Definition 1

— A bornology on a set $E$ is a subset $\mathscr{B}$ of the set of all subsets of $E$ satisfying the following conditions (cf. GT, X, § 1.2, Remark 2). (B1) Every subset of a set of $\mathscr{B}$ belongs to $\mathscr{B}$. (B2) Every finite union of a set of $\mathscr{B}$ belongs to $\mathscr{B}$.

We say that $\mathscr{B}$ is covering if every element of $E$ is contained in a set which belongs to $\mathscr{B}$, or, which is the same, if $\mathscr{B}$ is a cover of $E$.