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I'm a Junior Mathematics major at a small Liberal Arts college. I'm currently taking first semester topology (Munkres text). I feel like I'm barely able to tread water in this course. I was able to follow the material up until we began discussing limit points and convergence. I feel I should mention I have yet to take Real Analysis yet.

I've taken two courses over Abstract-Algebra, one of which was an independent study over Ring Theory. I find Algebra much more intuitive and I understand it much better than general topology.

The definitions seem to make sense to me, yet I find it difficult coming up with counter-examples and proofs for basic theorems. I really struggle with anything dealing with $R^{\omega}$.

I really hope to attend graduate school, or at the very least be involved in mathematics in some way throughout my life. I feel like I have hit a road block. Does anyone have any study tips or advice for understanding basic general topology?

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    $\begingroup$ Real Analysis is rather helpful; it helps provide some good intuition with dealing with the definitions. $\endgroup$ – Hayden Apr 4 '14 at 2:34
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    $\begingroup$ I consider myself fairly good at topology and I struggle with stuff like $\Bbb R^{\omega}$ too. Stuff like that is really, really bizarre and hard to get one's head around. (I feel that some other counterexamples, like the topologist's sine curve, are more intuitive.) The best thing I can suggest is to look at examples. Lots of them. Over and over. (In fact, don't just look at them, play with them yourself: prove lots of theorems about these specific objects!) Once you get a strong intuition for some simple spaces, proofs in the subject will come easier. $\endgroup$ – user98602 Apr 4 '14 at 2:34
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    $\begingroup$ vitamins for the memory, avoid long gaps between sessions of concentration $\endgroup$ – janmarqz Apr 4 '14 at 2:50
  • $\begingroup$ Sometimes I can't seem to find the motivation to study it because I struggle with some of the concepts. $\endgroup$ – Pubbie Apr 4 '14 at 3:16
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Don't beat yourself up over struggling so much that you doubt your ability to succeed in graduate school. Perseverance and hard work will go a long way. Don't give up. (But yes, consider how to learn.)

Be sure to understand the important things, and don't stress if some of it does not click in your first exposure. (In particular, I don't think the example you mention is central. Grasping continuity, compactness and connectedness is more important. Having a variety of examples that you can work with is also useful. For example, metric spaces are nice. When I'm thinking of a theorem, I have often create subspaces of $\mathbb{R}^2$ in my mind to work with. I suggest you talk with your professor to see what he/she thinks is most important for your class.

Also, do you have the chance to work together with some of your classmates to talk about the material?

Although Munkres' text most certainly has its strengths, from my little exposure to it, I'd guess that I would not have liked my first introduction to the subject to be that text. Rather, I used a book by George Simmons, titled "Introduction to Topology and Modern Analysis." You should try it out when you have time and/or check out some other book.

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  • $\begingroup$ I do have a friend with whom I am taking the class with, but he seems to grasp the concepts much more quickly. I think I will meet with my professor more often to discuss specific cases in $R$ or $R^{2}$, then generalize to infinite spaces. $\endgroup$ – Pubbie Apr 4 '14 at 3:17
  • $\begingroup$ David, how did the class finish out for you? Did you end up working with your friend? $\endgroup$ – Andrew Kelley May 28 '14 at 1:30
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    $\begingroup$ The course went much better than expected. My professor helped me through and I actually improved my grade on each exam! My final exam was an A- and I achieved a B+ in the course, which was above the average. $\endgroup$ – Pubbie Jun 26 '14 at 20:43
  • $\begingroup$ Good job on sticking with it and improving! $\endgroup$ – Andrew Kelley Jul 1 '14 at 21:36

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