Basis for Partition Topology
Let $S$ be a set.
Let $\PP$ be a partition of $S$.
Let $\tau$ be the partition topology on $S$ defined as:
Then $\PP$ forms a basis of $\tau$.
Checking the criteria for $\PP$ to be a synthetic basis for $\tau$:
We have that $\displaystyle S = \bigcup \PP$ from the definition of a partition.
Therefore, $\displaystyle S \subseteq \bigcup \PP$ 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 $B_1 \cap B_2 = \O$.
Hence $\PP$ is a synthetic basis for $\tau$.