A concept for resemblance in large scale geometry

Abstract

In this paper, we introduce the notion of large scale resemblance structure as a new large scale structure by axiomatizing the concept of `being alike in large scale' for a family of subsets of a set. We see that in a particular case, large scale resemblances on a set can induce a nearness on it, and as a consequence, we offer a relatively big class of examples to show that `not every near family is contained in a bunch'. Besides, We show how some large scale properties like asymptotic dimension can be generalized to large scale resemblance spaces.