"Closure" is a fuzzy term, used in many branches of mathematics, for functions that map sets to sets where it can be proved that
$f(A)$ is the smallest set that contain $A$ and satisfies such-and-such additional conditions
where the "additional conditions" vary between various kinds of closure. The usual way to this claim for a particular function $f$ is to prove each direction separately:
- $f(A)$ contains $A$ and satisfies the such-and-such conditions.
- Assume some set $C$ satisfies the such-and-such conditions. Then $f(B)\subseteq C$ for every $B\subseteq C$.
These two properties combine into the alternative characterization of $f$:
$f(A)$ is the intersection of all supersets of $A$ that satisfies such-and-such conditions.
This can be used as an alternative definition of $f$, but requires proof that (a) the such-and-such conditions are preserved by infinite intersection, and (b) there is at least one set (usually a "universal set" in some sense) that satisfies the conditions.
For any given closure concept, it is not uncommon that some texts will define it by one of the above characterization and others by a concrete construction that happens to provide the abstract closure property.