Basis for Partition Topology

Theorem
Let $S$ be a set.

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

Let $\tau$ be the partition topology on $S$ defined as:
 * $a \in \tau \iff a$ is the union of sets of $\mathcal P$

Then $\PP$ forms a basis of $\tau$.

Proof
Checking the criteria for $\PP$ to be a synthetic basis for $\tau$:

We have that $\ds S = \bigcup \PP$ from the definition of a partition.

Therefore, $\ds S \subseteq \bigcup \PP$ and $\PP$ is a cover for $S$.

Next, let $B_1, B_2 \in \PP$.

Then as $\PP$ is a partition of $S$, we have that $B_1 \cap B_2 = \O$.

But from Union of Empty Set we have that $\O$ is the (vacuous) union of sets of $\PP$.

Hence $\PP$ is a synthetic basis for $\tau$.