Definition:Closed Set/Closure Operator

Definition
Let $S$ be a set.

Let $\cl: \powerset S \to \powerset S$ be a closure operator.

Let $T \subseteq S$ be a subset.

Definition 1
The subset $T$ is closed (with respect to $\cl$) :
 * $\map \cl T = T$

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

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