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.)