A bornological set is a notion of space, where instead of considering open sets and continuous functions whose inverse images preserve open sets as one does for topological spaces, one considers bounded sets (which constitute a bornology) and bounded maps whose direct images preserve bounded sets. Bornological topological vector spaces, called bornological spaces, are important in functional analysis.
Let $X$ be a set. A bornology on $X$ is a collection $\mathcal{B} \subseteq P X$ of subsets of $X$ such that
$\mathcal{B}$ covers $X$: $\bigcup_{B \in \mathcal{B}} B = X$,
$\mathcal{B}$ is downward-closed: if $B \in \mathcal{B}$ and $A \subseteq B$, then $A \in \mathcal{B}$,
$\mathcal{B}$ is closed under finite unions: if $B_1 \ldots, B_n \in \mathcal{B}$, then $\bigcup_{1 \leq \i \leq n} B_i \in \mathcal{B}$.
A bornological set is a set $X$ equipped with a bornology. The elements of $\mathcal{B}$ are called the bounded sets of a bornological set.
If $X$, $Y$ are bornological sets, a function $f\colon X \to Y$ is said to be bounded if $f(B)$ is bounded in $Y$ for every bounded $B$ in $X$. One obtains a category of bornological sets and bounded maps.
If $X$ is any topological space, there is a bornology consisting of all precompact subsets of $X$ (subsets whose closure is compact). Any continuous map is bounded with respect to this choice of bornology.
If $X$ is any metric space, there is a bornology where a set is bounded if it is contained in some open ball. Any Lipschitz map is bounded with respect to this choice of bornology. A metric space is bounded if it's a bounded subspace of itself.
If $X$ is a measure space, then the subsets of the sets of finite measure form a bornology .
For linear operators between bornological spaces, a map is continuous if and only if it is bounded.
The category of bornological sets is a quasitopos, in fact a topological universe.
For a proof, see this article by Adamek and Herrlich.
Let $Alg_{\mathbb{C}}$ be the category of (noncommutative) finite-dimensional algebras over $\mathbb{C}$, the field of complex numbers. Let
be the functor that takes an algebra $A$ to the set ${|A|}$ equipped with the bornology of precompact sets. Then there is a canonical identification of the monoid $Born^{Alg_\mathbb{C}}(U, U)$ with the monoid of entire holomorphic functions.
This was proved by Schanuel.