A concept for resemblance in large scale geometry

TL;DR

This paper introduces large scale resemblance structures as a new way to formalize the idea of 'being alike' at large scales, generalizing concepts like asymptotic dimension to these spaces.

Contribution

It proposes a new large scale structure called large scale resemblance, axiomatizes it, and explores its implications, including inducing nearness and generalizing large scale properties.

Findings

Large scale resemblances can induce nearness on sets.

Not every near family is contained in a bunch.

Asymptotic dimension can be generalized to large scale resemblance spaces.

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.