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 !