# Definition:Quotient Topology/Quotient Space

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## 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.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**quotient space**