I've finished all the questions in Chapter 2 of Principles of Mathematical Analysis by Walter Rudin (self study), but I have a question about Q.28, which reads:
Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Corollary: Every countable closed set in $\mathbb{R}^k$ has isolated points.)
It's easy to answer the question, given what's proved in Q.27, but the corollary is a bit weird. It looks to me as though it is just an immediate consequence of the fact that non-empty perfect sets in $\mathbb{R}^k$ are uncountable, which is proved in the main text. I don't see what it has to do with what is proved in this question.
Given how meticulous the book is, I suspect the apparent non-sequiteur means I'm missing something.