Definition:Partition Topology

Definition
Let $S$ be a set.

Let $\mathcal P$ be a partition of $S$.

Let $\vartheta$ be the set of subsets of $S$ defined as:
 * $a \in \vartheta \iff a$ is the union of sets of $\mathcal P$

Then $\vartheta$ is a partition topology on $S$, and $\left({S, \vartheta}\right)$ is a partition (topological) space.

The partition $\mathcal P$ is the basis of $\left({S, \vartheta}\right)$.

Also see
The above statements are proved in:


 * Partition Topology is a Topology
 * Basis for Partition Topology