Definition:Closed Element
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Definition
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 $\cl$:
- $x \in \Img \cl$
Also see
- Results about closed elements can be found here.