Difference between revisions of "Pasting lemma"
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(Created page with "{{Stub page|grade=A*|msg=Create the closed pasting lemma and open pasting lemma pages. Do the proof, see page 58.9 in Lee's top manifolds if stuck, shouldn't be stuck}...") |
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* such that {{M|1=\forall \alpha,\beta\in I\forall x\in A_\alpha\cap A_\beta[f_\alpha(x)=f_\beta(x)]}} | * such that {{M|1=\forall \alpha,\beta\in I\forall x\in A_\alpha\cap A_\beta[f_\alpha(x)=f_\beta(x)]}} | ||
then{{rITTMJML}}: | then{{rITTMJML}}: | ||
− | * there exists a unique continuous map, {{M|f:X\rightarrow Y}}, such that {{M|f}}'s restriction to each {{M|A_\alpha}} is {{M|f_\alpha}} | + | * there exists a unique continuous map, {{M|f:X\rightarrow Y}}, such that {{M|f}}'s {{link|restriction|function}} to each {{M|A_\alpha}} is {{M|f_\alpha}} |
+ | |||
==Proof== | ==Proof== | ||
{{Requires proof|grade=A|msg=Do this, but remember it's the union of two other lemmas, so you can just write "by this, that" twice}} | {{Requires proof|grade=A|msg=Do this, but remember it's the union of two other lemmas, so you can just write "by this, that" twice}} |
Latest revision as of 07:07, 14 October 2016
Stub grade: A*
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Create the closed pasting lemma and open pasting lemma pages. Do the proof, see page 58.9 in Lee's top manifolds if stuck, shouldn't be stuck
Contents
[hide]- The closed pasting lemma and open pasting lemma are proved separately, this just unites the two.
Statement
Let (X,J) and (Y,K) be topological spaces, let {Aα}α∈I be either:
- An arbitrary open cover of X, or
- A finite closed cover of X
and let {fα:Aα→Y}α∈I be a family of continuous maps that agree where they overlap, formally:
- such that ∀α,β∈I∀x∈Aα∩Aβ[fα(x)=fβ(x)]
then[1]:
- there exists a unique continuous map, f:X→Y, such that f's restriction to each Aα is fα
Proof
Grade: A
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The message provided is:
The message provided is:
Do this, but remember it's the union of two other lemmas, so you can just write "by this, that" twice
References