Definition:Closed Set/Closure Operator

Definition
Let $S$ be a set.

Let $\operatorname{cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ be a closure operator.

Let $T \subseteq S$ be a subset.

Definition 1
The subset $T$ is closed (with respect to $\operatorname{cl}$) :
 * $\operatorname{cl} \left({T}\right) = T$

Definition 2
The subset $T$ is closed (with respect to $\operatorname{cl}$) $T$ is in the image of $\operatorname{cl}$:
 * $T \in \operatorname{im}(\operatorname{cl})$

Also see

 * Equivalence of Definitions of Closed Set under Closure Operator
 * Definition:Closure of Set under Closure Operator

Generalization

 * Definition:Closed Element under Closure Operator