Closed Elements Uniquely Determine Closure Operator

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Theorem

Let $\left({S, \preceq}\right)$ 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 $S$ (with respect to either $f$ or $g$) that succeed $x$.

By Closure is Smallest Closed Successor, $f \left({x}\right)$ and $g \left({x}\right)$ are smallest closed successors of $x$.

That is, $f \left({x}\right)$ and $g \left({x}\right)$ are smallest elements of $C \cap \bar\uparrow x$, where $\bar\uparrow x$ is the upper closure of $x$.

By Smallest Element is Unique, $f \left({x}\right) = g \left({x}\right)$.

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

$\blacksquare$