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I really need your help in understanding the following statement in the proof of the extension lemma in Lee's book: Let $A \subseteq M^n $ be a closed submanifold of dimension $k$ , and let $F:A \to \mathbb{R} $ be a smooth function. We want to extend this function to the entire $M$ . THe problem is that I can't understand how to do it locally- Let $ p \in A$ and let $ W_p $ be a neighborhood of $ p$ in $ M$ . I only know how to extend $F$ to a slice chart of $ p$ using projection . How can I do it for $ W_p$ ? The other problem is that the statement should also be true when $A$ is a closed subset of $M$, not necessarily a closed submanifold :(

i.e. how can I extend the function $F$ to a smooth function on $ W_p$ ?

Hope you'll be able to help me !

Thanks !

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    I assume, you are asking for smooth extension, I am assuming that $M$ is locally compact and hence $M$ admits Smooth partition of unity.2012-02-27
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    When $A$ is just an arbitrary closed subset of $M$, the statement follows from a partition of unity argument and [Whitney's extension theorem](http://en.wikipedia.org/wiki/Whitney_extension_theorem), which is a nontrivial result in harmonic analysis of Euclidean spaces. The submanifold case is much, much easier.2012-02-27
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    I'm not that sure about the Whitney's extension theorem, but thanks, I think I understand it now.2012-02-27

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