Partition Topology is Topology

Theorem
Let $S$ be a set.

Let $\mathcal P$ be a partition of $S$.

Let $\vartheta$ be the set of subsets of $S$ defined as:
 * $a \in \vartheta \iff a$ is the union of sets of $\mathcal P$

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

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

That's all we need to show.