Basis for Partition Topology

Theorem
Let $S$ be a set.

Let $\mathcal P$ 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 $\mathcal P$ forms a basis of $\tau$.

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

We have that $\displaystyle S = \bigcup \mathcal P$ from the definition of a partition.
 * B1

So $S$ is the union of sets of $\mathcal P$.

Let $B_1, B_2 \in \mathcal P$.
 * B2

Then as $\mathcal P$ is a partition of $S$, we have that $B_1 \cap B_2 = \varnothing$.

But from Union of Empty Set we have that $\varnothing$ is the union (vacuous) of no sets of $\mathcal P$.

Hence the result.