Quotient Topology of Partition Topology is Discrete Space

Theorem
Let $\PP$ be a partition of a set $S$.

Let $T = \struct {S, \tau}$ be the partition space formed from $\PP$.

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

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

Proof
Let $\BB$ be the set defined as:
 * $\BB = \set {\set A: A \in S / \PP}$

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

From Basis for Discrete Topology, $\BB$ forms a basis for the discrete topology on $S / \PP$.

Hence the result, by definition of quotient topology.