# Definition:Partition of Unity (Topology)

This page is about Partition of Unity in the context of topology. For other uses, see Partition of Unity.

## 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 $\set {\map \supp {\phi_\alpha}^\circ: \alpha \in A}$ 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 $X$.

### Subordinate to Cover

Let $X$ be a topological space.

Let $\AA = \set {\phi_\alpha : \alpha \in A}$ be a partition of unity.

Let $\BB = \set {U_\beta: \beta \in B}$ be an open cover of $X$.

Let the set $\set {\map \supp {\phi_\alpha}^\circ: \alpha \in A}$ of interiors of supports be a refinement of $\BB$.

Then $\AA$ is said to be subordinate to the cover $\BB$.