# The Unapologetic Mathematician

## Bump Functions, part 2

As an immediate application of our partitions of unity, let’s show that we can always get whatever bump functions we need.

Let $U$ be an open subset of $M$, and $V$ be a set whose closure $\bar{V}$ is contained within $U$. I say that there is a nonnegative smooth function $\phi:M\to\mathbb{R}$ which is identically $1$ on $\bar{V}$, and which is supported within $U$.

To find this function, we start with a cover of $M$. Specifically, let $U$ be one set of the cover, and let $M\setminus\bar{V}$ be the other set. Then we know that there is a countable smooth partition of unity subordinate to this cover. That is, for every $k$ we either have $\phi_k$ supported in $U$, or $\phi_k$ supported in $M\setminus\bar{V}$ (or possibly both).

In fact, no matter what countable partition we come up with, we can take all the $\phi_k$ supported within $U$ and add them all up into one function $\phi$, and then take all the remaining functions and add them all up into one function $\psi$. Then $\{\phi,\psi\}$ is a partition of unity subordinate to our cover, and I say that $\phi$ is exactly the function we’re looking for.

Indeed, as a part of a partition of unity, $\phi$ is a nonnegative smooth function, and we know it’s supported in $U$. The only thing we need to determine is if it’s identically $1$ on $\bar{V}$. But for $p\in\bar{V}$ we know that $\phi(p)+\psi(p)=1$, and yet we also know that $\psi(p)=0$, since $\psi$ is supported in $M\setminus\bar{V}$. Thus we must have $\phi(p)=1$, and $\phi$ is indeed our bump function.

March 16, 2011 - Posted by | Differential Topology, Topology