nLab
bornological set

Idea

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.

Definitions

Let $X$ be a set. A bornology on $X$ is a collection $ℬ\subseteq PX$ of subsets of $X$ such that

• $ℬ$ covers $X$: ${\bigcup }_{B\in ℬ}B=X$,

• $ℬ$ is downward-closed: if $B\in ℬ$ and $A\subseteq B$, then $A\in ℬ$,

• $ℬ$ is closed under finite unions: if $B_1\dots ,B_n\in ℬ$, then ${\bigcup }_{1\le i\le n}{B}_i\in ℬ$.

A bornological set is a set $X$ equipped with a bornology. The elements of $ℬ$ are called the bounded sets of a bornological set.

If $X$, $Y$ are bornological sets, a function $f:X\to Y$ 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.

Examples

• 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.

Properties

Theorem

The category of bornological sets is a quasitopos, in fact a topological universe.

For a proof, see this article by Adamek and Herrlich.

Theorem

Let ${\mathrm{Alg}}_{ℂ}$ be the category of (noncommutative) finite-dimensional algebras over $ℂ$, the field of complex numbers. Let

$$U:{\mathrm{Alg}}_{ℂ}\to \mathrm{Born}$$U \colon Alg_{\mathbb{C}} \to Born

be the functor that takes an algebra $A$ to the set $\mid A\mid$ equipped with the bornology of precompact sets. Then there is a canonical identification of the monoid ${\mathrm{Born}}^{{\mathrm{Alg}}_{ℂ}}(U,U)$ with the monoid of entire holomorphic functions.

This was proved by Schanuel.

References

• J. Adamek and H. Herrlich, Cartesian closed categories, quasitopoi, and topological universes. Comm. Math. Univ. Carol., Vol. 27, No. 2 (1986), 235-257. (web)

• S. Schanuel, Continuous extrapolation to triangular matrices characterizes smooth functions, J. Pure App. Alg. 24, Issue 1 (1982), 59–71. (web)