Complement of Clopen Set is Clopen

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

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

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

Then $\complement_S \left({H}\right)$ is also a clopen set of $T$.

Proof
By definition of clopen, $H$ is open in $T$.

By definition of closed set, $\complement_S \left({H}\right)$ is closed in $T$.

By definition of clopen, $H$ is closed in $T$.

By definition of closed set, $\complement_S \left({H}\right)$ is open in $T$.

Thus $\complement_S \left({H}\right)$ is both open in $T$ and closed in $T$.

Hence the result, by definition of clopen set.