Singleton Partition yields Indiscrete Topology

Theorem

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

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

Then the partition topology on $\mathcal P$ is the indiscrete topology.

Proof

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

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

$\blacksquare$

Sources

• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 5: \ 2$