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 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.
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