Equivalence of Definitions of Closed Element

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

Let $\cl$ be a closure operator on $S$.

Let $x \in S$.

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

Let $\cl: S \to S$ be a closure operator on $S$.

Let $x \in S$.

By the definition of closure operator, $\cl$ is idempotent.

Thus by Fixed Point of Idempotent Mapping:


 * An element of $S$ is a fixed point of $\cl$ it is in the image of $\cl$.

Thus the above definitions are equivalent.