# Constructing balleans

## TL;DR

This paper introduces new constructions of balleans, a type of coarse space, from families of balleans, and analyzes the compatible coarse structures on a set given a bornology.

## Contribution

It presents three novel constructions of balleans—bornological products, bouquets, and combs—and studies the extremal coarse structures compatible with a given bornology.

## Key findings

- Defined and analyzed bornological products, bouquets, and combs of balleans.
- Characterized smallest and largest compatible coarse structures on a set.
- Provided new methods for constructing and understanding balleans.

## Abstract

A ballean is a set endowed with a coarse structure. We introduce and explore three constructions of balleans from a pregiven family of balleans: bornological products, bouquets and combs. We analyze the smallest and the largest coarse structures on a set $X$ compatible with a given bornology on $X$.