Smooth Urysohn lemma - Imported Knowlpedia Content

Statement

Let $M$ be a smooth manifold and let $A,B\subseteq M$ be disjoint closed sets. There exists a smooth map $f:M\to[0,1]$ such that $f|_A=0$ and $f|_B=1$ . Equivalently, if $A\subseteq U$ with $A$ closed and $U$ open, there is a smooth $f:M\to[0,1]$ satisfying $f=1$ on $A$ and $\operatorname{supp}f\subseteq U$ . No compactness of $A$ is required, and the support in the equivalent formulation need not be compact. The equalities hold on the entire prescribed closed sets, not merely at selected points.

Construction

Choose a locally finite smooth partition of unity adapted to the two-set cover $M\setminus A$ and $M\setminus B$ . The sum of the partition functions assigned to $M\setminus A$ has value $0$ on $A$ and $1$ on $B$ . Local finiteness makes the sum smooth. For the cutoff formulation, first choose a neighborhood $V$ of $A$ whose closure lies in $U$ , then separate $A$ from $M\setminus V$ . This partition-of-unity proof appears in Lee, chapter on smooth functions and partitions of unity.

Relationship to bump functions

When $A$ is compact and $U$ is a prescribed neighborhood, the cutoff can be chosen with compact support in $U$ , hence as a bump function. For noncompact $A$ , a function equal to $1$ on all of $A$ cannot have compact support; the smooth Urysohn lemma still provides a cutoff whose support is closed and contained in $U$ .

Conventions and scope

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2012. DOI record. Relevant: the chapter on smooth functions, bump functions, and partitions of unity.
Loring W. Tu, An Introduction to Manifolds, 2nd ed., Universitext, Springer, 2011. DOI record. Relevant: Chapter 13, partitions of unity and smooth separation.