Basis for Partition Topology
Let $S$ be a set.
Let $\mathcal P$ be a partition of $S$.
Let $\tau$ be the partition topology on $S$ defined as:
Then $\mathcal P$ forms a basis of $\tau$.
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.
Therefore, $\displaystyle S \subseteq \bigcup \mathcal P$ and $\mathcal P$ is a cover for $S$.
Next, let $B_1, B_2 \in \mathcal P$.
Then as $\mathcal P$ is a partition of $S$, we have that $B_1 \cap B_2 = \varnothing$.
Hence $\mathcal P$ is a synthetic basis for $\tau$.