# Definition:Closed Element/Definition 1

## Definition

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

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

Let $x \in S$.

The element $x$ is a closed element of $S$ (with respect to $\cl$) if and only if $x$ is a fixed point of $\cl$:

$\map \cl x = x$