Closed Set iff Contains all its Limit Points

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

Let $H \subseteq S$.

Then $H$ is closed in $T$ iff every limit point of $H$ is also a point of $H$.

That is, by the definition of the derived set:
 * $H$ is closed in $T$ iff $H' \subseteq H$

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

Proof
Let $H^{\complement}$ denote the relative complement of $H$ in $S$.

Consider an arbitrary point $x \in H^{\complement}$.

Then the following equivalence holds:

Hence: