Saturated family

In mathematics, specifically in functional analysis, a family $$\mathcal{G}$$ of subsets a topological vector space (TVS) $$X$$ is said to be saturated if $$\mathcal{G}$$ contains a non-empty subset of $$X$$ and if for every $$G \in \mathcal{G},$$ the following conditions all hold:
 * 1) $$\mathcal{G}$$ contains every subset of $$G$$;
 * 2) the union of any finite collection of elements of $$\mathcal{G}$$ is an element of $$\mathcal{G}$$;
 * 3) for every scalar $$a,$$ $$\mathcal{G}$$ contains $$aG$$;
 * 4) the closed convex balanced hull of $$G$$ belongs to $$\mathcal{G}.$$

Definitions
If $$\mathcal{S}$$ is any collection of subsets of $$X$$ then the smallest saturated family containing $$\mathcal{S}$$ is called the of $$\mathcal{S}.$$

The family $$\mathcal{G}$$ is said to $$X$$ if the union $$\bigcup_{G \in \mathcal{G}} G$$ is equal to $$X$$; it is 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 $$\mathcal{G}$$ of subsets of $$X$$ containing at least one non-empty set, the saturated hull of $$\mathcal{G}$$ 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.