Definition:Partition Topology

Definition

Let $S$ be a set.

Let $\mathcal P$ 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 $\mathcal P$

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

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

Also see

The above statements are proved in:

• Partition Topology is Topology
• Basis for Partition Topology

• Results about partition topologies can be found here.

Sources

• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 5$