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 collection $\left\{{\operatorname{supp} \left({\phi_\alpha}\right): \alpha \in A}\right\}$ of supports is locally finite


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


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

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

Subordinate
If:
 * $\mathcal B = \left\{{U_\beta: \beta \in B}\right\}$ is an open cover of $X$

and
 * the set $\left\{{\operatorname{supp} \left({\phi_\alpha}\right): \alpha \in A}\right\}$ is a refinement of $\mathcal B$

then $\mathcal A$ is defined as subordinate to the cover $\mathcal B$.