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?