Let $\mathcal A$ be a collection of open sets in $\mathbb R^n$ and let $A$ be their union. A partition of unity subordinate to $\mathcal A$ is a countable collection of smooth functions ${\phi_i}_{i\in I}$ such that $\phi_i:\mathbb R^n \to \mathbb R$ such that
- each $\phi_i \geq 0$.
- each $\phi_i$ has compact support contained in some $U \in \mathcal A$.
- for all points $a$ of $A$, there exists an open neighborhood $V$ of $a$ such that all but finitely many of the $\phi_i$ are zero on $V$.
- for all points $a$ in $A$, $\sum\limits_{i\in I} \phi_i(a) = 1$.
Wikidata ID: Q191690