Bornology

In mathematics, especially functional analysis, a bornology on a set X is a collection of subsets of X satisfying axioms that generalize the notion of boundedness. One of the key motivations behind bornologies and bornological analysis is the fact that bornological spaces provide a convenient setting for homological algebra in functional analysis. This is because^{[1]pg 9} the category of bornological spaces is additive, complete, cocomplete, and has a tensor product adjoint to an internal hom, all necessary components for homological algebra.

Definitions[]

A bornology or boundedness on a set X is a collection ℬ of subsets of X such that

1. ℬ covers X (i.e. X = ${\displaystyle \cup _{B\in {\mathcal {B}}}B}$);
2. ℬ is stable under inclusion (i.e. if B ∈ ℬ then every subset of B belongs to ℬ);
3. ℬ is stable under finite unions (or equivalently, the union of any two sets in ℬ also belongs to ℬ).

in which case the pair (X, ℬ) is called a bounded structure or a bornological set.^[2] Elements of ℬ are called ℬ-bounded sets or simply bounded sets, if ℬ is understood. A subset