\begin{definition}[Definition:Quotient Topology/Definition 1]
Let $T = \struct {S, \tau}$ be a topological space.
Let $\RR \subseteq S \times S$ be an equivalence relation on $S$.
Let $q_\RR: S \to S / \RR$ be the quotient mapping induced by $\RR$.
Let $\tau_\RR$ be the identification topology on $S / \RR$ by $q_\RR$:
:$\tau_\RR := \set {U \subseteq S / \RR: q_\RR^{-1} \sqbrk U \in \tau}$
Then $\tau_\RR$ is the '''quotient topology on $S / \RR$ by $q_\RR$'''.
\end{definition}