Discrete Topology is Topology

Theorem
Let $S$ be a set.

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

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

Then by definition $\vartheta = \mathcal P \left({S}\right)$, 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, $\varnothing \in \mathcal P \left({S}\right)$ and $S \in \mathcal P \left({S}\right)$.
 * $(2): \quad$ From Power Set with Union is Monoid, $\mathcal P \left({S}\right)$ is closed under set union.
 * $(3): \quad$ From Power Set with Intersection is Monoid, $\mathcal P \left({S}\right)$ is closed under set intersection.