Components of Separation are Clopen

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

Let $A \mid B$ be a partition of $T$.

Then both $A$ and $B$ are both closed in $T$.

Proof
From Set with Relative Complement forms Partition it follows that:
 * $A = \complement_S \left({B}\right)$

and:
 * $B = \complement_S \left({A}\right)$

where $\complement_S$ denotes the complement relative to $S$.

As $A$ and $B$ are both open, it follows by definition that $A$ and $B$ are also both closed.