# Equivalence of Definitions of Closed Element

## Contents

## 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.

$\blacksquare$