I have a question over theorem 3-11 in Spivak (see below)
They mention 'some open set $U$' containing $A$. I have a simple question: is it correct that we could take $$ U=\bigcup_i D_i^o, $$ where $D_i^o$ is the interior of $D_i$. I am hesitating because they use the letter $U$ instead of $D$, but using the interiors of $D_i$ is what makes most sense to me.
Yes, you are right: you can take$$U=\bigcup_{i=1}^n\mathring{D_i}.$$That's almost certainly what Spivak had in mind. And please note that Michael Spivak is a human being, not a committee; I wrote this because you wrote “they” in your question. He he wrote several books and therefore perhaps that you could have told us which one it is (perhaps Calculus on manifolds).