Equivalence of Definitions of Closed Element

Theorem
The following definitions of closed element are equivalent:

Proof
Let $\left({S, \preceq}\right)$ be an ordered set.

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

Let $x \in S$.

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

Thus by Fixed Points of Idempotent Mapping:


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

Thus the above definitions are equivalent.