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.
Citation
Sh. Kalantari. B. Honari. "Asymptotic resemblance." Rocky Mountain J. Math. 46 (4) 1231 - 1262, 2016. https://doi.org/10.1216/RMJ-2016-46-4-1231
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