@Verty I agree; the definition should probably have been lumped in with others. And I also agree that this book isn't the best for everyone; I just think it's working well for me. Note that I haven't learned/taught/referenced from other books, so I'm not sure my opinion should be given much...
Yes. Page 35 on the third edition, remark 2.29: "... to be quite explicit, let us say that E is open relative to Y if to each p \in E there is associated an r> 0 such that q \in E whenever d(p,q) < r and q \in Y ."
I would also point out that in context it's quite clear what open...
@verty I was speaking specifically about missing steps in proofs. On the contrary with definitions, I have only thought twice that he didn't define something; both times when I looked back it turns out that the mistake was with me as I didn't read closely enough.
It might be helpful for someone somewhere to know that I didn't complete a book like Spivak before starting Rudin and it's going fine. On the other hand, I had built up math maturity from linear algebra and our differential equations course which, at a high point of abstraction proved...
Book isn't bad but isn't great either. Seems to straddle an unfortunate line in level of material between something like Apostol or Spivak and Stewart. Much better IMO to just sit down and work through Spivak etc. But it has an okay treatment of multivariate concepts and actually proves most...