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I'm an international student so please be kind for my poor English.

In my point set topology class I can understand the lecture notes and can explain proofs in detail if required.

However, I don't really do well on quizzes and that frustrates me a little.

I know point set topology is important and, whether or not I get good grade in the course, I need a solid understanding of it.

I want to be successful in my mathematical career so I ask for your advice on how my attitude should be towards my circumstances right now, and on which topics in point set topology one needs to know as a math researcher. For instance, in some book the author recommended Royden's Real Analysis book and said that topology chapter in the book is in his subjective opinion, is basically all point set topology most mathematicians need.

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I think this all depends upon what kind of mathematics you want to do. I think that most research level mathematicians should be well acquainted with the more basic notions of general topology. For example, all of the following should be internalized:

  • Compactness (esp. all its equivalent forms in metric spaces)
  • Connectedness (and arguments about how one can use connectedness--e.g. I want to show that $X$ is everything in my space $Y$ so I show $X$ is open and closed)
  • Metric Spaces (how all the general topology is nicer here)
  • Homeomorphisms
  • Product Topology
  • Urysohn's Lemma
  • Path Connectedness
  • Second Countability
  • Tychonoff's Theorem
  • Local Compactness
  • Paracompactness
  • Quotient topology
  • And it would be nice if you knew nets/filters

Those are the things that come immediately to mind (although I think that is extremely much for a first course, so don't stress not knowing some of it), and while there is undoubtedly things I left out, I really don't think there is much else. While I think every aspiring mathematician should see Nagata-Smirnov metrization theorem and the long-line at least once, I don't know if I'd say it's really super important. So, if you're getting caught up in a lot of the really set-theoretic arguments for so-and-so metrization theorem or when $T_i+T_j\leftrightarrow T_k$ I wouldn't worry too much. Just absorb as much as you can--I'm sure you'll encounter it either directly (through more courses that start with a reminder of point-set) or inadvertently (through analysis mostly) many more times in your career.

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Munkres's Topology covers all this and more and has nice exercises and problems. –  lhf Apr 18 '12 at 14:47
    
Indeed, Munkres is one of the few undeniably, universally great undergraduate text books. –  Alex Youcis Apr 18 '12 at 14:48
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One possibility is that your quizzes are too difficult or too "subject-matter removed" compared to the lecture notes. Another possibility is that you need more practice with writing mathematical proofs. An introduction to general topology usually requires little background knowledge and a lot of "mathematical maturity", and it can be an excellent subject to learn by the "Moore Method" (google "Moore Method" if you do not know what it is). Given that you understand your lecture notes and you can explain proofs in detail, I suspect one of the following two possibilities:

(1) The quizzes are not appropriate for the course.

(2) When you say you can explain proofs in detail, you mean something different than what I think of when someone says this. Perhaps the problem is the phrase "explain proofs in detail". If you mean "you have memorized the proofs and have memorized an explanation for each step in a proof", and not "you can easily reconstruct the proofs and explain them in detail in your own words", then maybe the problem is that you do not yet know how to construct proofs well enough by yourself.

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