Partition of Unity

Statement

Definition. Let $(X,\tau )$ be a topological space and let \{U_i\}_i$$ be an [[Open Set|open]] [[Topological Cover|topological cover|cover]] of X$.A_{p}artitionofunit{y}_{a}ssociatedwiththiscoverisacollectionof[[ContinuousFunction|continuousfunctions]]${\varphi_i}_i$ such that:

1. Each ${\phi }_i$ is a non-negative function with compact support (i.e., it is zero outside of some compact set).
2. The support of each ${\phi }_i$ is contained in $U_i$.
3. The sum of all ${\phi }_i$ over all $i$ is equal to 1, i.e., for every $x\in X$, $$\sum _i{\phi }_i(x)=1.$$

Remarks

Remark. Using a partition of unity, we can construct a smooth function on $X$ by taking a linear combination of the functions ${\phi }_i$ multiplied by smooth functions on $U_i$. Specifically, if $\{f_i{\}}_i$ is a collection of smooth functions on $U_i$, then the function

$$\begin{array}{rrcl}f:& X& \to & {\bigcup }_i{f}_i(U_i)\\ & x& \mapsto & {\sum }_i{\phi }_i(x)f_i(x)\end{array}$$

is a smooth function on $X$.

The partition of unity is a powerful tool in mathematics because it allows us to construct global objects (such as a smooth function on an entire manifold) by gluing together local data (the smooth functions on each open set in the cover). This technique is used in many areas of mathematics, including differential geometry, algebraic topology, and partial differential equations.

References

• ChatGPT
• Rowland, Todd. “Partition of Unity.” From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PartitionofUnity.html