Set in Discrete Topology is Clopen

Theorem
Let $S$ be a set.

Let $\tau$ be the discrete topology on $S$.
 * $\forall U \subseteq S: U$ is both closed and open in $\left({S, \tau}\right)$.

Proof
Let $T = \left({S, \tau}\right)$ be the discrete space on $S$.

Then by definition $\tau = \mathcal P \left({S}\right)$, that is, is the power set of $S$.

Let $U \subseteq S$.

Then by definition of power set, $U \in \tau$.

By definition of closed set, $\complement_S \left({U}\right)$ is closed in $T$, where $\complement_S \left({U}\right)$ is the relative complement of $U$ in $S$.

But from Set Difference is Subset:
 * $\complement_S \left({U}\right) = S \setminus U \subseteq S$

and so:
 * $\complement_S \left({U}\right) \in \tau$

That is, $\complement_S \left({U}\right)$ is both closed and open in $T$.

Then by Relative Complement of Relative Complement:
 * $\complement_S \left({\complement_S \left({U}\right)}\right) = U$

which is seen to be both closed and open in $T$.