# Some counterexamples in abstract coarse geometry

## TL;DR

This paper presents novel counterexamples in abstract coarse geometry, demonstrating limitations of coarse structures and asymptotic resemblance structures in inducing certain coarse proximity spaces.

## Contribution

It introduces counterexamples showing that some coarse proximity spaces cannot be derived from coarse structures or asymptotic resemblance structures, and provides a method to combine such spaces.

## Key findings

- Counterexample of a coarse proximity space not induced by any coarse structure
- Method to 'stitch' two coarse proximity spaces with homeomorphic boundaries
- Counterexample of a coarse proximity space not induced by any asymptotic resemblance structure

## Abstract

We construct an example of a coarse proximity space that is not induced by any coarse structure. We then show how to "stitch" two coarse proximity spaces with homeomorphic boundaries into one coarse proximity space. Finally, we construct a coarse proximity space that is not induced by any asymptotic resemblance structure.