I am trying to understand precisely the following paragraph:
Question
Why would he define the support $K$ of a form $\omega$ defined on an open set $U$ as a subset $K\subseteq M$ instead of a subset $K\subseteq U$? This doesn't make sense for me and can't be just a typo because he later says "assume that the support $K$ is contained in $U$", but this is sort of obvious from the definition, isn't it?
I'd appreciate any explanation. Thanks.
The set $K$ cannot be assumed to be contained in $U$ because it is a closure. In particular, if $\omega$ is nowhere zero on $U$, then $K=\overline{U}$, which probably is not a subset of $U$. This is why the condition $K \subseteq U$ is non-trivial; roughly speaking, it means $\omega$ is zero near the boundary of $U$.
On the other hand, it might have made more sense to define $A$ (the set that $K$ is the closure of) to be a subset of $U$, since the condition $\omega(p) \neq 0$ only really makes sense if $\omega$ is defined at $p$: that is, if $p \in U$. The way that paragraph is currently written, you should probably mentally read "$\omega(p) \neq 0$" as "$\omega(p)$ exists and is nonzero."