The normality of macrocubes and hyperballeans

TL;DR

This paper characterizes the normality of macrocubes and hyperballeans in terms of bornology properties, providing answers to open questions in the field of coarse geometry and bornology.

Contribution

It establishes a precise criterion for the normality of $ ext{macrocubes}$ based on the structure of the bornology, and shows that certain hyperballeans are not normal.

Findings

Macrocube normality iff bornology has a linearly ordered base

Hyperballean of bounded subsets of an ultradiscrete ballean is not normal

Answers to open questions in coarse geometry and bornology

Abstract

For a bornology $\mathcal B$ on a cardinal $\kappa$, we prove that the $\mathcal B$-macrocube is normal if and only if $\mathcal B$ has a linearly ordered base. As a corollary, we get that the hyperballean of bounded subsets of an ultradiscrete ballean is not normal. These answer Question 1 from \cite{b2} and Question 14.4 from \cite{b1}.