\begin{definition}[Definition:Partition Topology]
Let $S$ be a set.
Let $\PP$ be a partition of $S$.
Let $\tau$ be the set of subsets of $S$ defined as:
:$a \in \tau \iff a$ is the union of sets of $\PP$
Then $\tau$ is a '''partition topology''' on $S$, and $\struct {S, \tau}$ is a '''partition (topological) space'''.
The partition $\PP$ is the basis of $\struct {S, \tau}$.
\end{definition}