Set of Closed Elements wrt Closure Operator under Subset Operation is Complete Lattice

Theorem
Let $S$ be a set.

Let $\cl$ be a closure operator on the power set $\powerset S$ of $S$.

Let $\mathscr C$ be the set of all subsets $T$ of $S$ such that:
 * $\map \cl T = T$

Then the algebraic structure $\struct {\mathscr C, \subseteq}$ forms a complete lattice.

Proof
Recall the closure axioms:

First we note that from we have that:
 * $\map \cl S = S$

and so:
 * $S \in \mathscr C$

Let $\AA \subseteq \mathscr C$.

Thus $\AA$ is a set of subsets $T$ of $S$ for all of which $\map \cl T = T$.

From Intersection of Closed Sets is Closed:
 * $\ds \cap \AA \in \mathscr C$

The result follows from Set of Subsets which contains Set and Intersection of Subsets is Complete Lattice,