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I'm an undergrad with minimal experience in proof-based classes, and I'm in a pickle with a current course.

Normally, I have trouble understanding and/or retaining information during lectures, so I take notes and then read the textbook at home and practice doing problems/writing proofs. So far it has worked just fine. I think I've learned everything thoroughly and I tend to be at the top of my classes in terms of test grades.

I am now in a beginning topology course (only relevant previous course I've had is a beginning real analysis course) and it is taught very differently. There is no textbook, and the only notes are ultimate results (theorems, exercises, corollaries, etc.) and a few definitions. All of the direction on proofs are discussed in class, mainly between other students and the instructor. I get why this is a reasonable way to teach the material, but since I have trouble with understanding the concepts during real-time discussions, I feel VERY behind. I don't know if any of my proofs are adequate, and I can only ask so many questions in class or at office hours. I really miss having a discussion of concepts/proofs that establish concepts that I can read over a thousand times if I want to. One small thing gets covered in class and I spend some time working through it in my head, proving it to myself, and then when I am satisfied I understand it I zone back in and I've missed 10 more things.

So first, are there any textbooks that could help here? I've looked at Munkres, but I figured others here might have better suggestions. I'm also open to advice, if anyone's been in a similar situation before, or at least have good approaches to tackling topology in particular.

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  • $\begingroup$ Besides resources, here is a VERY important skill for every mathematician to master: suspension of disbelief. You mention how you worked through the proof of one small thing in your head, and missed 10 more things. You have to learn how to suspend your disbelief in that one small thing and keep paying attention to the 10 other things. Later, on your own time, you can come back and work through the proof of that one small thing. And on top of that, you'll have good notes on the 10 other things and you can work through their proofs too. $\endgroup$ – Lee Mosher Feb 9 at 17:52
  • $\begingroup$ @LeeMosher It's not even that I need a formal/written proof of everything, I just meant that I will hear something and, unless I think it through, it's just words. There is no mathematical meaning to me, it isn't just that I don't believe it right away. These things are sometimes not even written on the board so sometimes understanding it is a prerequisite to even being able to take notes on it! Perhaps I should take an audio recording of each class and write notes later? $\endgroup$ – Doug MacArthur Feb 9 at 18:00
  • $\begingroup$ @DougMacArthur frankly, that occurs at every level of mathematics, no matter what your experience level is. You just have to sometimes spend some quality time picking the material apart best you can. In class, my strategy would be to get an idea for the proof of something, take some notes on the general structure, and then pick it apart later when you are studying. This way, you do not miss material being covered. Don't get bogged down in the details in class, that's not the place for it. $\endgroup$ – rubikscube09 Feb 9 at 18:39
  • $\begingroup$ I was about to post a comment suggesting you take an audio recorder to the classes, but you've beaten me to it. $\endgroup$ – Calum Gilhooley Feb 9 at 18:53
  • $\begingroup$ Regarding "it's just words", sometimes you have to just record the words, either as notes, or as an audio recording, perhaps even as a video recording. Then you get the words into your head at a later time, memorizing them like a poem, even though you don't understand them. Then you can mull them over, let them swim around in your head, and try to figure them out. $\endgroup$ – Lee Mosher Feb 9 at 19:23
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Avoid Munkres , your best option is the Schaum's outline series text "General Topology" by Seymour Lipschutz .......................

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    $\begingroup$ I don't think this is appropriate as an answer; your opinion is made to sound like an edict. I recommend either editing to make this more of a suggestion, or reposting this as a comment. $\endgroup$ – Aweygan Feb 9 at 18:05
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    $\begingroup$ Munkres was my advisor at MIT and he was a great lecturer. Later, in grad school, we used two of his texts for classes. They were well-written and had many helpful exercises. Both his written and oral explanations were simple, direct and spot on. $\endgroup$ – B. Goddard Feb 9 at 18:12
  • $\begingroup$ @B.Goddard I'm essentially saying "Avoid Munkres, your best option is..." should be replaced with something like "One option is...". I am very fond of Munkres's book, and I don't think starting an answer on a topology book recommendation should ever begin with "Avoid Munkres". $\endgroup$ – Aweygan Feb 9 at 18:16
  • $\begingroup$ @Aweygan Yeah, as I was typing, my brain conflated you and the OP. My bit about "I don't understand" your comment was really my brain saying "I don't understand the OP's comment" but my fingers messed it up. $\endgroup$ – B. Goddard Feb 9 at 18:45
  • $\begingroup$ @B.Goddard That makes sense, given the nature of your comment :) $\endgroup$ – Aweygan Feb 9 at 19:03
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In my opinion, you should have a look at Chapter 2 and 3 of Munkres.

Another classic textbook is General Topology by J. L. Kelley, but it is about 50 years older than Munkres. As mathematician, Barry Simon, once wrote (in 1980) "For the reader who wishes to delve further into the realm of general point set topology, we recommend Kelley's General Topology most enthusiastically. The best way to read the book is to do all the problems; it is time consuming but well worth the effort if the reader can afford the time."

I also firmly believe in the advantages of having an assigned textbook for upper division mathematics. You should see the references/bibliography of your lecturer's notes (if it exists).

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I have two recommendations:

  • Introduction to Topology: Pure and Applied, by Colin Adams and Robert Franzosa. (This has lots of examples. It covers all the material one should see in a one-semester course in introductory topology. It also includes lots of interesting applications.)
  • Introduction to Topology, by Crump Baker. (This is a very underrated introductory book, in my opinion. It deserves more attention than it has received.)

The book by Munkres is great. However, if a person (like yourself) has been struggling with proofs, then the book by Munkres might be better for future reading.

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