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

• Results about partition topologies can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $5$. Partition Topology