# Extremal balleans

## TL;DR

This paper explores extremal properties of balleans, introducing concepts like ultranormal and extremely normal, and constructs examples illustrating these extremal conditions within coarse geometry.

## Contribution

It defines and investigates the properties of ultranormal and extremely normal balleans, establishing their relationships and providing concrete examples with extremal characteristics.

## Key findings

- Every maximal ballean is extremely normal.
- Every extremely normal ballean is ultranormal.
- A discrete ballean is ultranormal iff it is maximal.

## Abstract

A ballean (or coarse space) is a set endowed with a coarse structure. A ballean $X$ is called normal if any two asymptotically disjoint subsets of $X$ are asymptotically separated. We say that a ballean $X$ is ultranormal (extremely normal) if any two unbounded subsets of $X$ are not asymptotically disjoint (every unbounded subset of $X$ is large). Every maximal ballean is extremely normal and every extremely normal ballean is ultranormal, but the converse statements do not hold. A normal ballean is ultranormal if and only if the Higson$^{\prime}$s corona of $X$ is a singleton. A discrete ballean $X$ is ultranormal if and only if $X$ is maximal. We construct a series of concrete balleans with extremal properties.