Difference between revisions of "Pasting lemma"
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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
- The closed pasting lemma and open pasting lemma are proved separately, this just unites the two.
Statement
Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, let [ilmath]\{A_\alpha\}_{\alpha\in I} [/ilmath] be either:
- An arbitrary open cover of [ilmath]X[/ilmath], or
- A finite closed cover of [ilmath]X[/ilmath]
and let [ilmath]\{f_\alpha:A_\alpha\rightarrow Y\}_{\alpha\in I} [/ilmath] be a family of continuous maps that agree where they overlap, formally:
- such that [ilmath]\forall \alpha,\beta\in I\forall x\in A_\alpha\cap A_\beta[f_\alpha(x)=f_\beta(x)][/ilmath]
then[1]:
- there exists a unique continuous map, [ilmath]f:X\rightarrow Y[/ilmath], such that [ilmath]f[/ilmath]'s restriction to each [ilmath]A_\alpha[/ilmath] is [ilmath]f_\alpha[/ilmath]
Proof
Grade: A
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
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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
