# Definition:Quotient Topology/Quotient Space

## 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 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 $\tau_\RR$:

$T_\RR := \struct {S / \RR, \tau_\RR}$

## Also known as

A quotient space is also known as an identification space and a factor space.

However, note that an identification space is often (and on $\mathsf{Pr} \infty \mathsf{fWiki}$) used for a more general concept.

## Also see

• Results about quotient spaces can be found here.