Definition:Closed Element

Definition

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

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

Let $x \in S$.

Definition 1

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

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

Definition 2

The element $x$ is a closed element of $S$ (with respect to $\operatorname{cl}$) if and only if $x$ is in the image of $\operatorname{cl}$:

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

Also see

• Equivalence of Definitions of Closed Element
• Definition:Closure of Element under Closure Operator

Special case

• Definition:Closed Set under Closure Operator