So I am trying to solve this problem:
Given $M$ differentiable manifold, $P$ compact in $M$ and $U$ an open set such that $P \subset U$. Prove that there exists a differentiable functon $\phi: M \to [0,1]$ such that $\phi$ values $1$ in $P$, and its support is contained in $U$.
So I think I am missing a subtle detail, here is what I have been thinking:
For every $x \in P$, you can have local charts $(U_x,\phi_x)$, such that $U_x \subset U$ and $\phi_x(U_x)=B(\phi(x),\epsilon_x)$ for a $\epsilon_x > 0$, the ladder is just to make things easier. So the family of the open sets $U_x$ is an open cover of $P$, thus there exists $U_1,..,U_n$ (a little abuse of notation here) that make a subcover of $P$.
Without loss of generality I can suppose the $U_i$ are disjoint pairwise. and just define bump functions on neighborhoods of $\phi_i(U_i)$ for $i \leq n$, and since they are disjoint I can define a function that is $0$ outside those neighborhoods and values as the associated bump function inside its respective neigborhood . Like a bump function made with finite different bumps.
Now, the idea is to use this as a canvas to define sort of a composition of the local charts with this new function. The problem here is that it might not be well-defined, since some charts might send a point in $P$ outside of the neighboorhoods, in other words, this function is chart dependent.
So, if anybody can help me how to resolve from here or has a better suggestion it is welcomed, thanks.