Varieties of coarse spaces

A class $\mathfrak{M}$ of coarse spaces is called a variety if $\mathfrak{M}$ is closed under formation of subspaces, coarse images and products. We classify the varieties of coarse spaces and, in particular, show that if a variety $\mathfrak{M}$ contains an unbounded metric space then $\mathfrak{M}$ is the variety of all coarse spaces.

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 could be defined in terms of balls, see [7], [9]. 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 supposed to be connected. Given a coarse space (X, E), each subset Y ⊆ X has the natural coarse structure Let (X, E), (X ′ , E ′ ) be coarse spaces. A mapping f : X −→ X ′ is called coarse (or bornologous in terminology of [10]) if, for every ε ∈ E there exists 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), 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 P α<κ E α . If ε α ∈ E α , α < κ and x, y ∈ P α<κ X α , x = (x α ) α<κ , 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 formation of subspaces (SM ⊆ M), coarse images (QM ⊆ M) and products (PM ⊆ 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, and denote by M κ the variety of all κ-bounded coarse spaces.
We denote by M single and M bound the variety of singletons and the variety of all bounded coarse spaces.
Then we get the chain of varieties 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.

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 the coarse structure with the base 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 ( Lemma 1. If a coarse space (X, E) is free with respect to a class K then (X, E) is free with respect to SK, QK, PK.
Proof. We verify only the second statement. Let (X ′ , E ′ ) ∈ K, (X ′′ , E ′′ ) ∈ QK, and h : (X ′ , E ′ ) −→ (X ′′ , E ′′ ) be a coarse surjective mapping. We take an arbitrary f : is coarse so f is coarse as the composition of the coarse mappings h, h ′ . ✷ Lemma 2. Let X be a set and let K be a class of coarse spaces, K = M single . Then there exists a coarse structure E on X such that (X, E) ∈ SPK and (X, E) is free with respect to K.
To see that (X, E) is free with respect to K, it suffices to verify that, for each For every class K of coarse spaces, the smallest variety Var K containing K is QSPK.
Proof.The inclusion QSPK ⊆ K is evident. To prove the inverse inclusion, we suppose that K = M single (this case is evident) and take an arbitrary (X ′ , E ′ ) ∈ V ar (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) ∈ SPK, |X| = |X ′ | free with respect to K. By Lemma 1, any bijection f : To verify the inclusion M κ ⊆ M, we take a coarse space (X, E) ∈ M 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 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 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 a 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 id : (X, [X] <κ ) −→ (X, E ′ ) is coarse so (X, E ′ ) ∈ M. ✷ Remark 1. We note that M single is not closed under coarse equivalence because each bounded coarse space is coarsely equivalent to a singleton. Clearly, M bound is closed under coarsely 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 put 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 single and M bound , this is evident. Let (X, E) ∈ M κ and I is the ideal of all bounded subsets of (X, E). Since (X, [X] <κ ) is free with respect to M κ , the identity mapping id : (X, [X] <κ ) −→ (X, E) is coarse so [X] <κ ⊆ I and (X, E) ∈ M κ . Remark 2. Every metric d on a set X defines the 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 [9, 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 studding in Asymptotic Topology, see [1].
We assume that a variety M of 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 .

Comments
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 AB −1 ∈ I for all A, B ∈ I.
Let X be a G-space with the action G × X −→ X, (g, x) −→ gx. 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 base {ε A : A ∈ I, e ∈ A}, ε A = {(x, gx) : x ∈ X, g ∈ A}. Then B(x, ε A ) = Ax, Ax = {gx : g ∈ A}.
the chain.
but, we can not state that all inclusions are strict. In the case of course groups, this is so 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. The wide varieties and varieties are characterized syntactically by the limit laws [11] and filters [6]. In our coarse case, the part of filters play 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 is a 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 of the variety T (κ) of topological groups from [3], 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 [4]. I thank Miroslav Hušek for this reference.
6. On varieties of bornological spaces. An ideal I in P X is called a bornology if I = X. A set X, endowed with a bornology I is called a bornological space. Each A ∈ I is called bounded, (X, I) is bounded if X ∈ I. For an infinite cardinal κ, (X, I) is called κ-bounded if [X] <κ ⊆ I.
A mapping f : (X, I) −→ (X ′ , I ′ ) is called bornologous if f (A) ∈ I ′ for each A ∈ I. A class of bornological space closed under subspaces, products and bornologous images is called a variety. Repeating the first part of the proof of Theorem 2, we conclude that each variety of bornological spaces is either variety of singletons, or variety of all bounded spaces or variety of all κ-bounded spaces for some infinite cardinal κ.