Definition:Closed Element/Definition 2

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

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

Let $x \in S$.

Then $x$ is a closed element of $S$ (with respect to $\operatorname{cl}$) iff:


 * $x \in \operatorname{img} \left({ \operatorname{cl} }\right)$

That is, iff $x$ is in the image of $\operatorname{cl}$.