View source for Quotient topology

* Need to add [[Equivalent conditions to a map being a quotient map]]
{{Refactor notice|grade=A|msg=This page is an embarrassment}}
==Definition==
There are a few definitions of the quotient topology however they do not conflict. This page might change shape while things are put in place
===[[Quotient topology/Equivalence relation definition|Quotient topology via an equivalence-relation definition]]===
{{:Quotient topology/Equivalence relation definition}}
: '''Claim 1: ''' {{M|\mathcal{K} }} is indeed a topology on {{M|\frac{X}{\sim} }}
===[[Quotient topology/Mapping to a set definition|Quotient topology via a mapping to a set definition]]===
{{:Quotient topology/Mapping to a set definition}}
: '''Claim 2: ''' these definitions are equivalent
==Immediate theorems==
The next two theorems demonstrate the purpose, the job if you will, of the quotient topology. The second (''[[passing to the quotient (topology)|passing to the quotient]]'') is the most important.
{{Begin Inline Theorem}}
===[[Universal property of the quotient topology]]===
{{Begin Inline Proof}}
{{:Universal property of the quotient topology/Statement}}
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
===[[Passing to the quotient (topology)|Passing to the quotient]]===
{{Begin Inline Proof}}
{{:Passing to the quotient (topology)/Statement}}
{{End Proof}}{{End Theorem}}
==Proof of claims==

==Notes==
<references group="Note"/>
==References==
<references/>
{{Topology navbox|plain}}
{{Definition|Topology}}
=OLD PAGE=
'''Note:''' [[Motivation for quotient topology]] may be useful
==Definition of the Quotient topology==
<math>
\begin{xy}
\xymatrix{
X \ar[r]^p \ar[dr]_f & Q \ar@{.>}[d]^{\tilde{f}}\\
 & Y
}
\end{xy}
</math>

==(OLD)Definition of Quotient topology==
If <math>(X,\mathcal{J})</math> is a [[Topological space|topological space]], <math>A</math> is a set, and <math>p:(X,\mathcal{J})\rightarrow A</math> is a [[Surjection|surjective map]] then there exists '''exactly one''' topology <math>\mathcal{J}_Q</math> relative to which <math>p</math> is a quotient map. This is the '''quotient topology''' induced by <math>p</math>

{{Begin Theorem}}
The quotient topology is actually a topology
{{Begin Proof}}
{{Todo|Easy enough}}
{{End Proof}}
{{End Theorem}}

==Quotient map==
Let {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} be [[Topological space|topological spaces]] and let {{M|p:X\rightarrow Y}} be a [[Surjection|surjective]] map.


{{M|p}} is a quotient map<ref>Topology - Second Edition - James R Munkres</ref> if we have <math>U\in\mathcal{K}\iff p^{-1}(U)\in\mathcal{J}</math>

That is to say <math>\mathcal{K}=\{V\in\mathcal{P}(Y)|p^{-1}(V)\in\mathcal{J}\}</math>


Also known as:
* Identification map

===Stronger than continuity===
If we had {{M|1=\mathcal{K}=\{\emptyset,Y\} }} then {{M|p}} is automatically continuous (as it is surjective), the point is that {{M|\mathcal{K} }} is the [[Topology#Finer.2C_Larger.2C_Stronger|largest topology]] we can define on {{M|Y}} such that {{M|p}} is continuous

===Theorems===
{{Begin Theorem}}
Theorem: The quotient topology, {{M|\mathcal{Q} }} is the largest topology such that the quotient map, {{M|p}}, is continuous. That is to say any other topology such on {{M|Y}} such that {{M|p}} is continuous is contained in the quotient topology
{{Begin Proof}}
For a map {{M|p:X\rightarrow Y}} where {{M|(X,\mathcal{J})}} is a [[Topological space]] we will show that the topology on {{M|Y}} given by:
* <math>\mathcal{Q}=\{V\in\mathcal{P}|p^{-1}(V)\in\mathcal{J}\}</math>
is the largest topology on {{M|Y}} we can have ''such that'' {{M|p}} is [[Continuous map|continuous]]

'''Proof method:''' suppose there's a larger topology, reach a contradiction.

Suppose that {{M|\mathcal{K} }} is any topology on {{M|Y}} and that {{M|p:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})}} is continuous.

Suppose that {{M|\mathcal{K}\ne\mathcal{Q} }}


Let {{M|V\in\mathcal{K} }} such that {{M|V\notin \mathcal{Q} }}

By continuity of {{M|p}}, {{M|p^{-1}(V)\in\mathcal{J} }}

This contradicts that {{M|V\notin\mathcal{Q} }} as {{M|\mathcal{Q} }} contains all subsets of {{M|Y}} whose inverse image (preimage) is open in {{M|X}}


Thus any topology on {{M|Y}} where {{M|p}} is continuous is contained in the quotient topology
{{End Proof}}
{{End Theorem}}
This theorem hints at the [[Characteristic property of the quotient topology]]

==Quotient space==
Given a [[Topological space]] {{M|(X,\mathcal{J})}} and an [[Equivalence relation]] {{M|\sim}}, then the map: <math>q:(X,\mathcal{J})\rightarrow(\tfrac{X}{\sim},\mathcal{Q})</math> with <math>q:p\mapsto[p]</math> (which is a quotient map) is continuous (as above)

The topological space {{M|(\tfrac{X}{\sim},\mathcal{Q})}} is the ''quotient space''<ref>Introduction to topological manifolds - John M Lee - Second edition</ref> where {{M|\mathcal{Q} }} is the topology induced by the quotient


Also known as:
* Identification space

==See also==
* [[Open and closed maps]]

==References==
<references/>


{{Definition|Topology}}