The Lattice of Bornologies on a Set

We study the complete distributive lattice of bornologies $\mathfrak{B}_X$ on a nonempty set $X$, with particular attention given to principal bornologies. These are precisely the complemented elements of the lattice, and as such, form a Boolean algebra. Each Boolean algebra can be Boolean embedded in the Boolean algebra of principal bornologies on some set. Remarkably, if a principal bornology $\mathcal {A}$ is a subset of an arbitrary bornology $\mathcal {B}$, then $\mathcal {A}$ must be way below $\mathcal {B}$ in the sense of continuous lattices.