While applying for PhD in mathematics, am I better off dropping a course I'm performing poorly in? 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.
(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  E.g., will it show up on your transcript as a withdrawn course?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.