Difference between revisions of "Quotient topology"
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| + | <math>\require{AMScd}</math> | ||
| + | <math>\begin{equation}\begin{CD} | ||
| + | S^{{\mathcal{W}}_\Lambda}\otimes T @>j>> T\\ | ||
| + | @VVV @VV{\End P}V\\ | ||
| + | (S\otimes T)/I @= (Z\otimes T)/J | ||
| + | \end{CD}\end{equation}</math> | ||
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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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====Stronger than continuity==== | ====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 | 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 | ||
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| + | See [[Motivation for quotient topology]] for a discussion on where this goes. | ||
==Definition== | ==Definition== | ||
Revision as of 09:22, 7 April 2015
\require{AMScd} \begin{equation}\begin{CD} S^{{\mathcal{W}}_\Lambda}\otimes T @>j>> T\\ @VVV @VV{\End P}V\\ (S\otimes T)/I @= (Z\otimes T)/J \end{CD}\end{equation}
Quotient map
Let (X,\mathcal{J}) and (Y,\mathcal{K}) be topological spaces and let p:X\rightarrow Y be a surjective map.
p is a quotient map[1] if we have U\in\mathcal{K}\iff p^{-1}(U)\in\mathcal{J}
Notes
Stronger than continuity
If we had \mathcal{K}=\{\emptyset,Y\} then p is automatically continuous (as it is surjective), the point is that \mathcal{K} is the largest topology we can define on Y such that p is continuous
See Motivation for quotient topology for a discussion on where this goes.
Definition
If (X,\mathcal{J}) is a topological space, A is a set, and p:(X,\mathcal{J})\rightarrow A is a surjective map then there exists exactly one topology \mathcal{J}_Q relative to which p is a quotient map. This is the quotient topology induced by p
TODO: Munkres page 138
References
- Jump up ↑ Topology - Second Edition - James R Munkres
