I'm working through some lecture notes on differential geometry and I have a question in one of the proofs there, however I can't give precisely the data using in those lecture notes because it will be very long post, so I will just give the idea of the context of the proof.
Let $M$ be a compact oriented manifold. Let $\alpha$ be a closed differential form on $M$. Suppose that there exists another closed differential form $\beta$ which vanishes on a subset $C$ of $M$ and which satisfies $\int_M \alpha = \int_M \beta$.
When the author wants to use this information he says: let $U$ be an open set which contains $C$, then we have $\int_M \alpha = \int_U \beta$ and he continues the proof.
My question is why it wouldn't be possible to say $\int_M \alpha =\int_C \beta$, I don't understand why he works with an open set of $C$ rather than $C$ ?