---- Legacy notes: [[partition of unity (deprecated)]], [[Partition of Unity on a manifold (deprecated)]] > [!definition] Definition. ([[partition of unity]]) > > Let $M$ be a [[smooth manifold]]. For any [[cover|open cover]] $\{ U_{\alpha} \}$, there exists a countable collection of nonnegative functions $\rho_{i} \in C^{\infty}(M)$, $i=1,2,\dots$, such that the following hold: > > 1. For a given $i$, $\text{supp }\rho_{i}= \overline{\{ x \in M: \rho_{i}(x) \neq 0 \}}$ is [[compact]] and contained in some $U_{\alpha}$ > 2. The collection is [[locally finite]]: each $x \in M$ has a [[neighborhood]] $W_{x}$ such that $\rho_{i}(W_{x})=0$ for all but finitely many $i$ > 3. $\sum_{i} \rho_{i}(x)=1$ for all $x \in M$ (this sum is finite by $(2)$) > > The collection $\{ \rho_{i} \}$ is called a **partition of unity subordinate to $\{ U_{\alpha} \}$** ---- #### (topological partition of unity much easier than smooth one) [[closure]] ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```