The normality of macrocubes and hyperballeans

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}.

Keywords: coarse structure, ballean, macrocube, hyperballeans, Ramsey ultrafilter. Keywords: coarse structure, ballean, macrocube, hyperballeans, Ramsey ultrafilter