# Definition:Quotient Topology

## Definition

Let $T = \left({S, \tau}\right)$ 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 $\mathcal R$.

### Definition 1

Let $\tau_\mathcal R$ be the identification topology on $S / \mathcal R$ by $q_\mathcal R$:

$\tau_\mathcal R := \left\{{U \subseteq S / \mathcal R: q_\mathcal R^{-1} \left({U}\right) \in \tau}\right\}$

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

## Quotient Space

The quotient space of $S$ by $\mathcal R$ is the topological space whose points are elements of the quotient set of $\mathcal R$ and whose topology is $\tau_\mathcal R$:

$T_\mathcal R := \left({S / \mathcal R, \tau_\mathcal R}\right)$