Bump Functions, part 2
Let be an open subset of , and be a set whose closure is contained within . I say that there is a nonnegative smooth function which is identically on , and which is supported within .
To find this function, we start with a cover of . Specifically, let be one set of the cover, and let be the other set. Then we know that there is a countable smooth partition of unity subordinate to this cover. That is, for every we either have supported in , or supported in (or possibly both).
In fact, no matter what countable partition we come up with, we can take all the supported within and add them all up into one function , and then take all the remaining functions and add them all up into one function . Then is a partition of unity subordinate to our cover, and I say that is exactly the function we’re looking for.
Indeed, as a part of a partition of unity, is a nonnegative smooth function, and we know it’s supported in . The only thing we need to determine is if it’s identically on . But for we know that , and yet we also know that , since is supported in . Thus we must have , and is indeed our bump function.