Partition Topology is Topology

Theorem
Let $S$ be a set.

Let $\PP$ be a partition of $S$.

Let $\tau$ be the set of subsets of $S$ defined as:
 * $a \in \tau \iff a$ is the union of sets of $\PP$

Then $\tau$ is a topology on $S$.

Proof
From Basis for Partition Topology, we have that $\PP$ is a basis for the partition topology.

The result follows.