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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.

Mittens
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  • To understand lots of analysis you need to understand limits and continuity, and those are topological ideas. – Elchanan Solomon Oct 04 '22 at 17:18
  • Aren't these ideas used in the next chapter? – Randall Oct 04 '22 at 17:21
  • I can see he's using the concept of a function to talk about ideas around countability but I haven't gone beyond that yet. But towards the end he talks about metric spaces. So just trying to piece that together in terms of high level ideas and how they connect. –  Oct 04 '22 at 17:23
  • @Randall are you saying that I should just take the ideas on board on faith now and then I will see how they're used in chapter 3. That is, I won't see the point of chapter 2 until chapter 3? –  Oct 04 '22 at 17:25
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    Most of Chapter 2 is overkill for what's needed in a beginning analysis course. But if you choose to learn from Rudin, you choose to deal with his complete lack of pictures and geometric intuition. And if you make it to the end of the book, please learn multivariable analysis somewhere else. – Ted Shifrin Oct 04 '22 at 17:26
  • @TedShifrin Why is it that Rudin gets dislike for his multivariable analysis chapters? – FShrike Oct 04 '22 at 18:11
  • @FShrike Because he is an anti-geometer and multivariable calculus/analysis is inherently geometric. No pictures. Yeah, right. – Ted Shifrin Oct 04 '22 at 18:16
  • @FShrike I've been told that chatpers 10,11, and 15 in Apostol's Mathematical Analysis is better for that. My plan is to do those first and then come back to Rudin again. –  Oct 04 '22 at 18:21
  • @TedShifrin: It's been a few years since I looked at Baby Rudin. Does it really have no pictures? – Brian Tung Oct 05 '22 at 01:51
  • @BrianTung You really have no recollection? Go back and see if I’m fibbing. – Ted Shifrin Oct 05 '22 at 02:36
  • @TedShifrin: LOL yeah I'll check. I read it in Scribd, since I don't have the text myself. (I don't remember which text I used for my analysis course—it was a long time ago!—but it was thin and mostly supplemented by the professor's own nearly illegible notes.) ETA: I mean to say, I don't remember it having any diagrams, but that doesn't mean I remembered it having absolutely no diagrams. :-) – Brian Tung Oct 05 '22 at 03:54
  • @BrianTung I can confirm it has NO diagrams :)))) –  Oct 05 '22 at 10:57

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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.

BaronVT
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