= Solution
Let $\Omega\subset\mathbb R^n$ be an <open set> and $(U_\alpha)_{\alpha\in A}$ an <open cover> of $\Omega$. There exist nonnegative <smooth functions> $(\chi_j)_{j\in J}$ and choices $\alpha(j)$ such that
$$
\boxed{\operatorname{supp}_\Omega\chi_j\subset U_{\alpha(j)},\qquad
\sum_{j\in J}\chi_j(x)=1\quad(x\in\Omega),}
$$
and the family of <supports> is locally finite: every point has a neighborhood meeting only finitely many of them. This is a smooth <partition of unity> subordinate to the cover, allowing a refinement of its indexing. In Euclidean <open sets>, one can choose $J$ countable and the individual <supports> compact in their assigned cover members. Local finiteness ensures that the sum is smooth and that only finitely many terms meet any given <compact set>.
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