Coarse structures and fuzzy metrics

M.Zarichnyi

It is known that every fuzzy metric on a set generates a uniform structure on this set. The aim of this note is to show that every fuzzy metric on a set generates a coarse structure on this set. In the case of fuzzy non-Archimedean metric space, the obtained coarse space turns out to be asymptotically zero-dimensional in the sense of Gromov.

1. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. 28 (1942), 535-537.

2. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983.

3. V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy sets and systems, 115 (2000), 485-489.

4. J. Roe, Index theory, coarse geometry, and the topology of manifolds, Regional Conference Series on Mathematics, vol. 90, CBMS Conference Proceedings, American Mathematical Society, 1996.

5. M. Gromov, Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993.

6. A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), 395-399.

7. D. Mihe t, Fuzzy $\psi$-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets and Systems archive Volume 159 , Issue 6 (March 2008) , 739-744.

8. P. D. Mitchener, Coarse homology theories, Algebr. Geom. Topol. 1 (2001), 271-297 (electronic). Addendum. Algebr. Geom. Topol. 3 (2003), 1089–1101.

9. B. Grave, Asymptotic dimension of coarse spaces, New York J. Math. 12 (2006), 249–256.

10. G. Skandalis, J.L. Tu, G. Yu, The coarse Baum.Connes conjecture and groupoids, Topology 41 (2002), 807-834.

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

12. O. Kramosil, J. Mich\'alek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), 326-334.

	Coarse structures and fuzzy metrics
Author	M.Zarichnyi mzar@litech.lviv.ua Department of Mechanics and Mathematics, Lviv National University,
Abstract	It is known that every fuzzy metric on a set generates a uniform structure on this set. The aim of this note is to show that every fuzzy metric on a set generates a coarse structure on this set. In the case of fuzzy non-Archimedean metric space, the obtained coarse space turns out to be asymptotically zero-dimensional in the sense of Gromov.
Keywords	coarse structure, fuzzy metric, non-Archimedean metric space
DOI	doi:10.30970/ms.32.2.180-184
Reference	1. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. 28 (1942), 535-537. 2. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983. 3. V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy sets and systems, 115 (2000), 485-489. 4. J. Roe, Index theory, coarse geometry, and the topology of manifolds, Regional Conference Series on Mathematics, vol. 90, CBMS Conference Proceedings, American Mathematical Society, 1996. 5. M. Gromov, Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993. 6. A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), 395-399. 7. D. Mihe t, Fuzzy $\psi$-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets and Systems archive Volume 159 , Issue 6 (March 2008) , 739-744. 8. P. D. Mitchener, Coarse homology theories, Algebr. Geom. Topol. 1 (2001), 271-297 (electronic). Addendum. Algebr. Geom. Topol. 3 (2003), 1089–1101. 9. B. Grave, Asymptotic dimension of coarse spaces, New York J. Math. 12 (2006), 249–256. 10. G. Skandalis, J.L. Tu, G. Yu, The coarse Baum.Connes conjecture and groupoids, Topology 41 (2002), 807-834. 11. I.Protasov, M.Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., Vol. 12, VNTL, Lviv, 2007. 12. O. Kramosil, J. Mich\'alek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), 326-334.
Pages	180-184
Volume	32
Issue	2
Year	2009
Journal	Matematychni Studii
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