Definition:Partition of Unity (Topology)

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

Definition

Let $X$ be a topological space.

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

$(1): \quad$ The set $\left\{{\operatorname{supp} \left({\phi_\alpha}\right)^\circ: \alpha \in A}\right\}$ of interiors of the 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 $X$.


Subordinate to Cover

Let $X$ be a topological space.

Let $\left\{{\phi_\alpha : \alpha \in A}\right\}$ be a partition of unity.

Let $\mathcal B = \left\{{U_\beta: \beta \in B}\right\}$ be an open cover of $X$.

Suppose the set $\left\{{\operatorname{supp} \left({\phi_\alpha}\right)^\circ : \alpha \in A}\right\}$ of interiors of supports is a refinement of $\mathcal B$


Then $\mathcal A$ is said to be subordinate to the cover $\mathcal B$.