# Definition:Quotient Topology/Definition 1

## Definition

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 $q_\RR$.