Partition of Singletons yields Discrete Topology
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Theorem
Let $S$ be a set which is non-empty.
Let $\PP$ be the (trivial) partition of singletons on $S$:
- $\PP = \set {\set x: x \in S}$
Then the partition topology on $\PP$ is the discrete topology.
Proof
From Basis for Discrete Topology it is shown that $\PP$ as defined here forms the basis of the discrete topology.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$