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.
Jiří Adámek and H. Herrlich, Cartesian closed categories, quasitopoi, and topological universes, Comm. Math. Univ. Carol., Vol. 27, No. 2 (1986), 235-257. (web)
Stephen Schanuel, Continuous extrapolation to triangular matrices characterizes smooth functions, J. Pure App. Alg. 24, Issue 1 (1982), 59–71. (web)
Fabienne Prosmans, Jean-Pierre Schneiders, A homological study of bornological spaces, December 2000, Prepublications Mathematiques de l’Universite Paris 13, 46 (pdf)
Review includes