# 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

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