Minmax bornologies

TL;DR

This paper characterizes minmax bornologies on sets through ultrafilter properties and constructs examples demonstrating their diversity, linking bornology structures with ultrafilter isomorphism classes.

Contribution

It provides a characterization of minmax bornologies via ultrafilter structures and constructs explicit examples with infinite ultrafilter sets.

Findings

Minmax bornologies correspond to ultrafilters that are pairwise non-isomorphic.

Existence of a minmax bornology on ω with an infinite set of ultrafilters.

Connection established between bornology structures and ultrafilter topology in βω.

Abstract

A bornology $\mathcal{B}$ on a set $X$ is called minmax if the smallest and the largest coarse structures on $X$ compatible with $\mathcal{B}$ coincide. We prove that $\mathcal{B}$ is minmax if and only if the family $\mathcal B^\sharp=\{p\in\beta X:\{X\setminus B:B\in\mathcal B\}\subset p\}$ consists of ultrafilters which are pairwise non-isomorphic via $\mathcal B$-preserving bijections of $X$. Also we construct a minmax bornology $\mathcal B$ on $\omega$ such that the set $\mathcal B^\sharp$ is infinite. We deduce this result from the existence of a closed infinite subset in $\beta\omega$ that consists of pairwise non-isomorphic ultrafilters.