> [!NOTE] Theorem > Let $A \subset \mathbb{R}^n$ and $\mathcal{O}$ be an open cover of $A$. Then there exists a set $\Phi$ of $C^\infty$ functions defined on $U= \cup_{O\in \mathcal{O}} O$ (an open set containing $A$), with the following properties > (1) For each $x\in U$ and $\varphi \in \Phi$, we have $0 \leq \varphi(x) \leq 1$. > (2) For each $x\in A$, there is an open neighbourhood $V_{x}$ of $x$ such that all but finitely many $\varphi \in \Phi$ are $0$ on $V_{x}$ > (3) For each $x\in A$, we have $\sum_{\varphi\in \Phi}\varphi(x) = 1$ (by (2) this sum is finite on $V_{x}$) > (4) For each $\varphi \in \Phi$, there exists an open set $O \in \mathcal{O}$ such that $\varphi = 0$ outside some closed set contained in $O$. > > A collection $\Phi$ satisfying (1)-(3) is called a smooth partition of unity for A. If Φ also satisfies (4), it is said to be subordinate to the cover $\mathcal{O}$. Proof: