Definition:Closed Element

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:
 * $\operatorname{cl} \left({x}\right) = x$

That is, iff $x$ is a fixed point of $\operatorname{cl}$.

Since a closure operator is idempotent, Fixed Points of Idempotent Mapping shows that the set of closed elements of $S$ with respect to $\operatorname{cl}$ is $\operatorname{cl} \left({S}\right)$, the image of $S$ under $\operatorname{cl}$.