nLab
bornological set

Contents

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 XX be a set. A bornology on XX is a collection PX\mathcal{B} \subseteq P X of subsets of XX such that

  • \mathcal{B} covers XX: BB=X\bigcup_{B \in \mathcal{B}} B = X,

  • \mathcal{B} is downward-closed: if BB \in \mathcal{B} and ABA \subseteq B, then AA \in \mathcal{B},

  • \mathcal{B} is closed under finite unions: if B 1,B nB_1 \ldots, B_n \in \mathcal{B}, then 1inB i\bigcup_{1 \leq \i \leq n} B_i \in \mathcal{B}.

A bornological set is a set XX equipped with a bornology. The elements of \mathcal{B} are called the bounded sets of a bornological set.

If XX, YY are bornological sets, a function f:XYf\colon X \to Y is said to be bounded if f(B)f(B) is bounded in YY for every bounded BB in XX. One obtains a category of bornological sets and bounded maps.

Examples

  • If XX is any topological space, there is a bornology consisting of all precompact subsets of XX (subsets whose closure is compact). Any continuous map is bounded with respect to this choice of bornology.

  • If XX 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 XX 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 Alg Alg_{\mathbb{C}} be the category of (noncommutative) finite-dimensional algebras over \mathbb{C}, the field of complex numbers. Let

U:Alg BornU \colon Alg_{\mathbb{C}} \to Born

be the functor that takes an algebra AA to the set A{|A|} equipped with the bornology of precompact sets. Then there is a canonical identification of the monoid Born Alg (U,U)Born^{Alg_\mathbb{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)

Revised on June 9, 2013 12:52:04 by Toby Bartels (173.190.139.188)