Definition:Partition of Unity (Topology)

Definition
Let $X$ be a topological space.

Let $\mathcal A = \{\phi_\alpha : \alpha \in A\}$ be a collection of $C^\infty$ functions $X \to \R$ such that:


 * 1. The collection $\{\operatorname{supp}\phi_\alpha : \alpha \in A\}$ of supports is locally finite


 * 2. For all $x \in X$, and for each $\alpha \in A$ $\phi_\alpha(x) \geq 0$


 * 3. For all $x \in X$, $\displaystyle \sum_{\alpha \in A} \phi_\alpha(x) = 1$.

Then $\{\phi_\alpha : \alpha \in A\}$ is a partition of unity on $X$.

If $\mathcal B = \{ U_\beta : \beta \in B \}$ is an open cover of $X$, and the collection $\{\operatorname{supp}\phi_\alpha : \alpha \in A\}$ is a refinement of $\mathcal B$ we say that $\mathcal A$ is subordinate to the cover $\mathcal B$.