Partition of Singletons yields Discrete Topology

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Theorem

Let $S$ be a set which is non-empty.

Let $\mathcal P$ be the (trivial) partition of singletons on $S$:

$\mathcal P = \left\{{\left\{{x}\right\}: x \in S}\right\}$


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


Proof

From Basis for Discrete Topology it is shown that $\mathcal P$ as defined here forms the basis of the discrete topology.

$\blacksquare$


Sources