Saturated Family

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

Definitions

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

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

See also

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References

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{{Functional analysis

Functional analysis