Representations of bornologies

[EN] Bornologies abstract the properties of bounded sets of a metric space. But there are unbounded bornologies on a metric space like $P(\backslash RR)$ 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.