I have just completed chapter 1 of Rudin's PMA and done the questions. I'm looking at chapter 2 and I am wondering why he's introducing topology.
Why does the chapter on Basic Topology appear before sequences? Generally I would like to know at a high level (intuitively) what is the thread of thought that flows through Baby Rudin? I would like to know the high-level concepts and how they connect together and the order so I can have that as a mental model when going through the book.
Epsilon-delta notions of limits will get you fairly far in an elementary analysis course, but they're pretty dependent on the intuitive understanding you probably already have about the topology of the reals. Defining these ideas in terms of open sets will give you a much more flexible and general understanding of what it means for a sequence to converge, what it means to be a limit point, what it means for a set to be compact, etc. which will make the following material a lot easier to understand in its full generality vs. needing to constantly resort to intervals in the reals.