Definition:Partition Topology

Definition
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 $\struct {S, \tau}$.

Also see
The above statements are proved in:


 * Partition Topology is Topology
 * Basis for Partition Topology