Interesting areas of study in point-set topology

Question

I'm undertaking a little self-study in point-set topology, because my undergraduate course does not have a module in Topology.

I have a copy of Topology by James Munkres, but do not have the time to cover the material in its entirety. I'm looking for advice on key areas/theorems/examples that I may find interesting.

The main purpose of this is to essentially produce a set of notes mirroring lecture notes for my own use. My other main modules this year focus on groups and set theory, so hopefully this will give you an overview of my interests.

Thanks for your time!

Answer 1

Tim's answer is great for any typical introduction into Point Set Topology. I think that Chapters 2-4 and the first section of Chapter 5 are usually assumed knowledge in more mature courses and lectures, and as such form a good foundation for further studies. With your interest in algebra you might also enjoy skimming the material on Algebraic Topology in Part II of the book.

My main goal with this answer, however, is to perhaps prevent a pitfall with making lecture style notes for yourself. That is what I did when I self-studied with Munkres' book, and all that time spent writing/typing out the theorems and proofs, as one would during a lecture, is time wasted! It is not an efficient way to self-study, and you could always reference the book later if needed.

What is crucial is reading and digesting theorems and definitions by understanding the pertinent examples and their proofs through and through. You can achieve this without writing down every theorem and proof in a book: just take the book and some paper and read the proof while you work out anything confusing on the side. Exotic examples or examples Munkres does not work through are ones you may wish to work yourself and have in your own notes.

In my opinion the minimal amount of material which you should save for posterity, perhaps in a neatly formatted $\LaTeX$ document, are worked examples from the book and any exercises you've solved (you should attempt 90% of the essential exercises which are usually the first ten problems or so at the end of each section). Working the examples and exercises is the only way to understand the context and applications of the theorems and definitions. I think this will help you save time and quickly work through more material.

Of course you should do whatever is comfortable for you. I hope this is good advice, so good luck.

Answer 2

Since you are learning set theory, you won't need to read Chapter 1. You'll probable need to skim over it, to see what's covered, the notation etc. Judging by the edit you made to your post, I'm assuming that you're primary goal is to be able to use basic theorems and methods in general topology, as they apply to other subjects. This is as opposed to learning general topology for it's own sake. If this is the case, you should read Chapters 2 and 3 at the very minimum, and ideally also Chapter 4. Because you're interested in algebra, you may want to take a look at the exercises on topological groups.