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 the 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.
Abstract. We introduce the notion of large scale inductive dimension for asymptotic resemblance spaces. We prove that the large scale inductive dimension and the asymptotic dimensiongrad are equal in the class of r-convex metric spaces. This class contains the class of all geodesic metric spaces and all finitely generated groups. This leads to an answer to a question asked by E. Shchepin, concerning the relation between the asymptotic inductive dimension and the asymptotic dimensiongrad, for r-convex metric spaces.
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