Definition:Quotient Topology

Definition
Let $\left({S, \tau_1}\right)$ be a topological space.

Let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$.

Let $q_\mathcal R: S \to S / \mathcal R$ be the quotient mapping induced by $\mathcal R$.

Let $\tau_2$ be the identification topology on $S / \mathcal R$ by $q_\mathcal R$:
 * $\tau_2 = \left\{{U \subseteq S / \mathcal R: q_\mathcal R^{-1} \left({U}\right) \in \tau_1}\right\}$.

Then $\tau_2$ is called the quotient topology on $S / \mathcal R$ by $q_\mathcal R$.

Quotient Space
Thus we have that $\left({S / \mathcal R, \tau_2}\right)$ is a topological space.

It is called the quotient space of $\left({S, \tau_1}\right)$ by $\mathcal R$.

Also see

 * Quotient space