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


Generalization