Difference between revisions of "Quotient topology"
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| + | ==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. | ||
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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> | ||
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| + | ===Notes=== | ||
| + | ====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 | ||
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==Definition== | ==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> | 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> | ||
{{Todo|Munkres page 138}} | {{Todo|Munkres page 138}} | ||
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| + | ==References== | ||
| + | <references/> | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 08:43, 7 April 2015
Quotient map
Let (X,J) and (Y,K) be topological spaces and let p:X→Y be a surjective map.
p is a quotient map[1] if we have U∈K⟺p−1(U)∈J
Notes
Stronger than continuity
If we had K={∅,Y} then p is automatically continuous (as it is surjective), the point is that K is the largest topology we can define on Y such that p is continuous
Definition
If (X,J) is a topological space, A is a set, and p:(X,J)→A is a surjective map then there exists exactly one topology JQ 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
