\begin{definition}[Definition:Partition of Unity (Topology)]
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 $\set {\map \supp {\phi_\alpha}^\circ: \alpha \in A}$ 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 $X$.
\end{definition}