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Or do you read a lot of books on the same subject (say Topology)? Consider these two cases:

Case 1: Suppose you want to study point set topology. You pick up a good book and work thoroughly on it.

Case 2: You pick up many books on point set topology and read them passively (to get different viewpoints).

I think Case 1 is the better learning method. What do you think?

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I think Math Overflow is not a discussion forum. –  j.c. Jan 16 '10 at 4:47
As written, I think this question is a little too subjective for MO. –  Qiaochu Yuan Jan 16 '10 at 4:55
I think it should be treated as advice, not discussion. Agreed that it's not the best question, but there is something to say here, I think. –  Pete L. Clark Jan 16 '10 at 4:57
I agree with Pete. There is no reason to turn it into a discussion of merits and demerits and whatnot. –  davidk01 Jan 16 '10 at 5:19
I'm voting to close, mostly because the person asking the question didn't bother to make any arguments to explain or justify his position. What sort of response is he expecting? –  S. Carnahan Jan 16 '10 at 9:09
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closed as not constructive by Qiaochu Yuan, S. Carnahan, Charles Siegel, Anton Geraschenko Jan 18 '10 at 7:55

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1 Answer

The choice of subject and level influences the answer, I think.

If you want to learn general topology [as in a thread in meta-MO, I claim this is the same thing as point-set topology but sounds less old-fashioned] from scratch, then yes, I think it is preferable to pick up a good book -- e.g. Munkres, Kelley, Willard [not Bourbaki, IMHO] -- and work steadily through it.

However, if you go farther in general topology, it does become beneficial to compare different sources. (I think the word "passively" in Case 2 above is put there to make this case sound bad. Comparing different treatments of the same subject and trying to figure out whether they are really different is a quite active process.) I myself decided a couple of years ago that I wanted to revisit general topology (which I hadn't thought about since I was a 19 year-old undergraduate), and it has been very helpful to me to compare different sources. For instance, in my study of convergence I was quite baffled by the fact that any one book I looked at took a side on "nets versus filters" and then vaguely indicated that whichever one they didn't choose resulted in an equivalent theory. Only by comparing several different sources (and some research articles) was I able to figure out what was going on to my satisfaction: see Section 6 of


for what I learned.

For a different subject, flipping around might be a better approach from the start. Indeed you might not have a choice: as you go on in your study of mathematics, you find that it is very often the case that there is no unique text that is squarely focused on what you want to know (the bright side of this is that it is very exciting when a text comes out serving this purpose whereas previously there was none, e.g. Silverman's Arithmetic of Elliptic Curves).

Of course I agree that to really learn something you have to spend some time exploring it linearly. E.g., in order to internalize (even moderately) complicated definitions, you need to work out some proofs in which these definitions appear. Flipping around for comparison is not going to help you if you don't already have some sense of what you're reading.

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Very well said. +1 –  Zev Chonoles Jan 16 '10 at 5:23
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