Closed Elements Uniquely Determine Closure Operator

Theorem
Let $\struct {S, \preceq}$ be an ordered set.

Let $f, g: S \to S$ be closure operators on $S$.

Suppose that $f$ and $g$ have the same closed elements.

Then $f = g$.

Proof
Let $x \in S$.

Let $C$ be the set of closed elements of $S$ ( either $f$ or $g$) that succeed $x$.

By Closure is Smallest Closed Successor, $\map f x$ and $\map g x$ are smallest closed successors of $x$.

That is, $\map f x$ and $\map g x$ are smallest elements of $C \cap \bar \uparrow x$, where $\bar \uparrow x$ is the upper closure of $x$.

By Smallest Element is Unique:
 * $\map f x = \map g x$

Since this holds for all $x \in S$, $f = g$ by Equality of Mappings.