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    $\begingroup$ I'd think a (typical) student learning point-set topology before real analysis is not going to get a more unified approach but a more confused approach. Doing analysis in R^n and in metric spaces gives the intuition for topology. Otherwise you have nothing down to earth to dig your teeth into to generate examples and counterexamples to understand what things mean. Studying topology before analysis seems analogous to trying to "learn" what a base for a topology is before you learn what a topology is. At the very least study analysis at the same time as topology if the latter is the goal. $\endgroup$ Commented Jul 13, 2010 at 4:02
  • $\begingroup$ @KConrad: I think we agree. $\endgroup$ Commented Jul 13, 2010 at 5:09
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    $\begingroup$ @KConrad,Daniel I second the motion. $\endgroup$ Commented Jul 13, 2010 at 5:17
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    $\begingroup$ @Andrew L: Aren't you thirding the motion? $\endgroup$ Commented Jul 13, 2010 at 15:26
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    $\begingroup$ I took my first topology course and my first analysis course concurrently. The pacing of the courses was such that all the topics that overlapped were done first in topology. Although there were a few proofs that I did not follow very well at the time (e.g., connectedness and compactness of the interval), I did not find the point-set topology confusing, and in fact found a lot of the analysis redundant and unnecessarily complicated. [I should note that this topology course emphasized nice spaces rather than counterexamples.] $\endgroup$ Commented Jul 13, 2010 at 21:47