Theorem: Let $A\subseteq \mathbb{R}^d$ be compact, and let $O$ be an open cover of $A$. Then there is a partition of unity subordinate to $O$.
Below I give Spivak's proof and my slight simplification. Is my proof (the simplified version of Spivak's) correct?
Spivak's proof:
My Proof:
Let $U_1,\ldots, U_n$ be a finite subcover of $A$ in $O$. By Lemma 1 there are compact sets $D_i$ such that $$A\subseteq \bigcup_{i=1}^nD_i^{\circ} \ \ \ \ \text{and} \ \ \ \ D_i\subseteq U_i.$$ Let $U:=U_1\cup \ldots \cup U_n$. By Lemma 2 there are functions $\psi_i:U\to[0,1]$ so that
- $\psi_i$ is $C^{\infty}$.
- $\psi_i$ is $1$ on $D_i$.
- $\psi_i$ is $0$ outside of some closed set contained in $U_i$. Define $$\phi_i : U\to [0,1] : x\mapsto \frac{\psi_i(x)}{\psi_1(x) + \ldots + \psi_n(x)}.$$ The partition of unity $\Phi:=\{\phi_1,\ldots ,\phi_n\}$ is subordinate to $O$.
Lemma 1 (akin to Spivak's Problem 1-22):
- Let $U$ be open and $K\subseteq U$ compact. Then there is a compact set $D$ such that $$K\subseteq D^{\circ} \ \ \ \ \text{and} \ \ \ \ D\subseteq U.$$
- Let $\{U_1, \ldots , U_n\}$ be an open cover of the compact set $K$. Then there are compact sets $D_i$ such that $$K \subseteq \bigcup_{i=1}^nD_i^{\circ} \ \ \ \ \text{and} \ \ \ \ D_i\subseteq U_i.$$
Lemma 2 (akin to Spivak's Problem 2-26): Let $U$ be open and $D\subseteq U$ be compact. Then there is a function $f:\mathbb{R}^n\to\mathbb{R}^n$ such that
- $f$ is $C^{\infty}$.
- $f$ is $1$ on $D$.
- $f$ is $0$ outside of some closed set contained in $U$.