variety of groups different from the variety of singletons. We denote by $G b o u n d$ the variety of right coarse groups $( G , P G ) , f o r G ∈ G$. For an infinite cardinal $κ$, we denote by $G κ$, the variety of all $κ$-bounded right coarse groups $( G , I )$, for $G ∈ G$.
Let $M$ be a variety of right coarse groups such that $M ♭ = G$. In contrast to Theorem 2, we do not know if $M$ lies in the chain:
$G b o u n d ⊂ … ⊂ G κ ⊂ … ⊂ G ω .$
If G is a group of cardinality $κ$ and $G ∈ G$ then $( G , [ G ] < κ ) ∈ G κ \ G κ ′$ for each $κ ′ > κ$. Hence, all inclusions in the above chain are strict.
3. Let $Ω$ be a signature, A be an $Ω$-algebra, and $E$ be a coarse structure on A. We say that A is a coarse Ω-algebra if every n-ary operation from $Ω$ is coarse, for example the mapping $( A , E ) n ⟶ ( A , E )$. We note that each coarse group is a right coarse group but the converse statement need not be true, see (, Section 6.1).
A class $M$ of a coarse $Ω$-algebra is called a variety if $M$ is closed under formation of subalgebras, coarse homomorphic images, and products. Given a variety $M$ of coarse algebras, the class $M ♭$ of all $Ω$-algebras A, such that $( A , E ) ∈ M$, is a variety of $Ω$-algebras. Let A be a variety of $Ω$-algebras different from the variety of singletons. We let $A b o u n d$ be the variety of coarse algebras $( A , P A )$, for $A ∈ A$. For an infinite cardinal $κ$, we denote $A κ$ as the variety of all $κ$-bounded $Ω$-algebras $( A , E )$ such that $A ∈ A$, and get the chain:
$A b o u n d ⊆ … ⊆ A κ ⊆ … ⊆ A ω ,$
however, we can not state that all inclusions are strict. In the case of course groups, this is because each non-trivial variety of groups contains some Abelian group A, of cardinality $κ$, and the coarse group $( A , [ A ] < κ )$ is $κ$-bounded but not $κ +$-bounded.
4. A class $M$ of topological $Ω$-algebras (with regular topologies) is called a variety (a wide variety) if $M$ is closed under formation of closed subalgebras (arbitrary subalgebras), continuous homomorphic images, and products. Wide varieties and varieties are characterized syntactically by the limit laws  and filters . In our coarse case, the part of filters is played by the ideals $[ X ] < κ$.
There are only two wide varieties of topological spaces, the variety of singletons and the variety of all topological spaces, but there are plenty of varieties of topological spaces. The variety of coarse spaces $M κ$ is a twin of the varieties of topological spaces in which every subset of cardinality $< κ$ is compact. We note also that $G κ$ might be considered as a counterpart to the variety $T ( κ )$ of topological groups from , where $G ∈ T ( κ )$ if and only if each neighborhood of e contains a normal subgroup of index strictly less then $κ$.
5. A class $M$ of uniform spaces is called a variety if $M$ is closed under formation of subspaces, products and uniformly continuous images. For an infinite cardinal $κ$, a uniform space X is called $κ$-bounded if X can be covered by $< κ$ balls of arbitrary small radius. Every variety of uniform spaces different from varieties of singletons and all spaces coincides with the variety of $κ$-bounded spaces for some $κ$, see . I thank Miroslav Hu$s ˇ$ek for this reference.

## Conflicts of Interest

The authors declare no conflict of interest.

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