Closure Operator from Closed Sets

Theorem
Let $S$ be a set.

Let $\mathcal C$ be a set of subsets of $S$.

Suppose that $\forall \mathcal K \in \mathcal P (\mathcal C): \bigcap \mathcal K \in \mathcal C$, where $\bigcap \varnothing$ is taken to be $S$.

That is, suppose that $\mathcal C$ is closed under arbitrary intersections.

Define $\operatorname{cl}: \mathcal P(S) \to \mathcal C$ by letting
 * $\operatorname{cl} (T) = \bigcap \{ C \in \mathcal C: T \subseteq C \}$

Then $\operatorname{cl}$ is a closure operator whose closed sets are the elements of $\mathcal C$.

Proof
First we will show that $\operatorname{cl}$ is a closure operator.

Inflationary
Follows from Set Intersection Preserves Subsets/General Result/Corollary.