Category:Definitions/Closed Elements

This category contains definitions related to Closed Elements.
Related results can be found in Category: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 $\cl$:

$x \in \Img \cl$