Closed Set iff Contains all its Limit Points

Theorem
Let $\left({T, \vartheta}\right)$ be a topological space.

Then $H$ is a closed subset of $T$ iff $H$ contains all its limit points.

By the definition of derived set, it follows that:
 * $H$ is closed in $T$ iff $H' \subseteq H$

where $H'$ is the derived set of $H$.

Proof
Let $H$ be a closed subset of $T$.

Then by definition of closed set, $T \setminus H$ is open in $T$.

Let $x \in T \setminus H$.

Then:

Hence, by definition: