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.
covers : ,
is downward-closed: if and , then ,
is closed under finite unions: if , then .
A bornological set is a set equipped with a bornology. The elements of are called the bounded sets of a bornological set.
If , are bornological sets, a function is said to be bounded if is bounded in for every bounded in . One obtains a category of bornological sets and bounded maps.
If 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 is a measure space, then the subsets of the sets of finite measure form a bornology .
The category of bornological sets is a quasitopos, in fact a topological universe.
For a proof, see this article by Adamek and Herrlich.
Let be the category of (noncommutative) finite-dimensional algebras over , the field of complex numbers. Let
be the functor that takes an algebra to the set equipped with the bornology of precompact sets. Then there is a canonical identification of the monoid with the monoid of entire holomorphic functions.
This was proved by Schanuel.