Construction of a partition of unity by Warner I'm reading the proof on the existence of a partition of unity, given by Warner. The proof is constructive, and to start it, he chose a coordinate system $(V,\tau)$ on which many restrictions are imposed:

I feel awkward about such an artificial setting. Is it always possible to make $\tau$ centered at a specific point? Is it always can be done to restrict $V$ within a certain set? To make our discussion efficient, some terminologies are given as follows.



Any advice is welcome. Thank you.
 A: It is possible. The general situation is that you have $p\in M$ and an open neigborhood $W$ of $p$. The claim is that there exists a coordinate system ($V,\tau)$ centered at $p$ such that $V \subset W$ and $\overline{C(2)} \subset \tau(V)$.
Take any coordinate system $(U,\varphi)$ in $\mathcal F$ such that $p \in U$. The map $\varphi : U \to U'$ is a homeomorphism onto an open $U' \subset \mathbb R^d$. Then also $(U \cap W, \psi : U \cap W \stackrel{\varphi}{\rightarrow} \varphi(U \cap W))$ is a coordinate system in $\mathcal F$ with $p \in U \cap W$. Thus we can assume w.l.o.g. that $U \subset W$.
Let $q = \varphi(p)$. The translation $T_q : \mathbb R^d \to \mathbb R^d, T_q(x) = x - q$, is a diffeomorphism and $(U,\varphi_q)$ with $\varphi_q : U \stackrel{\varphi}{\rightarrow} U' \stackrel{T_q}{\rightarrow} T_q(U')$ is a new coordinate system in $\mathcal F$ which satisfies $\varphi_q(p) = 0$. Since $T_q(U')$ is an open neighborhood of $0$, there exists $r > 0$ such that $\overline{C(r)} \subset T_q(U')$. The map $S_r : \mathbb R^d \to \mathbb R^d, S_r(x) = \frac{2}{r}x$, is a diffeomorphism such that $S_r(\overline{C(r)}) = \overline{C(2)}$. Now define
$\tau : U \stackrel{\varphi_q}{\rightarrow} T_q(U') \stackrel{S_r}{\rightarrow} S_r(T_q(U'))$. Then $(U,\tau) \in \mathcal F$.
What is done here is something like "norming" coordinate systems $(U,\varphi)$ with $p \in U$ with respect to their range $U' =\varphi(U)$. In fact you may achieve that $\varphi(p)$ is any prescribed point $q \in \mathbb R^d$ and that $U'$ is any prescribed open neighborhood of $q$. Frequently one takes $q = 0$ and $U' = \mathbb R^d$ or $U' =$ open unit ball etc.
