View source for Quotient topology/Equivalence relation definition

<noinclude>
{{Requires references|grade=A|msg=See the notes page, the books are plentiful I just don't have them to hand.}}
==Definition==
</noinclude>Given a [[topological space]], {{M|(X,\mathcal{J})}} and an [[equivalence relation]] on {{M|X}}, {{M|\sim}}<ref group="Note"><!--

-->Recall that for an [[equivalence relation]] there is a [[natural map]] that sends each {{M|x\in X}} to {{M|[x]}} (the [[equivalence class|equivalence class containing {{M|x}}]]) which we denote here as {{M|\pi:X\rightarrow\frac{X}{\sim} }}. 
Recall also that {{M|\frac{X}{\sim} }} denotes the [[set of all equivalence classes|set of all equivalence classes of {{M|\sim}}]].<!--

--></ref>, the ''quotient topology'' on {{M|\frac{X}{\sim} }}, {{M|\mathcal{K} }} is defined as:
* The set {{M|\mathcal{K}\subseteq\mathcal{P}(\frac{X}{\sim})}} such that:
** {{M|\forall U\in\mathcal{P}(\frac{X}{\sim})[U\in\mathcal{K}\iff \pi^{-1}(U)\in\mathcal{J}]}} or equivalently
* {{M|1=\mathcal{K}=\{U\in\mathcal{P}(\frac{X}{\sim})\ \vert\ \pi^{-1}(U)\in\mathcal{J}\} }}
In words:
* The topology on {{M|\frac{X}{\sim} }} consists of all those sets whose [[pre-image]] (under {{M|\pi}}) are [[open set|open]] in {{M|X}}<noinclude>
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Topology}}
</noinclude>