Partition of Singletons yields Discrete Topology

Theorem
Let $S$ be a set which is non-empty.

Let $\PP$ be the (trivial) partition of singletons on $S$:
 * $\PP = \set {\set x: x \in S}$

Then the partition topology on $\PP$ is the discrete topology.

Proof
From Basis for Discrete Topology it is shown that $\PP$ as defined here forms the basis of the discrete topology.