Definition 3

Given a family $(A_i)_{i \in I}$ of subsets of a topological space $X$, a family $(f_i)_{i \in I}$ of real-valued functions defined on $X$ is said to be subordinate to the family $(A_i)_{i \in I}$ if we have $\operatorname{Supp}(f_i) \subset A_i$ for each index $i \in I$.

A continuous partition of unity on $X$ is any family $(f_i)_{i \in I}$ of real-valued continuous functions $\geq 0$ on $X$ whose supports form a locally finite family and which are such that $\sum_{i \in I} f_i(x) = 1$ for all $x \in X$.