Corollary

Given any locally finite open covering $(A_i)$ of a closed set $F$ in a normal space $X$, there exists a family $(f_i)$ of continuous real-valued functions $\geq 0$ on $X$, subordinate to the covering $(A_i)_{i \in I}$ and such that $\sum_{i \in I} f_i(x) = 1$ for all $x \in F$ and $\sum_{i \in I} f_i(x) \leq 1$ for all $x \in X$.

The family of sets consisting of $CF$ and the $A_i$ is a locally finite open covering of $X$. There is therefore a continuous partition of unity subordinate to this covering, consisting of a family $(f_i)_{i \in I}$ such that $\operatorname{Supp}(f_i) \subset A_i$ for each $i \in I$, and a function $g$ whose support is contained in the complement of $F$. The family $(f_i)$ clearly satisfies the required conditions.