Definition:Closed Element
(Redirected from Definition:Closed Element under Closure Operator)
Definition
Let $\left({S, \preceq}\right)$ be an ordered set.
Let $\operatorname{cl}$ be a closure operator on $S$.
Let $x \in S$.
Definition 1
The element $x$ is a closed element of $S$ (with respect to $\operatorname{cl}$) 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 $\operatorname{cl}$:
- $x \in \operatorname{im} \left({\operatorname{cl} }\right)$
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