Warner - Proof of existence of a partition of unity I'm studying the proof of existence of a partition of unity given by Warner, and a statement(underlined with red ink) confuses me:


Why do those supports form a locally finite collection of sets? Thanks.

 A: Although it is not said in the question what the $G_i$ are, it is obvious from the context that we have $G_i$ open, $\overline{G}_i$ compact, $\overline{G}_i \subset G_{i+1}$ and $\bigcup_i G_i = M$.
Let us first recall the construction of the $\psi_p$ for $p\in M$. With a little change of notation we start by defining $r(p)$ as the largest integer such that $p \in M \setminus \overline{G}_{r(p)}$. Then after some constructions we end with $\psi_p$ whose
support is contained in $G_{r(p) +2} \setminus \overline{G}_{r(p)}$ plus open neigborhoods $W_p$ of $p$ on which $\psi_p$ takes the value $1$.
Let us now have a look at the $\psi_j$. For each $i$ the authors chooses finitely many $p_i^k \in \overline{G}_i \setminus G_{i-1}$ such that the $W_i^k = W_{p_i^k}$ cover the compact set $\overline{G}_i \setminus G_{i-1}$. The support $S_i^k$ of $\psi_i^k = \psi_{p_i^k}$ is contained in $G_{r(p_i^k) +2} \setminus \overline{G}_{r(p_i^k)}$. Since $p_i^k \in \overline{G}_i \setminus G_{i-1} \subset \overline{G}_i \setminus \overline{G}_{i-2}$ we conclude that $i-2  \le r(p_i^k) < i$. Hence $S_i^k \subset G_{i+1} \setminus \overline{G}_{i-2}$.
Now let us check that the $S_i^k$ form a locally finite family. Consider $q \in M$. Then $q \in G_\ell$ for some $\ell$. But $G_\ell$ does not intersect $S_i^k$ for $i-2 \ge \ell$. Thus $U_\ell \cap S_i^k \ne \emptyset$ is only possible when $i < \ell +2$ so that the number of $S_i^k$ intersecting $G_\ell$ is finite.
