Quotient Topology of Partition Topology is Discrete Space

Theorem
Let $\mathcal P$ be a partition of a set $S$.

Let $T = \left({S, \tau}\right)$ be the partition space formed from $\mathcal P$.

Let $S / \mathcal P$ be the quotient set of $S$ by $\mathcal P$.

Then the quotient topology $\tau_{S / \mathcal P}$ is a discrete topology.

Proof
Let $\mathcal B$ be the set defined as:
 * $\mathcal B = \left\{ {\left\{ {A}\right\}: A \in S / \mathcal P}\right\}$

From Basis for Partition Topology, $\mathcal B$ forms a basis for a partition space on $S$.

From Basis for Discrete Topology, $\mathcal B$ forms a basis for the discrete topology on $S / \mathcal P$.

Hence the result, by definition of quotient topology.