Definition:Partition of Unity (Topology)

Definition
Let $X$ be a topological space.

Let $\AA = \set {\phi_\alpha : \alpha \in A}$ be a collection of continuous mappings $X \to \R$ such that:


 * $(1): \quad$ The set $\set {\map {\operatorname {supp} } {\phi_\alpha}^\circ: \alpha \in A}$ of interiors of the supports is a locally finite cover of $X$


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


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

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