Set Closure is Smallest Closed Set/Closure Operator

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Theorem

Let $S$ be a set.

Let $\cl: \powerset S \to \powerset S$ be a closure operator.

Let $T \subseteq S$.

Then $\map \cl T$ is the smallest closed set (with respect to $\cl$) containing $T$ as a subset.


Proof

By definition, $\map \cl T$ is closed.

Let $C$ be closed.

Let $T \subseteq C$.

By the definition of closure operator, $\cl$ is $\subseteq$-increasing.

So:

$\map \cl T \subseteq \map \cl C$

Since $C$ is closed, $\map \cl C = C$.

So:

$\map \cl T \subseteq C$

Thus $\map \cl T$ is the smallest closed set containing $T$ as a subset.

$\blacksquare$