Open Access
2016 Asymptotic resemblance
Sh. Kalantari, B. Honari
Rocky Mountain J. Math. 46(4): 1231-1262 (2016). DOI: 10.1216/RMJ-2016-46-4-1231


\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.


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Sh. Kalantari. B. Honari. "Asymptotic resemblance." Rocky Mountain J. Math. 46 (4) 1231 - 1262, 2016.


Published: 2016
First available in Project Euclid: 19 October 2016

zbMATH: 1355.53019
MathSciNet: MR3563180
Digital Object Identifier: 10.1216/RMJ-2016-46-4-1231

Primary: 18B30 , 51F99 , 53C23 , 54C20

Keywords: Asymptotic dimension , asymptotic resemblance , coarse structure , Higson compactification , proximity

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 4 • 2016
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