# Category:Closed Elements

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This category contains results about **Closed Elements**.

Definitions specific to this category can be found in Definitions/Closed Elements.

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

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

Let $x \in S$.

### Definition 1

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

- $\map \cl x = x$

### Definition 2

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

- $x \in \Img \cl$

## Pages in category "Closed Elements"

The following 2 pages are in this category, out of 2 total.