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One of my supervisors once mentioned that when he was learning analysis he learnt it backwards. He learnt topology first and then saw analysis after, instead of the usual approach of doing everything with deltas and epsilons just to see the sexy topology proofs later. He said it was a very fun way to learn. I was wondering if there are any good books on analysis/topology that pursue this approach: i.e. they start with topology and then introduce, motivate, and prove analysis results.

In terms of background: I have graduate level discrete math, combinatorics, and linear algebra (mostly from the theoretical computer science perspective). I also have an undergraduate level physicsy-math: basic ODEs, PDEs, calc, and baby analysis.

Can you recommend a good book for learning topology as a precursor to analysis?

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    Munkres can be used for this. In fact, you could just learn topology from any textbook, most topology books will cover proofs of analysis theorems in the sections that they study metric spaces. Normally the worry is that a lot of this seems unmotivated and unnatural if one hasn't seen analysis before, but you seem to be ok with that.2011-08-26
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    What exactly do you mean by analysis?2011-08-26
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    @Ragib I would really prefer if it was a text that motivated the analysis through topology instead of me working through exercises to prove theorems that I don't know why I should care about.2011-08-26
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    @Mark part of the problem is that I have a really shallow grasp of analysis, so for me in means the standard analysis stream a math student would take: Series of functions, Riemann integration, metric spaces, implicit and inverse function theorems, Lebesque measure, Fubini's theorem. Abstract measure and integration. Convergence theorems. Fourier integrals. Standard things that a honours math student following an analysis stream should know by grad school.2011-08-26
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    @Artem: That is quite a list! I would say that topics like metric spaces, continuity, and convergence of sequences of functions are most relevant to your question. You should note that Riemann integration and the implicit/inverse function theorems are specific to $\mathbb{R}^n$, and that "abstract measure and integration" and Fubini's Theorem have nothing to do with topology at all.2011-08-26
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    @Jesse yeah, in the usual analysis stream it seems that folks are usually motivated by $\mathbb{R}^n$, so I would still like to know those results if possible, but I was hoping there would be a way to get them for free after learning topology. I always thought that Kuratowski-Ulam theorem was sort of the topology version of Fubini. Thanks for the advice though. I am not trying to learn this for a specific application, just really to cover up gaping holes in my knowledge.2011-08-26
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    You might want to take on measure theory along with the topology.2011-08-26
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    Some of these topics cannot be studied before some prior understanding of at least $\mathbb{R}$, simply because they rely on various properties of $\mathbb{R}$. For instance, $\mathbb{R}$ already appears in the definition of a metric space and indeed one has to appeal to its properties (like completeness) from time to time. Another example is the definition of path connectedness in topology, which uses closed, bounded intervals in $\mathbb{R}$, so one often needs to know that such intervals are connected and compact. You can't start building a pyramid from the top.2011-08-27
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    @Artem, I believe you asked this question at least partly because you wanted to save yourself some time. If so, then I would recommend not studying Riemann integration at all, in my opinion it's value in pure mathematics is largely historical. If you read this you can make up your own mind: http://mathoverflow.net/questions/52708/why-should-one-still-teach-riemann-integration/52764#52764 .2011-08-28
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    In principle, Bourbaki does introduce the real numbers as a consequence of copleting a uniformity, but that is no way to learn analysis. You are better off using a sophisticated approach to elementary analysis. The book "Foundations of Mathematical Analysis" by Truss contains a lot of "elementary" analysis on a graduate level.2011-12-27

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