\begin{definition}[Definition:Quotient Topology/Quotient Space]
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 quotient topology on $S / \RR$ by $q_\RR$:
:$\tau_\RR := \set {U \subseteq S / \RR: \map {q_\RR^{-1} } U \in \tau}$
The '''quotient space of $S$ by $\RR$''' is the topological space whose points are elements of the quotient set of $\RR$ and whose topology is $\tau_\RR$:
:$T_\RR := \struct {S / \RR, \tau_\RR}$
\end{definition}