Best resources for learning/practicing beginning topology?

Question

I'm an undergrad with minimal experience in proof-based classes, and I'm in a pickle with a current course.

Normally, I have trouble understanding and/or retaining information during lectures, so I take notes and then read the textbook at home and practice doing problems/writing proofs. So far it has worked just fine. I think I've learned everything thoroughly and I tend to be at the top of my classes in terms of test grades.

I am now in a beginning topology course (only relevant previous course I've had is a beginning real analysis course) and it is taught very differently. There is no textbook, and the only notes are ultimate results (theorems, exercises, corollaries, etc.) and a few definitions. All of the direction on proofs are discussed in class, mainly between other students and the instructor. I get why this is a reasonable way to teach the material, but since I have trouble with understanding the concepts during real-time discussions, I feel VERY behind. I don't know if any of my proofs are adequate, and I can only ask so many questions in class or at office hours. I really miss having a discussion of concepts/proofs that establish concepts that I can read over a thousand times if I want to. One small thing gets covered in class and I spend some time working through it in my head, proving it to myself, and then when I am satisfied I understand it I zone back in and I've missed 10 more things.

So first, are there any textbooks that could help here? I've looked at Munkres, but I figured others here might have better suggestions. I'm also open to advice, if anyone's been in a similar situation before, or at least have good approaches to tackling topology in particular.

Answer 1

Avoid Munkres , your best option is the Schaum's outline series text "General Topology" by Seymour Lipschutz .......................

Answer 2

In my opinion, you should have a look at Chapter 2 and 3 of Munkres.

Another classic textbook is General Topology by J. L. Kelley, but it is about 50 years older than Munkres. As mathematician, Barry Simon, once wrote (in 1980) "For the reader who wishes to delve further into the realm of general point set topology, we recommend Kelley's General Topology most enthusiastically. The best way to read the book is to do all the problems; it is time consuming but well worth the effort if the reader can afford the time."

I also firmly believe in the advantages of having an assigned textbook for upper division mathematics. You should see the references/bibliography of your lecturer's notes (if it exists).

Answer 3

I have two recommendations:

• Introduction to Topology: Pure and Applied, by Colin Adams and Robert Franzosa. (This has lots of examples. It covers all the material one should see in a one-semester course in introductory topology. It also includes lots of interesting applications.)
• Introduction to Topology, by Crump Baker. (This is a very underrated introductory book, in my opinion. It deserves more attention than it has received.)

The book by Munkres is great. However, if a person (like yourself) has been struggling with proofs, then the book by Munkres might be better for future reading.