Let $X$ be a locally compact Hausdorff space that is $\sigma$-compact, and let $\{U_\alpha\}_{\alpha \in A}$ be an open cover of $X$. Then there exists a partition of unity subordinate to this cover: a family $\{\psi_\alpha\}_{\alpha \in A}$ of continuous functions $\psi_\alpha: X \to [0, 1]$ such that:

1. Each $\psi_\alpha$ has compact support contained in $U_\alpha$: $\operatorname{supp}(\psi_\alpha) \subset U_\alpha$. 2. The collection $\{\operatorname{supp}(\psi_\alpha)\}$ is **locally finite**: every point $x \in X$ has a neighbourhood intersecting only finitely many of the supports. 3. $\sum_{\alpha \in A} \psi_\alpha(x) = 1$ for all $x \in X$ (the sum is finite at each point by local finiteness).