Topology Notes

I’ve been talking about writing a topology textbook introductory notes on topology for years. Basically since I wrote my Rethinking Topology (or a Personal Topologodicy) post 2 years ago — it’s hard to believe it’s been that long!

In any case, I finally started writing it. I’ve done a mild review of existing introductions to general topology (ie. I skimmed through the first few chapters of a dozen topology textbooks), so I feel somewhat comfortable contrasting my work to existing literature. It’s quite a different approach.

Topological Anatomy: Closure, Interior, and Boundary

Topological Anatomy: Closure, Interior, and Boundary

I initially develop topology based on closures and adherant points. Kuratowski’s closure axioms are then built up with natural explanations. Emphasis is given to the variety of possible definitions (along the lines of Lakatos et al’s Proofs and Refutations) and exercises encourage the reader to explore the variety of possible definitions. I attempt to justify the axiomatic approach in a manner similar to Pinter’s wonderful A Book of Abstract Algebra, though I may fall very short. From here, we build intuition for closure, boundary, and interior with some diagrams and proofs of identities. Finally, we wrap up the first chapter with a visual interpretation of the closure axioms.

Arrows represent closure and lines superset in a visualization of the indiscrete closure operator on {1,2}.

The indiscrete closure operator on {1,2}

(You can find the most recent version of the book on github.)

As a relatively informal and low pressure way to get feedback, I posted it to r/math. The feedback was pretty negative, if sympathetic.

I’ve been struggling to figure out how seriously I should take this criticism. On the one hand, I’m strongly biased to be dismissive of it, because I’m so invested in my work. On the other hand, I worry that there was a significant element of group think (people who were more supportive and my explanations were down voted) and that I accidentally primed a negative response by opening the book with a Preface beginning:

I call myself an aspiring mathematician because I’m painfully aware of how little I know compared to, for example, research mathematicians. I’m also aware that many introductions to topology have been written by people with much deeper understandings of the subject than me…

(This was largely because I’ve been made very uncomfortable by news paper articles calling me a “math genius” or “math savant” and am nervous that the mathematical community might perceive me as arrogant, despite the fact that I’ve asked news organizations not to do that.. What I know is a little bit impressive for my age, and nothing more.)

I’m concerned that this might have made the readers not take me very seriously, or look for reasons to say “it’s good that you’re trying, but you’re doing X wrong…”, because that’s an easy response to give to someone you think you know better than. Perhaps I should have prepared them to expect a very different approach to topology than the standard one…

The criticism largely fell into a few categories:

  1. It’s different. It takes 15 pages to get to content that is usually on page 2. Topology should be focused on continuous maps. Focuses too much on closure. And so on.
  2. It’s too hard/easy/trivial. Readers won’t learn anything. Readers will struggle to read it. It explains too much, readers need to struggle to understand the motivation behind definitions to learn.
  3. It’s too verbose.
  4. Poor Quality of Writing.

For (1), I feel like no one ever actually justified why the standard way to teach topology is best: there was some argument that continuous maps are the most important concept in topology, but it wasn’t very compelling to me, and it doesn’t follow that most fundamental perspective is how one should enter the field of topology. (2) is kind of self-contradicting. (3), I suspect, arises from explanatory and motivational sections being tedious for some of those that are very mathematically mature. (4) is probably true.

But I may just be being very dismissive, as I’m biased to do.

I think that, perhaps, I really need feedback from mathematicians I already know and respect. There’s a few readers of this blog that I’d be thrilled to get comments from. Beyond that, I’m considering sending this to one of the topoligists I know, or perhaps reaching out to Michael Henle, who’s Introduction to Combinatorial Topology includes a few similar elements (and I really enjoyed the half of it I read!).

I’d also appreciate feedback in how to determine how seriously to take criticism.


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9 Responses to “Topology Notes”

  1. doctorkiwano Says:

    On one hand, I can see where the interest in covering continuous maps early comes from: they’re absolutely essential to e.g. compare topological spaces. On the other hand, I have a recollection of Armstrong not getting into homomorphisms terribly early, and I rather enjoyed his treatment of group theory when I first encountered it in second year.

    (Sadly, I haven’t yeat had the opportunity to give your book a good read, and make meaningful comments of my own.)

    • colah Says:

      Thanks, Kris!

      Clearly continuous maps are very important. They’re just not obviously, to me, the best way to introduce the subject.

      It’s comforting that you know of an intro to group theory that you liked that didn’t emphasize the morphisms early on…

      If you have time to read and comment on the book, I’d be thrilled to have your feedback.

  2. Anonymous Says:

    M-m-m… It will take me some time to explain how I feel about this. In fact, this comment is going to match the length of your textbook, probably.

    I won’t lie to you: the idea to write a topology textbook is ridiculously, insanely ambitious. Imagine if I came to you one day and told you:

    “Hey, have you ever noticed that this Windows 8 thing you are using tends to crash all the time? I think I can solve this problem! I will write a brand new operating system, more stable one. I even know a really awesome programming language suited for the task, it’s called BASIC. It has all kinds of fancy features, such as variables, and loops too! Do you realize just how badass loops are? They let you perform the same action as many times as you want, and you can put loops inside other loops as well! With my programming skill and the awesome power of loops I can’t possibly fail to write a better OS than Windows!”

    What would you tell me in response to that? This is roughly how your idea is going to sound to the majority of mathematicians.

    Don’t get me wrong: if I worked really, really hard and studied a lot, I would eventually manage to slap together some sort of an operating system. And the whole experience would dramatically improve my programming skill. So it wouldn’t be a complete waste of time. Still, it would be unrealistic for me to expect to eventually create on operating system that a significant number of people would actually want to use.

    This was an indirect, meta-level criticism of the whole writing-a-textbook idea. I’ll now try to provide some bits of object-level criticism.

    I skimmed through the book. It was an easy read. The way you demonstrated what your axioms mean graphically on those little lattices was cute. I don’t think I learned anything new. I don’t think any person with a math degree and an interest in topology would learn anything new.
    I really like talky math books. I like books that take time to motivate. I think Hatcher’s Algebraic Topology is awesome in that regard. (Have you read it? Read it.) I’ve seen many, many math books that commit the sin of explaining and motivating too little. Yours is the first one that comes close to committing the opposite sin of explaining more than necessary.

    Here’s my main point: if you are going to write this, you must have a firm idea of what your target audience is.
    It can’t be guys with PhDs: for them your book in it’s current form is going to be nearly worthless. Even, say, experts in applied math, who secretly fear topology will probably want something a bit more dense with more meat in it.
    University freshmen is one possible audience for you. I knew people who might have benefited from reading a book such as yours. But then you will probably have to switch to the standard set of axioms. (I think it’s only slightly less intuitive than the one you use.) Having to digest two different sets of axioms would be too confusing for many. When it comes to different axiomatics of the same theory, I think it’s better to stick to the existing ones. They are all equivalent, and trying to push new ones is going to be more trouble than it’s worth. Remember that xkcd comic about new standards?
    I know that if someone gave me your textbook when I was 16 I would really enjoy it and learn a great deal from it. Even then though, I would feel insulted reading the following sentence:

    “One of the most famous mathematicians of all time is Euclid. He wrote a book, Euclid’s Elements, proving a large number of geometric results.”
    Euclid wrote a book?? You just blew my mind! And it contained geometric results too? I can feel my entire world-view crushing around me.

    The reference to the category theory, on the other hand, would completely fly over my head. Anyone with enough mathematical maturity to wrap their head around categories is going to find your book too elementary. And schoolchildren, to whom this style would be far more appropriate, don’t even know what a continuous function is. Maybe a good target audience for your book would be curious, bright schoolchildren with a lot of mathematical erudition. Still, you wouldn’t be able to rely on them knowing what a group is.

    I think that you should either dumb this text down, to the schoolchildren level – this will involve explaining continuity, limits, algebraic structures and stuff like that and will probably double it’s length, or, – you could try to write something aimed to the college students, something like “Topology 101”, by making it more conventional and a little denser. Just a little, mind you. No need for the Postnikov’s neutron-star levels of denseness. But you will probably need to cut the length in half. A good amount of your explanations will not actually be necessary for them.

    So, to summarize: form a firm idea of your target audience. Don’t assume they are idiots on one page and then go on to assume they are geniuses on the next one. Pick one level and stick with it. For geniuses xor for idiots. I recommend the latter. It’s not a bad book, but it’s a bit unfocused and in need of a lot of beta-testing and editing. If you could find some actual teenagers to read it, it might help to improve it immensely.

    A style advice: if you have an important definition, separate it from the text. Put it in a little box. It’s a good idea, trust me.

    I am really curious about the next chapter about homotopy stuff. I wonder how will you handle it.

    – George G

    • colah Says:

      Hey George,

      Thanks for taking the time to write such a thorough response.

      Your meta-criticism is particularly ironic because, of course, one of the first posts on this blog was about me writing my own operating system kernel. (Though I feel like there’s some issue with your analogy in that a modern operating system is the work of thousands or tens of thousands of people, where as most topology books are primarily the work of a single person.)

      In any event, would do you think you and others would be more comfortable if I referred to it as “introductory topology notes” instead of a textbook? Honestly, when I’ve tried to outline how the standard undergraduate intro to topology curriculum would fit into this, I’ve run into lots of things I can’t give a sufficiently intuitive explanation of (like compactness, what is it *really*? “topological finitness”?). In any event, a present, I only have a few chapters of content that I feel capable of writing to the standards I want.

      I was going to respond that to your object level criticisms as well, but I think it might be better for me to postpone that until I’ve reflected longer. I agree with several points you’ve made, but there’s a few points that I disagree quite strongly with, and I’d like to make sure that I’m not just being defensive when I respond.

      • Anonymous Says:

        I’ll wait for you to formulate a longer response, meanwhile I’ll say this much:

        “would do you think you and others would be more comfortable if I referred to it as “introductory topology notes” instead of a textbook?”

        YES. Do that. “Intuitive Explanation of Fundamentals of Topology”, you could call it as well, I guess. Anything without the word ‘textbook’.


    • colah Says:

      Hey George,

      Sorry about the very delayed reply.

      Basically, my perspective boils down to me having very different intuitions about high school/university students than you. I’m not sure which one of our intuitions to trust more. The correct way for me to move forward would be empirical investigation, and I don’t have the energy for that.

      The first chapter got a lot of my motivation for writing the notes out and I haven’t felt motivated to work on it since. Maybe I will at some point in the future, but for now I’m just going to let this rest.

      Thanks again for taking the time to comment!


  3. Person Says:

    I enjoyed this, perhaps because my experience with topology is limited to a first course at the undergraduate level. Also, the first two exercises were particularly entertaining. Thanks.

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