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.
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.
(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:
- 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.
- 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.
- It’s too verbose.
- 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.