Asymptotic Resemblance
Geometric Topology
2021-11-12 v3 General Topology
Abstract
Uniformity and proximity are two different ways for defining small scale structures on a set. 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 Higson compactification. At the end we show that how the asymptotic dimension of a metric space can be generalized to a set equipped with an asymptotic resemblance relation.
Cite
@article{arxiv.1310.6313,
title = {Asymptotic Resemblance},
author = {Sh. Kalantari and B. Honari},
journal= {arXiv preprint arXiv:1310.6313},
year = {2021}
}