# Definition:Closed Set/Closure Operator

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## 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 $\operatorname{cl}$) if and only if:

- $\operatorname{cl} \left({T}\right) = T$

### Definition 2

The subset $T$ is **closed** (with respect to $\operatorname{cl}$) if and only if $T$ 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