I am having trouble understanding this claim about the existence of the sets $U_j$ in the following theorem from Spivak's differential geometry textbook. I tried proving like this: Firstly, we may take a covering which has no subcover and refine it to produce open sets $U_j$ diffeomorphic to $\...

Let’s talk about the status tags. Every single user who visits a Meta believes status tags mean something. They’re not wrong, of course. But what, really, do the status tags mean? Do users’ beliefs about status tags match how they’re used? What steps do we need to take to ensure status tags are r...