Example

— Let E be a metric space; the set of all bounded subsets of E (GT, IX, § 2, No. 2) is a covering bornology on E. Let G be the group of isometries of E; the set of all subsets M of G such that for every $x \in E$, the set $M.x$ is a bounded subset of E, is a covering bornology on G.

If $\mathscr{B}$ is a bornology on a set E, a subset $\mathscr{B}_1$ of $\mathscr{B}$ is said to be a base of $\mathscr{B}$ if every set of $\mathscr{B}$ is contained in a set of $\mathscr{B}_1$.

The intersection of a family of bornologies on E is a bornology; consequently for every subset $\mathfrak{S}$ of $\mathfrak{P}(E)$, there exists a smallest bornology containing $\mathfrak{S}$; this bornology is said to be generated by $\mathfrak{S}$ and admits as a base the set of finite unions of sets of $\mathfrak{S}$. If E and E' are two sets, and $\mathscr{B}$ (resp. $\mathscr{B}'$) a bornology on E (resp. E'), the product bornology is the bornology on $E \times E'$ which admits the sets $M \times M'$ as a base, where $M \in \mathscr{B}$ and $M' \in \mathscr{B}'$.