Description:
<jats:p>Bornologies abstract the properties of bounded sets of a metric space. But there are unbounded bornologies on a metric space like $\mathcal{P}(\RR)$ with the Euclidean metric. We show that by replacing $[0,\infty)$ with a partially ordered monoid every bornology is the set of bounded subsets of a generalized metric mapped into a partially ordered monoid. We also prove that the set of bornologies on a set is the join completion of the equivalence classes of a relation on the power set of the set.</jats:p>