# Definition:Closed Set/Closure Operator

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## 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 $\cl$) if and only if:

$\map \cl T = T$

### Definition 2

The subset $T$ is closed (with respect to $\cl$) if and only if $T$ is in the image of $\cl$:

$T \in \Img \cl$