Clopen Set and Complement form Separation

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

Let $H \subseteq S$ be a clopen set of $T$.

Let $\complement_S \left({H}\right)$ be the complement of $H$ relative to $S$.

Then $H$ and $\complement_S \left({H}\right)$ form a separation of $T$.

Proof
By Set with Relative Complement forms Partition, $H$ and $\complement_S \left({H}\right)$ form a partition of $S$.

By Complement of Clopen Space is Clopen, $\complement_S \left({H}\right)$ is also a clopen set of $T$.

By definition of clopen set, both $H$ and $\complement_S \left({H}\right)$ are open in $T$.

Thus $H$ and $\complement_S \left({H}\right)$ are a pair of open in $T$ forming a partition of $S$.

Hence the result, by definition of separation.