Discrete Topology is Topology

Theorem
Let $S$ be a set.

Let $\tau$ be the discrete topology on $S$.
 * $\tau$ is a topology on $S$.

Proof
Let $T = \struct {S, \tau}$ be the discrete space on $S$.

Then by definition $\tau = \powerset S$, that is, is the power set of $S$.

We confirm the criteria for $T$ to be a topology:
 * $(1): \quad$ By definition of power set, $\O \in \powerset S$ and $S \in \powerset S$.
 * $(2): \quad$ From Power Set with Union is Monoid, $\powerset S$ is closed under set union.
 * $(3): \quad$ From Power Set with Intersection is Monoid, $\powerset S$ is closed under set intersection.