Material for preparation for Topology exam

Question

I'm a student of Mathematical Physics and at the end of this year I need to take one exam in Topology which covers the following topics:

Metric spaces, topological spaces, continuous functions, product space, quotient space. Convergence of nets and filters. Hausdorff spaces, regular and normal spaces. Compactness and connectedness. Homotopy, fundamental group. Covering spaces.

The topics are said to be those of chapters 2-7 and 9-11 from Munkres book.

The point is that I have to self study for this exam, and although I have already had contact with topology some times during my undergrad course, I still feel somehow uneasy about this exam. The exam is also very important, because I really do need to pass it in order to finish the undergraduate course this year.

Now, preparing alone for one exam like this is being a quite complicated task. I know a great deal of the definitions, I also know some important results and know how to prove basic facts (like that the image of compact/connected spaces by continuous functions are also compact/connected). Things like separation axioms, metrization theorems, nets/filters, algebraic topology I admit I never studied.

Also looking at some problems, mainly those dealing with specific topological spaces, rather than general properties, I feel that I still need more preparation.

My greatest problem is time. Although I had contact with topology because I needed to study differential geometry for Physics, there are many topics in the list that I never studied formally. Unfortunately I was only notified that I would be allowed to take the exam recently, and now there's not much time. In that case I'm looking for some way to prepare the most I can in the least time.

In that case, my question is: how can I prepare for this exam in a good way? What are good materials that can allow me to prepare to this exam and cover as much as possible in not so much time?

Answer 1

You should attempt solving problems on the topics of your exam. Some good sources are:

• Lee's "Introduction to Topological Manifolds" which covers all of your topics
• Hatcher's point set topology notes (links to the pdf)
• Hatcher's "Algebraic Topology" for problems on homotopy, fundamental group and covering spaces

Select some problems that seem particularly relevant to you and then try solving them. If you get stuck, ask a question on MSE. As you go along, you might want to compile a list of (counter)examples and useful tricks or facts.