Learning analysis through topology

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. –  Ragib Zaman Aug 26 '11 at 1:53
What exactly do you mean by analysis? –  Mark Aug 26 '11 at 1:57
@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. –  Artem Kaznatcheev Aug 26 '11 at 1:58
You might want to take on measure theory along with the topology. –  Asaf Karagila Aug 26 '11 at 6:18
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. –  Mark Aug 27 '11 at 1:01