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Axioms 2018, 7(2), 32; doi:10.3390/axioms7020032
Varieties of Coarse Spaces
Faculty of Computer Science and Cybernetics, Kyiv University, Academic Glushkov pr. 4d, 03680 Kyiv, Ukraine
Received: 27 March 2018 / Accepted: 10 May 2018 / Published: 14 May 2018
A class of coarse spaces is called a variety if 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 contains an unbounded metric space then is the variety of all coarse spaces.
Keywords:coarse structure; coarse space; ballean; varieties of coarse spaces
Following , we say that a family of subsets of is a coarse structure on a set X if:
- Each contains the diagonal , ;
- If then and , where and ;
- And if and then .
Each is called an entourage of the diagonal. A subset is called a base for if, for every there exists such that .
The pair is called a coarse space. For and , we denote and say that 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 is called connected if, for any , there exists such that . A subset Y of X is called bounded if there exist and such that . The coarse structure is the unique coarse structure such that is connected and bounded. In what follows, all coarse spaces under consideration are assumed to be connected.
Given a coarse space , each subset has the natural coarse structure , where is called a subspace of . A subset Y of X is called large (or coarsely dense) if there exists an such that where .
Let , and be coarse spaces. A mapping is called coarse (or bornologous in the terminology of ) if, for every there exists an such that, for every , we have . If f is surjective and coarse then is called a coarse image of . If f is a bijection, such that f and are coarse mappings, then f is called an asymorphism. The coarse spaces , are called coarsely equivalent if there exist large subsets , and such that and are asymorphic.
To conclude the coarse vocabulary, we take a family of coarse spaces and define the product as the set endowed with the coarse structure with the base set . If , for and , where , and then if and only if for every .
Let be a class of coarse spaces closed under asymorphisms. We say that is a variety if is closed under the formation of subspaces , coarse images , and products .
For an infinite cardinal , we say that a coarse space is κ-bounded if every subset , such that , is bounded. Additionally, we denote as the variety of all -bounded coarse spaces. We denote by and the variety of singletons and the variety of all bounded coarse spaces, respectively. Thus we have the chain of varieties:
We recall that a family of subsets of a set X is an ideal in the Boolean algebra of all subsets of X, if is closed under finite unions and subsets. Every ideal defines a coarse structure with the base where . Therefore, if and if . We denote the obtained coarse space by . For a cardinal , denotes the ideal . If is a coarse space, the family of all bounded subsets of X is an ideal. The coarse space is called the companion of .
Let be a class of coarse spaces. We say that a coarse space is free with respect to if, for every every mapping is coarse. For example, is free with respect to the variety . Since but for each ; the inclusion is strict.
If a coarse space is free with respect to a class then is free with respect to , , .
We verify only the second statement. Let , , and be a coarse surjective mapping. We take an arbitrary and choose such that . Since is free with respect to , is coarse so f is coarse as the composition of the coarse mappings h, and . ☐
Let X be a set and let be a class of coarse spaces, . Then there exists a coarse structure on X such that and is free with respect to .
We take a set S of all pairwise non-asymorphic coarse spaces , such that , and enumerate all possible triplets , such that and . Then we consider the product and define by . Since , f is injective and so we can identify X with and consider the subspace of . Clearly, .
To see that is free with respect to , it suffices to verify that, for each , every mapping is coarse. We take such that and . Then is the restriction to X of the projection . Hence, is coarse. ☐
For every class of coarse spaces, the smallest variety, Var , containing is .
The inclusion is evident. To prove the inverse inclusion, we suppose that (this case is evident) and take an arbitrary . Then can be obtained from by means of some finite sequence of operations . We use Lemma 2 to choose a coarse space , with , which is free with respect to . By Lemma 1, any bijection is coarse so . ☐
Let be a variety of coarse spaces such that , and . Then there exists a cardinal κ such that .
Since and , there exists a minimal cardinal such that contains an unbounded space of cardinality ; so .
To verify the inclusion , we take a coarse space , which is free with respect to , and show that is free with respect to . We prove that . If then is bounded and the statement is evident. Assume that but . Assume that, for every , , the set is bounded in . By the choice of , and for all . It follows that . Then there exists an such that the set is unbounded in .
We choose a maximal, by inclusion, subset such that for all distinct . We observe that Y is unbounded so . We take an arbitrary and choose a mapping such that for each and f is injective on . Since is free with respect to , the mapping must be coarse. Hence, there exists an such that for each . It follows that is bounded in . We note that so contains a bounded subset Z such that . Since is free with respect to , every is -bounded and we get a contradiction with the choice of . To conclude the proof, we take an arbitrary and note that the identity mapping is coarse so . ☐
We note that is not closed under coarse equivalence because each bounded coarse space is coarsely equivalent to a singleton. Clearly, is closed under coarse equivalence. We show that the same is true for every variety . Let be a coarse space, Y be a large subset of . We assume that but . Then X contains an unbounded subset Z such that . We choose such that and . For each , we pick such that . We let . Since , is bounded in . It follows that Z is bounded in , a contradiction with the choice of Z.
We note also that every variety of coarse spaces is closed under formations of companions. For and , this is evident. Let and be the ideal of all bounded subsets of . Since is free with respect to , the identity mapping is coarse. Hence, and .
Every metric d on a set X defines a coarse structure on X with the base , . A coarse structure on X is called metrizable if there exists a metric d on X such that . By (, Theorem 2.1.1), is metrizable if and only if has a countable base. From the coarse point of view, metric spaces are important in Asymptotic Topology, see .
We assume that a variety of a coarse space contains an unbounded metric space and show that . We choose a countable unbounded subset Y of X and note that for so , and the variety generated by is .
1. Let G be a group with the identity e. An ideal in is called a group ideal if and for all .
Let X be a G-space with the action , and . We assume that G acts on X transitively, take a group ideal on G, and consider the coarse structure on X with the bases , where . Then , where .
By (, Theorem 1), for every coarse structure on X, there exist a group G of permutations of X and a group ideal in such that . Now let such that G acts on X by left shifts, for . We denote by and say that is a right coarse group. If then 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 is a right coarse group if and only if, for every , there exists such that for all . For group ideals and coarse structures on groups see (, Chapter 6) and .
2. A class of right coarse groups is called a variety if is closed under formation of subgroups, coarse homomorphic images, and products.
Let be a class of right coarse groups, and G be a group generated by a subset . We say that a right coarse group is free with respect to if, for every , any mapping extends to the coarse homomorphism . Then Lemmas 1 and 2 and Theorem 1 hold for the right coarse groups in place of coarse spaces.
Let be a variety of right coarse groups. We take an arbitrary , delete the coarse structure on G and the class of the groups. If then . It follows that is a variety of groups.
Now let be a variety of groups different from the variety of singletons. We denote by the variety of right coarse groups . For an infinite cardinal , we denote by , the variety of all -bounded right coarse groups , for .
Let be a variety of right coarse groups such that . In contrast to Theorem 2, we do not know if lies in the chain:
If G is a group of cardinality and then for each . Hence, all inclusions in the above chain are strict.
3. Let be a signature, A be an -algebra, and be a coarse structure on A. We say that A is a coarse Ω-algebra if every n-ary operation from is coarse, for example the mapping . 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 of a coarse -algebra is called a variety if is closed under formation of subalgebras, coarse homomorphic images, and products. Given a variety of coarse algebras, the class of all -algebras A, such that , is a variety of -algebras. Let A be a variety of -algebras different from the variety of singletons. We let be the variety of coarse algebras , for . For an infinite cardinal , we denote as the variety of all -bounded -algebras such that , and get the chain: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 , and the coarse group is -bounded but not -bounded.
4. A class of topological -algebras (with regular topologies) is called a variety (a wide variety) if 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 .
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 is a twin of the varieties of topological spaces in which every subset of cardinality is compact. We note also that might be considered as a counterpart to the variety of topological groups from , where if and only if each neighborhood of e contains a normal subgroup of index strictly less then .
5. A class of uniform spaces is called a variety if is closed under formation of subspaces, products and uniformly continuous images. For an infinite cardinal , a uniform space X is called -bounded if X can be covered by balls of arbitrary small radius. Every variety of uniform spaces different from varieties of singletons and all spaces coincides with the variety of -bounded spaces for some , see . I thank Miroslav Huek for this reference.
Conflicts of Interest
The authors declare no conflict of interest.
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