Let $(X,\tau)$ be a [topological space](/page/Topological%20Space), let $\mathcal U \subset \tau$ be a locally finite [open cover](/page/Open%20Cover) of $X$, and let $(\phi_U)_{U \in \mathcal U}$ be a family of functions $\phi_U:X \to \mathbb R$ such that, for every $U \in \mathcal U$ and every $x \in X \setminus U$, one has $\phi_U(x)=0$. Then, for every $x \in X$, there exists an open neighbourhood $W \in \tau$ of $x$ such that the set

\begin{align*} \{U \in \mathcal U : \text{there exists } y \in W \text{ with } \phi_U(y) \neq 0\} \end{align*}

is finite.