I'm reading "Introduction to Manifolds" 2nd edition by Loring W. Tu and have a question.
Remark 8.2 of this book says that
"If $U$ is an open set containing $p$ in $M$, then the algebra $C_p^{\infty}(U)$ of germs of $C^{\infty}$ functions in $U$ at $p$ is the same as $C_p^{\infty}(M)$. Hence, $T_pU$ = $T_pM$."
But I can't understand this sentence. $C_p^{\infty}(U)$ is clearly contained in $C_p^{\infty}(M)$ but I don't know the other inclusion.
So I browsed "Introduction to Smooth Manifolds" by John M.Lee and found proposition 3.9 saying that
"Let $M$ be a smooth manifold with or without boundary, let $U \subset M$ be an open subset, and let $i:U{\hookrightarrow}M$ be an inclusion map. For every $p\in U$, the differential $di_p: T_pU {\hookrightarrow} T_pM$ is an isomorphism." followed by the sentences of proof of this proposition.
The key point of this proposition is same as Remark 8.2. of Tu's book in my thought. And I think that Remark 8.2 of Tu's book is less rigorous. Am I right?
This is another question. Lee's book contains exercises that require to prove that real projective space is a manifold by showing it is second countable and Hausdorff. Is that short and obvious? Tu's book assign much more pages to prove that.
