Equality between support for a function and closed union of elements of a partition of unity (Proof from John Lee's Smooth Manifolds)?

Question

I have a minor question in the following proof from John Lee's Intro to Smooth Manifolds:

At the end, there is the equality $\mathrm{supp}\tilde{f}=\overline{\bigcup_{p\in A}\mathrm{supp}\psi_p}$. It's clear that $\tilde{f}(x)\neq 0$ requires that $x\in\bigcup\mathrm{supp}\psi_p$, so the $\subseteq$ containment is fine, which gives the desired result of the lemma. Why is there equality though? Is it possible you could have a point in that closure, but the values of $\tilde{f}_p(x)$ is the sum defining $\tilde{f}$ somehow cancel things out?

Answer 1

You're right, this is a mistake. I've added a correction to my online errata list. Thanks for pointing it out.

Answer 2

I agree that this is a mistake in the text (that doesn't affect the truth of the lemma, as you noted). For instance, if you start with the constant function $f=0$ and apply the construction, you get $\operatorname{supp}\widetilde{f}=\emptyset$, which can't be equal to the right side unless $A =\emptyset$ as well.