Lemma 1

Let U be an open subset of $\mathbf{R}^n$.

a) Let K be a compact subset of U and V an open neighbourhood of K in U. There exists a function $\varphi \in \mathscr{C}^{\infty}(U)$ with compact support contained in V such that $0\leqslant \varphi \leqslant 1$ and such that $\varphi (x) = 1$ for every $x\in K$;

b) For every locally finite open covering of U, there exists a partition of unity formed of functions of class $C^{\infty}$ which is subordinate to it.

This results from VAR, R1, p. 40, 5.3.6.