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