Rocky Mountain Journal of Mathematics

Asymptotic resemblance

Sh. Kalantari and B. Honari

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

\textit {Uniformity} and \textit {proximity} are two different ways of defining small scale structures on a set. \textit {Coarse structures} are large scale counterparts of uniform structures. In this paper, motivated by the definition of proximity, we develop the concept of asymptotic resemblance as a relation between subsets of a set to define a large scale structure on it. We use our notion of asymptotic resemblance to generalize some basic concepts of coarse geometry. We introduce a large scale compactification which, in special cases, agrees with the \textit {Higson compactification}. At the end of the paper we show how the \textit {asymptotic dimension} of a metric space can be generalized to a set equipped with an asymptotic resemblance relation.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 4 (2016), 1231-1262.

Dates
First available in Project Euclid: 19 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1476884581

Digital Object Identifier
doi:10.1216/RMJ-2016-46-4-1231

Mathematical Reviews number (MathSciNet)
MR3563180

Zentralblatt MATH identifier
1355.53019

Subjects
Primary: 18B30: Categories of topological spaces and continuous mappings [See also 54-XX] 51F99: None of the above, but in this section 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 54C20: Extension of maps

Keywords
Asymptotic dimension asymptotic resemblance coarse structure Higson compactification proximity

Citation

Kalantari, Sh.; Honari, B. Asymptotic resemblance. Rocky Mountain J. Math. 46 (2016), no. 4, 1231--1262. doi:10.1216/RMJ-2016-46-4-1231. https://projecteuclid.org/euclid.rmjm/1476884581


Export citation

References

  • G. Bell and A. Dranishnikov, Asymptotic dimension, Topol. Appl. 155 (2008), 1265–1296.
  • A. Dranishnikov, On asymptotic inductive dimension, J. Geom. Topol. 1 (2001), 239–247.
  • J. Dydak and C.S. Hoffland, An alternative definition of coarse structures, Topol. Appl. 155 (2008), 1013–1021.
  • V.A. Efremovic, Infinitesimal space, Dokl. Akad. Nauk. 76 (1951), 341–343.
  • V.A. Efremovic, The geometry of proximity, I, Mat. Sbor. 31 (1951), 189–200.
  • S.A. Naimpally and B.D. Warrack, Proximity spaces, Cambr. Tract Math. 59, Cambridge University Press, Cambridge, UK, 1970.
  • I. Protasov, Normal ball structures, Mat. Stud. 20 (2003), 3–16.
  • J. Roe, Lectures on coarse geometry, Univ. Lect. Series 31, American Mathematical Society, Providence, RI, 2003.
  • J.W. Tukey, Convergence and uniformity in topology, Ann. Math. Stud. 2, Princeton University Press, Princeton, 1940.
  • A. Weil, Sur les espaces a structure uniforme et sur la topologie generale, Herman, Paris, 1937.
  • S. Willard, General topology, Addison-Wesley, Reading, MA, 1970.
  • N. Wright, Simultaneous metrizability of coarse spaces, Proc. Amer. Math. Soc. 139 (2011), 3271–3278.