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I may or may not 'drop' my Point Set Topology course since I performed very poorly in it (also I am partially inclined to 'drop' the course so as to avoid a low GPA ), but I do not have any idea whether it will be a problem during my PhD application or interview. I want to know how much it matters.

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    How poorly is poorly? Are we talking about a B+ or a D or something in between? What other courses are you currently taking? Can you give us an idea what factors have contributed to your not doing well in this class? What are you planning to take in the spring semester? – aparente001 Nov 21 '16 at 6:39
  • Why was my comment deleted? It was a legitimate comment and adhered to stackexchange comment guidelines. – Mars Nov 21 '16 at 21:11
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(I am the graduate coordinator of the math department at UGA.)

Point set topology is a good course to take for an undergraduate hoping to go to graduate school in mathematics, but it is not a required one for admission to any program that I know of.

If you are asking whether it looks better to have a transcript containing point set topology with a poor grade vs. a transcript not containing point set topology at all, I would say: if it doesn't appear at all, it looks much better. Point set topology is an introduction to the abstraction / generality of graduate level mathematics, perhaps more so than any other undergraduate course because nowadays we don't really have separate undergraduate versus graduate offerings in the subject: the point set topology that you learn as an undergraduate is in many cases the point set topology that you take with you through your graduate career. So doing well in that course looks like a good predictor of success in graduate level coursework. Thus by doing badly in it, you're showing a clear data point which is not in your favor.

I should say though that dropping the course is not helping out your education or graduate preparedness at all: at best, it's papering over the problem. Are you sure you can't redouble your efforts and actually do well in this important course? Or the flip side of the same question: are you sure that you can truly drop it with no stigma whatsoever? E.g., will it show up on your transcript as a withdrawn course? Will faculty who write your letters know that you really struggled in this important course and responded by dropping it? (If I were writing a letter for such a student and had this knowledge, I would probably feel honorbound to mention it.) These are things to think carefully about and discuss with faculty advisors.

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    A "W" appearing on the transcript for this course would (in my opinion at least) look worse than an A, B, or even a C in the course. My experience in advising students on this is that for some reason many students think they're better off with a "W" than a "B" in a course. – Brian Borchers Nov 21 '16 at 3:15
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    I appreciate @BrianBorchers' opinion about a W, although mine is different. If I saw a W on a transcript, I would not hold it against a student, since I've seen so many students withdraw from courses for external or neutral reasons, such as health problems or having taken on too large a course load. Also, some schools allow students to drop a course very late in the semester, while others would call it a withdrawal. – Ellen Spertus Nov 21 '16 at 11:19
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Is there an extenuating reason (e.g. illness) why point set topology was difficult for you? If not, I'd rather be blunt about this: you're probably not cut out for a Ph.D. in mathematics. One of the purposes of grades is to figure these things out.

Depending on your program, point-set topology is likely one of the first courses in which you are expected to be fluent in the level of abstraction that mathematicians work in all of the time. If you were able to adapt quickly and more or less enjoyed it, that's good. If you were not able to adapt or did not enjoy it much at all, then you probably do not want a Ph.D. in mathematics and would not do well in it.

There are many other things, Ph.D. and non-Ph.D., to do with one's life. Undergraduate education should inform you for which of these things you will want to do. Large struggle in point-set topology does not indicate a good fit with mathematics research.

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    This sounds a bit extreme. At least in my university, point-set topology is a first-year undergraduate course (it's also mandatory, unlike in some universities, based on previous replies). I don't think a student's potential to do graduate research can really be judged by their performance in their first year(s). Also, apart from illness, there could be reasons such as laziness which can be dealt with later (I think user JeffE likes telling about his low undergraduate GPA which he overcame later). – Pandora Nov 21 '16 at 11:16
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    @Pandora: I would be interested to learn where (e.g. in what country) point set topology is a required course for first year undergraduates. – Pete L. Clark Nov 21 '16 at 18:36
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    Nothing wrong with bluntness when it's called for, but I agree with @Pandora. I think many well known mathematicians struggled with one or more of their math classes as undergraduates, so this answer is needlessly discouraging and negative in its tone. (In addition, it doesn't address OP's actual question, but assumes he/she is asking for advice about whether he/she should pursue a PhD in math.) – Dan Romik Nov 21 '16 at 20:51
  • @PeteL.Clark Israel. – Pandora Nov 21 '16 at 21:10
  • @Pandora I think that's a fair criticism and I didn't edit my answer because I think that something like this needs to be said. I haven't found a less harsh way to say something that will necessarily be somewhat harsh so will just accept the answer with mixed votes and a detracting comment. – user18072 Nov 27 '16 at 15:52
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Does time permit retaking the course? I remember from my undergraduate days that I failed Calc II, but when I repeated the class the next semester I got an A. Clearly you need a little more time and effort to master the course material.

So much of mathematics is foundational: you have to master A before moving on to B, and so forth. If you think you understand foundational topics early in a course, and then find out in mid-course that nothing is making sense, then you know you're in trouble; I've been in that situation myself.

My advice, then, is to repeat the course if you can, and do what you have to do to ace it.

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