View source for Passing to the quotient (topology)/Statement

<noinclude>
{{Stub page|grade=A}}
{{Requires references|grade=A}}
==Statement==
</noinclude><div style="float:right;margin-left:0.05em;overflow:hidden;">
{| style="margin-top:0px;margin-bottom:0px;" class="wikitable" border="1"
|-
| <center><span style="font-size:1.1em;"><m>\xymatrix{ X \ar[d]_\pi \ar[dr]^f & \\ \frac{X}{\sim} \ar@{.>}[r]^{\overline{f} }& Y}</m></span></center>
|-
! {{M|f}} descends to the quotient
|}
</div>
Suppose that {{Top.|X|J}} is a [[topological space]] and {{M|\sim}} is an [[equivalence relation]], let {{Top.|\frac{X}{\sim}|Q}} be the resulting [[quotient topology]] and {{M|\pi:X\rightarrow\frac{X}{\sim} }} the resulting [[quotient map (topology)|quotient map]], then:
* Let {{Top.|Y|K}} be any topological space and let {{M|f:X\rightarrow Y}} be a continuous map that is constant on the [[fibre|fibres]] of {{M|\pi}}<ref group="Note"><!--

-->
That means that: 
* {{M|1=\forall x,y\in X[\pi(x)=\pi(y)\implies f(x)=f(y)]}} - as mentioned in [[passing to the quotient (function)|passing-to-the-quotient for functions]], or
* {{M|1=\forall x,y\in X[f(x)\ne f(y)\implies \pi(x)\ne\pi(y)]}}, also mentioned
* See :- [[Equivalent conditions to being constant on the fibres of a map]] for details
<!--

--></ref> then:
* there exists a unique continuous map, {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}} such that {{M|1=f=\overline{f}\circ\pi }}
We may then say {{M|f}} ''descends to the quotient'' or ''passes to the quotient''
: '''Note: ''' this is an instance of ''[[passing to the quotient (function)|passing-to-the-quotient for functions]]''<div style="clear:both;"></div>
<noinclude>
==Notes==
<references group="Note"/>
==References==
<references/>
{{Theorem Of|Topology}}
</noinclude>