Definition:Closed Set/Closure Operator
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Let $S$ be a set.
Let $\cl: \powerset S \to \powerset S$ be a closure operator.
Let $T \subseteq S$ be a subset.
$T$ is closed (with respect to $\cl$) if and only if:
- $\map \cl T = T$
$T$ is closed (with respect to $\cl$) if and only if $T$ is in the image of $\cl$:
- $T \in \Img \cl$
- Equivalence of Definitions of Closed Set under Closure Operator
- Definition:Closure of Set under Closure Operator
- Results about closed elements can be found here.