Complement of Clopen Set is Clopen

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

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

Let $\relcomp S H$ denote the complement of $H$ relative to $S$.

Then $\relcomp S H$ is also a clopen set of $T$.

Proof

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

By definition of closed set, $\relcomp S H$ is closed in $T$.

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

By definition of closed set, $\relcomp S H$ is open in $T$.

Thus $\relcomp S H$ is both open in $T$ and closed in $T$.

Hence the result, by definition of clopen set.

$\blacksquare$