---
tag: 0V8046R0P9
kind: definition
name: Quotient topology
msc: [54B15]
defines: 0NQ1H2X495
uses: [0CNQXVBHCX, 0SGTMBEWRF, 0ZVHXDSZC7, 0FGCFY1AZX]
sources:
  - url: https://en.wikipedia.org/wiki/Quotient_space_(topology)
    locator: Definition
---

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]] and let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on $X$.
Let $q \colon X \to X / {\sim}$ be the map to the [[quotient set|0ZVHXDSZC7]] that sends $x$ to its class.
The quotient topology on $X / {\sim}$ is
$$\{ V \subseteq X / {\sim} : q^{-1}(V) \in \tau \}.$$

## Remarks

It is the finest topology on $X / {\sim}$ for which $q$ is continuous.
A [[preimage|0FGCFY1AZX]] $q^{-1}(V)$ is a union of classes. Thus $V$ is open if and only if the union of the classes in $V$ is open in $X$.
Quotients need not keep good properties: a quotient of a Hausdorff space need not be [[Hausdorff|0MXB4CKHFW]].
Examples are the [[circle|0WH1GVBJNH]], the [[torus|03NM7DZXXC]] and the [[Möbius strip|04AJE0M271]].

## Explanations

### L1 human

You make a quotient space by gluing the points of each class into one point.
For example, gluing the two ends of $[0, 1]$ gives a circle.