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I may be overlooking something very simple, but I am reading the proof of Lemma 2.26 in Lee's Smooth Manifolds 2nd ed., and am a bit confused by the following statement:

For each $p\in A$, choose a neighborhood $W_p$ of $p$ and a smooth function $\tilde f_p:W_p\rightarrow \mathbb{R}^k$ that agrees with $f$ on $W_p\cap A$.

How do we know such a neighborhood/function exists? I'm not sure how you would construct such a function, since you don't have a definition for $f$ away from $A$. Any insight here would be greatly appreciated.

r0b
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  • I don't have Lee's book, but I can tell you what I've seen this mean in other cases. We say that $f$ is of class $C^k$ on $A$ if for each $p \in A$ there exists a neighborhood $W_p$ of $p$ and a $C^k$ function $\tilde{f}_p$ on $W_p$ whose restriction to $A$ is $f$, i.e., $\left. \tilde{f}_p \right|_A = f$. – Jon Warneke Dec 20 '15 at 17:44
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    It might be helpful for those who don't have the book with them to state Lemma 2.26 and state what $A$ is. – TomGrubb Dec 20 '15 at 17:47

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It's always a good idea to read the definitions. In the paragraph immediately before that lemma, I wrote:

Suppose $M$ and $N$ are smooth manifolds with or without boundary, and $A\subseteq M$ is an arbitrary subset. We say that a map $F\colon A\to N$ is smooth on $\boldsymbol A$ if it has a smooth extension in a neighborhood of each point: that is, if for every $p\in A$ there is an open subset $W \subset M$ containing $p$ and a smooth map $F\colon W\to N$ whose restriction to $W \smallsetminus A$ agrees with $F$.

(As a postscript, I should note that the definition above works fine in the situation considered in Lemma 2.26. However, I later realized because of this question that my definition needed to be modified in the case that the target manifold $N$ has nonempty boundary. A corrected definition in that case is given in my list of corrections.)

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Jack Lee
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  • Why does the definition need to be modified in the boundary case, @Jack Lee? I tried thinking through it but didn't succeed – Martin Geller Nov 23 '22 at 14:10
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    @MartinGeller: My answer to the question I linked to above gave an example of a map into a manifold with boundary that should be smooth, but wouldn't be smooth by the definition I originally gave. – Jack Lee Nov 23 '22 at 16:50