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 John Armstrong | Differential Topology, Topology

3 Comments »

[…] of homeomorphic to a ball in , and we can certainly find within such a neighborhood. Anyhow, we know that there exists a bump function which is identically on and supported within . We can thus […]

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[…] each . We let be a neighborhood of whose closure is contained in . We know we can find a smooth bump function supported in and with on […]

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[…] of then . Equivalently (given linearity), if in a neighborhood of then . But then we can pick a bump function which is on a neighborhood of and outside of . Then we have […]

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