Clopen Set and Complement form Separation

Theorem

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

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

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

Then $H$ and $\relcomp S H$ form a separation of $T$.

Proof

By Set with Relative Complement forms Partition, $H$ and $\relcomp S H$ form a partition of $S$.

By Complement of Clopen Set is Clopen, $\relcomp S H$ is also a clopen set of $T$.

By definition of clopen set, both $H$ and $\relcomp S H$ are open in $T$.

Thus $H$ and $\relcomp S H$ are a pair of open sets in $T$ forming a partition of $S$.

Hence the result, by definition of separation.

$\blacksquare$