Definition:Partition of Unity (Topology)

From ProofWiki
Jump to navigation Jump to search

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 $\set {\map {\operatorname {supp} } {\phi_\alpha}^\circ: \alpha \in A}$ of interiors of the supports is a locally finite cover of $X$
$(2): \quad \forall x \in X: \forall \alpha \in A: \map {\phi_\alpha} x \ge 0$
$(3): \quad \displaystyle \forall x \in X: \sum_{\alpha \mathop \in A} \map {\phi_\alpha} x = 1$


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


Subordinate to Cover

Let $X$ be a topological space.

Let $\left\{{\phi_\alpha : \alpha \in A}\right\}$ 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 $\mathcal B$.