Sparse and thin metric spaces

I. V. Protasov

We define thin and sparse metric spaces as asymptotic counterparts of discrete and very close to discrete metric spaces respectively. We characterize sparse spaces in terms of prohibited subspaces, and classify thin metric spaces up to coarse equivalence. We introduce the types of sparse spaces and construct the spaces of distinct types. We unify the notions of sparse spaces and sparse subsets of a group in context of balleans.

1. T. Banakh, I. Protasov, S. Slobodianiuk, Scattered subsets of groups, preprint available at arxiv: submit/0877498.

2. T. Banakh, I. Zarichnyi, Characterizing the Cantor bi-cube in asymptotic categories, Groups, Geometry and Dynamics, 5 (2011), 691–728.

3. A. Dranishnikov, M. Zarichnyi, Universal spaces for asymptotic dimention, Topology Appl., 140 (2004), 203–225.

4. Ie. Lutsenko, I.V. Protasov, Sparse, thin and other subsets of groups, Intern. J. Algebra Computation, 19 (2009), 491–510.

5. Ie. Lutsenko, I.V. Protasov, Thin subsets of balleans, Appl. Gen. Topology, 1 (2010), 89–93.

6. Ie. Lutsenko, I.V. Protasov, Relatively thin and sparse subsets of groups, Ukr. Math. J., 63 (2011), 216–225.

7. O.V. Petrenko, I.V. Protasov, Balleans and filters, Mat. Stud., 38 (2012), 3–11.

8. O.V. Petrenko, I.V. Protasov, Balleans and G-spaces, Ukr. Math. J., 64 (2012), 344–350.

9. I.V. Protasov, Morphisms of ball structures of groups and graphs, Ukr. Math. J., 54 (2002), 847–855.

10. I.V. Protasov, Balleans of bounded geometry and G-spaces, Mat. Stud., 30 (2008), 61–66.

11. I.V. Protasov, S. Slobodianiuk Ultracompanions of subsets of groups, preprint available at arxiv: 1308.1497.

12. I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., V.12, VNTL Publishers, Lviv, 2007.

13. J. Roe, Lectures on Coarse Geometry, Amer. Math. Soc., Providence, R.I., 2003.

	Sparse and thin metric spaces
Author	I. V. Protasov i.v.protasov@gmail.com Department of Cybernetics, Kyiv University
Abstract	We define thin and sparse metric spaces as asymptotic counterparts of discrete and very close to discrete metric spaces respectively. We characterize sparse spaces in terms of prohibited subspaces, and classify thin metric spaces up to coarse equivalence. We introduce the types of sparse spaces and construct the spaces of distinct types. We unify the notions of sparse spaces and sparse subsets of a group in context of balleans.
Keywords	asymptotically isolated balls; sparse and thin spaces; balleans
Reference	1. T. Banakh, I. Protasov, S. Slobodianiuk, Scattered subsets of groups, preprint available at arxiv: submit/0877498. 2. T. Banakh, I. Zarichnyi, Characterizing the Cantor bi-cube in asymptotic categories, Groups, Geometry and Dynamics, 5 (2011), 691–728. 3. A. Dranishnikov, M. Zarichnyi, Universal spaces for asymptotic dimention, Topology Appl., 140 (2004), 203–225. 4. Ie. Lutsenko, I.V. Protasov, Sparse, thin and other subsets of groups, Intern. J. Algebra Computation, 19 (2009), 491–510. 5. Ie. Lutsenko, I.V. Protasov, Thin subsets of balleans, Appl. Gen. Topology, 1 (2010), 89–93. 6. Ie. Lutsenko, I.V. Protasov, Relatively thin and sparse subsets of groups, Ukr. Math. J., 63 (2011), 216–225. 7. O.V. Petrenko, I.V. Protasov, Balleans and filters, Mat. Stud., 38 (2012), 3–11. 8. O.V. Petrenko, I.V. Protasov, Balleans and G-spaces, Ukr. Math. J., 64 (2012), 344–350. 9. I.V. Protasov, Morphisms of ball structures of groups and graphs, Ukr. Math. J., 54 (2002), 847–855. 10. I.V. Protasov, Balleans of bounded geometry and G-spaces, Mat. Stud., 30 (2008), 61–66. 11. I.V. Protasov, S. Slobodianiuk Ultracompanions of subsets of groups, preprint available at arxiv: 1308.1497. 12. I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., V.12, VNTL Publishers, Lviv, 2007. 13. J. Roe, Lectures on Coarse Geometry, Amer. Math. Soc., Providence, R.I., 2003.
Pages	92-100
Volume	41
Issue	1
Year	2014
Journal	Matematychni Studii
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