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

$T$ is closed (with respect to $\cl$) if and only if:

$\map \cl T = T$

Definition 2

$T$ is closed (with respect to $\cl$) if and only if $T$ 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

• Results about closed elements can be found here.

Generalization

• Definition:Closed Element under Closure Operator