Singleton Partition yields Indiscrete Topology

Theorem

Let $S$ be a set which is not empty.

Let $\PP$ be the (trivial) singleton partition $\set S$ on $S$.

Then the partition topology on $\PP$ is the indiscrete topology.

Proof

By definition, the partition topology on $\PP$ is the set of all unions from $\PP$.

This is (trivially, and from Union of Empty Set) $\set {\O, S}$ which is the indiscrete topology on $S$ by definition.

$\blacksquare$

Sources

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