Singleton Partition yields Indiscrete Topology
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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: $2$