Pasting lemma
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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:
Do this, but remember it's the union of two other lemmas, so you can just write "by this, that" twice
References