I have a question on the proof of Lemma 10.29 in Prof. John M Lee's book "Introduction to Smooth Manifolds" (second edition), called the bundle homomorphism characterization lemma. My questions are essentially related to the use of the extension lemma for smooth functions (Lemma 2.26) and for vector bundles (Lemma 10.12).
Let me briefly recall the part of the proof I am having trouble with. In the second paragraph of page 263, let $\tau\in\Gamma(M, E)$ be a smooth section of the smooth vector bundle $E\to M$ (of rank $k$) satisfying $\tau(p)=0$, and $(\sigma_1, \ldots, \sigma_k)$ a smooth local frame for $E\to M$ over a neighborhood $U$ of $p$. Write $\tau=u^i\sigma_i$ for some unique $u^i\in C^\infty(U)$. Then $\tau(p)=0$ implies $u^i(p)=0$ for every $1\leq i\leq k$. Then by the extension lemmas for vector bundles and for functions (I presume the author is talking about Lemma 2.26 and Lemma 10.12), there exist smooth global sections $\tilde{\sigma}_i\in\Gamma(M, E)$ that agree with $\sigma_i$ is a neighborhood of $p$, and smooth functions $\tilde{u}^i\in C^\infty(M)$ that agree with $u^i$ in some neighborhood of $p$.
The questions I have are:
The smooth functions $u^i$ and smooth local sections $\sigma^i$ are defined on $U$, an open subset of $M$. I am not sure how to apply Lemma 2.26 and Lemma 10.12 (if I did not misunderstood what the author said in the book) as these lemmas are stated for smooth functions and sections defined on a closed subset of $M$, say $A$.
Moreover, when it says the smooth extensions agree with $u^i$ and $\sigma^i$ in a neighborhood of $p$, I think the author meant open neighborhood. But it seems that Lemma 2.26 and Lemma 10.12 state that the resulting extension agree with the original one on the closed subset $A$. Of course, one could take the interior of the closed subset $A$ in question, but I am not sure if this is what the author meant.
I was thinking if it is possible that the argument is like the one given in Proposition 8.7. That is, since $u^i(p)=0$ and $\sigma^i(p)$ is a nonzero vector, by Lemma 2.26 and Lemma 10.12 one can find smooth extensions of the functions $\{p\}\to\mathbb{R}$ and sections $\{p\}\to E$, where the closed subset $A$ is the singleton $p$. But then by 2 above, I don't see how these smooth extensions agree with $u^i$ and $\sigma^i$ in a (open) neighborhood of $p$.
Please let me know if I miss something here. Thanks~
It seems one can use a general fact for a Hausdorff and locally compact space $M$, namely, in such a space, for every point $x$ and every open neighborhood $U$ of $x$, there exists a precompact open neighborhood $V$ of $x$ such that $\overline{V}\subseteq U$. The compactness of $\overline{V}$ implies that it is closed in $M$.
Well, seems like I forgot all the point-set topology