I am an undergraduate student currently learning differential forms to be used in the context of multivariable calculus, namely to prove the generalized Stokes' Theorem.
I'm studying by the book "Differential forms and applications" by Manfredo do Carmo and my question takes place in the section where he defines integration of differential forms.
So, let $M$ be a $n$-dimensional differentiable manifold and let $\omega$ be a $n$-differential form over an open subset $U\subseteq M$. Before he defines integration, he defines the $\textbf{support}$ $K$ of the $n$-form $\omega$ as the closure of the subset \begin{equation} A=\{p\in M\:|\:\omega(p)\neq0\}. \end{equation}
My question basically is:
Would it be the same to define the support $K$ of $\omega$ as the closure of the subset \begin{equation} A_{U}=\{p\in U\:|\:\omega(p)\neq0\}? \end{equation}
I am asking this because I feel that, logically speaking, since $\omega$ is defined only in $U$, the points $q$ on the manifold $M$ such that $\omega$ is not defined on $q$ would belong to the support since I feel like $\omega(q)\neq 0$ is vacuously true (not sure, though). I think that this was a typo in the book and he meant to define the form $\omega$ in the whole manifold $M$. However, if I want to define it in an open $U\subseteq M$, I would do it like in the second equation?
Does my question make sense? Sorry if this is kind of naive, but I really do want to understand things correctly and with no ambiguity. Thank you in advance.
As you said, the first definition doesn't make sense since $\omega$ is not defined on all of $M$. So the good definition is indeed your $A_U$. However, for differential form over $M$, the first definition is the good definition of the support.