Definition:Closed Element/Definition 1

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

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

Let $x \in S$.

The element $x$ is a closed element of $S$ (with respect to $\operatorname{cl}$) $x$ is a fixed point of $\operatorname{cl}$:
 * $\operatorname{cl} \left({x}\right) = x$

Also see

 * Equivalence of Definitions of Closed Element