4.15 Partitions of Unity

Definition 4.15.1 (Partition of Unity). Let $X$ be a topological space and $E \subset X$, then a partition of unity on $E$ is a family $\seqi{f}\subset C(X; [0, 1])$ such that:

1. For each $x \in X$, there exists $U \in \cn(x)$ such that $\bracs{i \in I|f_i|_U \ne 0}$ is finite.

2. $\sum_{i \in I}f_{i}|_{E} = 1$.

For any open cover $\mathcal{U}$ of $X$, $\seqi{f}$ is subordinate to $\mathcal{U}$ if for every $i \in I$, there exists $U \in \mathcal{U}$ such that $\supp{f_i}\subset \mathcal{U}$.