I have 3 different proposition and I really have problem to see in what they are that different.
Let $M$ a smooth manifold.
Prop 1 : Let $U\subset M$ an open and $C\subset U$ a compact. Then, there is a $f\in\mathcal C^\infty (M)$ s.t. $f|_C\equiv 1$ and $supp(f)\subset U$.
Prop 2 : Let $p\in M\backslash \partial M$. For all neighborhood $U_p$, there is $f\in\mathcal C^\infty (U)$ s.t. $f|_V\equiv 1$ where $V\subset U$ is a smaller neighborhood and $supp(f)\subset U$.
Prop 3 : Let $K\subset M$ a compact and $W\subset M$ an open s.t. $W\supset K$. Then, there is a $f\in\mathcal C^\infty (M)$ s.t. $f|_K\equiv 1$ and $supp(f)\subset W$.
Excepted some very little details, theses 3 propositions looks exactly the same to me (and almost equivalent). What are the subtlety between them ?
To me P1 and P3 are exactly the same. The difference is between P2 and P3.
P2) $f$ depend of $p$.
P3) You can take the same $f$ for all $p$ in $K$.