Equivalence of Definitions of Closed Element

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Theorem

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

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

Let $x \in S$.


The following definitions of the concept of Closed Element are equivalent:

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 $\operatorname{cl}$:

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


Proof

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

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

Let $x \in S$.

By the definition of closure operator, $\operatorname{cl}$ is idempotent.

Thus by Fixed Point of Idempotent Mapping:

An element of $S$ is a fixed point of $\operatorname{cl}$ if and only if it is in the image of $\operatorname{cl}$.

Thus the above definitions are equivalent.

$\blacksquare$