Definition:Partition of Unity (Topology)

Definition
Let $X$ be a topological space.

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


 * $(1): \quad$ The set $\left\{{\operatorname{supp} \left({\phi_\alpha}\right): \alpha \in A}\right\}$ of supports is a locally finite cover of $X$


 * $(2): \quad \forall x \in X: \forall \alpha \in A: \phi_\alpha \left({x}\right) \ge 0$


 * $(3): \quad \displaystyle \forall x \in X: \sum_{\alpha \mathop \in A} \phi_\alpha \left({x}\right) = 1$

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