In which order should I read Pete Clark's general topology or what is the prerequisite for Pete Clark's general topology?

Question

Recently,I start to read Prof. Pete Clark's lecture notes on general topology,and first I read the appendix,however I find it very hard for me to learn. There are some objects with no definition in the appendix.In fact,I am a self learner,I have only learned calculus and linear algebra,and know some basic real analysis objects,such as countable sets,uncountable sets,mappings,etc.Many people tell me that if you have knowledge of calculus,you can start to learn general topology.However,for now,I think that Pete Clark's general topology is hard for me.

Answer 1

As I write this answer now (December 2020) I am coming up on the end of the second time I have taught general topology; the first was in Spring 2015. When I taught the course the first time I already had my notes on convergence (i.e., nets and filters) but that was it -- in the first half of the course I was largely following Kaplansky's Set Theory and Metric Spaces and in the second half of the course I was trying to cover as much of what I think of as the "standard stuff" (my notions for what this is are probably formed from Munkres's Topology more than anything else) on connectedness, compactness, separation axioms and so forth before getting to the big theorems at the end: Tietze Extension Theorem/Urysohn's Lemma, Tychonoff's Theorem and applications to Urysohn's Embedding and Metrization Theorems. Much of the rest of the textbook came from my own notes on these lectures. Then I decided that I wanted to enrich the text by including lots of things on metric spaces, so over the years I went back to add more material there.

The next thing to say is that I regard my notes as being very much incomplete as a general topology text. There is on the other hand enough there to read through, and in this run through of the semester course I covered perhaps 2/3 of the material in the text.

In the course of teaching the 2020 course I had the occasion to add to the notes a bit: the latest copy is available here. From my perspective the most significant addition is a discussion of Arrow's Impossibility Theorem and its infinite (non?-)analogue using ultrafilters. What can I say -- at the end of 2020 this material felt timely. The introduction (which, appropriately, is also unfinished) contains some indications of what I wish would be there -- for me personally the lack of discussion of the Stone-Cech compactification is the most outstanding. Also, in the (at long last!) published version of my instructor's guide to real induction I have a fairly systematic discussion of order topologies, and for some reason most of this does not appear in the text yet. When I taught the course this time around I did briefly introduce the topology on an ordered space before going on to metric spaces and I think this might be a good idea but my thoughts about the pedagogical merits of this approach are still a little fuzzy.

And now I will address your question directly. The course that these notes accompany is an undergraduate course, but it is the most advanced undergraduate course that I have ever taught (at UGA or elsewhere). This course is for students in transition from a study of undergraduate level mathematics to a study of graduate level mathematics. It is in order to facilitate this transition that I start with the topology of the real line (Real Induction is, more than anything else, an excuse to begin by thinking deeply about intervals on the real line, whereas a more standard approach to this would seem "unfresh" to many students and maybe not get their full attention) and then transition to and linger for quite a while on metric spaces, which again the majority of my students have seen in a previous course, but not in much depth.

So you get from this that the majority of my students -- and I think even more of the students who really engage with the course material -- certainly do have a background beyond calculus. All of the students have had a course on sequences and series of real numbers and of real functions, and the majority have had an undergraduate course in real analysis at approximately the level of (some initial segment of) Rudin's Principles of Mathematical Analysis. Also students are expected to have prior exposure to reading, understanding and writing proofs (there is a prior course on precisely this at UGA) and the course is "fully theoretical" in the sense that the material is definitions, theorems and proofs and whatever else helps to motivate and aid understanding of them, and most of the exercises are proofs.

In terms of self study, I would say that a student who just finished an undergraduate math major (or equivalent content) but hasn't had a general topology course yet would be at the right level to read these notes.

Finally, in terms of the set theory: yes, I agree with Brian M. Scott. No one puts something in an appendix that they want you to read first! I think the title "Very Basic Set Theory" is accurate in the sense that set theory is an entire mathematical field and the material here is something like the first 3-4 weeks of material in that field. On the other hand, nowadays very few working mathematicians know more set theory than this, and a lot of us can get away with knowing even less. (But it is nice to know these 3-4 weeks. I have a paper on the border of commutative algebra and elliptic curve theory where the main result is proven using transfinite induction. I was so excited to prove a real theorem using transfinite induction that I bounded giddily around my department for a while until I ran into a topologist colleague of mine and told them the news. Their reply: "That's great! [pause] What's transfinite induction?")

I forgot to mention this before, but the set theory notes were also written much earlier and not for this particular course -- I just figured it couldn't hurt to lump them in. In fact the basic narrative of the course notes requires only the trichotomy finite / countably infinite / uncountably infinite. Cardinal numbers appear only in corners, and ordinal numbers -- which do have applications in topology -- appear as pointers to those who care but are not really developed. (This might change a bit: in an exam that I wrote today there is an optional problem introducing the long line. If I write up solutions to that problem, I might as well stick it in the text!)

I hope these remarks are helpful. By the way, anyone is more than welcome to contact me with questions or comments about the notes. Unfortunately I mostly have time to make changes when I am teaching the corresponding course -- I believe that someone wrote me years ago with pages and pages of (good!) comments, and I implemented the first 20 or so of them before I had to get back to what I was doing at the time. But I would be very interested to hear about what people like or don't like or want to see in the notes. After all these years I am still really excited that if I post a couple hundred pages of mathematics on the internet, some people will actually read it!