Saturated family


In mathematics, specifically in functional analysis, a collection ? of subsets a topological vector space X is said to be saturated if ? contains a non-empty subset of X and if the following conditions all hold:
  1. for every G ∈ ?, ? contains every subset of G;
  2. the union of any finite collection of elements of ? is an element of ?;
  3. for every G ∈ ? and scalar a, ? contains aG;
  4. for every G ∈ ?, the closed, convex, balanced hull of G belongs to ?.
If ? is any collection of subsets of X then the smallest saturated family containing ? is called the saturated hull of ?.
? is said to cover X if the union equals X;
it is total if the linear span of this set is a dense subset of X.

Examples

The intersection of an arbitrary family of saturated families is a saturated family.
Since the power set of X is saturated, any given non-empty family ? of subsets of X containing at least one non-empty set, the saturated hull of ? is well-defined.
Note that a saturated family of subsets of X that covers X is a bornology on X.
The set of all bounded subsets of a topological vector space is a saturated family.