Pasting lemma

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

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:

1. An arbitrary open cover of [ilmath]X[/ilmath], or
2. 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:

• 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

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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