Definition:Closed Set under Closure Operator/Definition 1

Definition
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 $\cl$) :
 * $\map \cl T = T$

Also see

 * Equivalence of Definitions of Closed Set under Closure Operator