Difference between revisions of "Quotient topology"
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| + | '''Note:''' [[Motivation for quotient topology]] may be useful | ||
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| + | ==Definition== | ||
| + | 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> | ||
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| + | {{Begin Theorem}} | ||
| + | The quotient topology is actually a topology | ||
| + | {{Begin Proof}} | ||
| + | {{Todo|Easy enough}} | ||
| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
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==Quotient map== | ==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. | 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. | ||
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{{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> | {{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> | ||
| − | + | ===Stronger than continuity=== | |
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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 | 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 | ||
| − | + | {{Todo|Now we can explore the characteristic property (with {{M|\text{Id}:\tfrac{X}{\sim}\rightarrow\tfrac{X}{\sim} }} ) for now}} | |
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==References== | ==References== | ||
Revision as of 11:13, 7 April 2015
Note: Motivation for quotient topology may be useful
Definition
If [math](X,\mathcal{J})[/math] is a topological space, [math]A[/math] is a set, and [math]p:(X,\mathcal{J})\rightarrow A[/math] is a 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]
The quotient topology is actually a topology
TODO: Easy enough
Quotient map
Let [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] be topological spaces and let [ilmath]p:X\rightarrow Y[/ilmath] be a surjective map.
[ilmath]p[/ilmath] is a quotient map[1] if we have [math]U\in\mathcal{K}\iff p^{-1}(U)\in\mathcal{J}[/math]
Stronger than continuity
If we had [ilmath]\mathcal{K}=\{\emptyset,Y\}[/ilmath] then [ilmath]p[/ilmath] is automatically continuous (as it is surjective), the point is that [ilmath]\mathcal{K} [/ilmath] is the largest topology we can define on [ilmath]Y[/ilmath] such that [ilmath]p[/ilmath] is continuous
TODO: Now we can explore the characteristic property (with [ilmath]\text{Id}:\tfrac{X}{\sim}\rightarrow\tfrac{X}{\sim} [/ilmath] ) for now
References
- ↑ Topology - Second Edition - James R Munkres
