The Math Sorceror on Topology by Munkres

Here's my comment:

I studied an undergraduate course based on Sutherland's book An Introduction to Metric and Topological Spaces. For some reason I really liked that book, but I never understood what the course I was studying was actually supposed to be about. It was just a bunch of proofs of theorems that seemed kind of random to me. Like we were just making these definitions and proving theorems about them without having any real goal. Since then I've started to see that what it is really about is analysis on sets and Poincaré's and Dirichlet's attempts to make this rigorous. But the course I did didn't have any of this historical context, so it just seemed like a vague association of disconnected ideas all called "topology" and not always for the same reasons!

The other day I saw a really interesting Biography of Poincaré in a bookshop. It may have been this one: Henri Poincaré: A Scientific Biography by Jeremy Gray. The course I studied was Open University M435 An Introduction Metric and Topological Spaces and Geometric Topology.

Maybe students could try writing solutions to exercises as proofs in Lean, then they would at least know that they hadn't done anything really stupid in their solutions. See https://leanprover-community.github.io/undergrad.html and Daniel Tubbenhauer on Computer Aided Mathematics.

Here is a playlist of his videos about Topology.

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Here's Norman Wildberger explaining why the Fundamental Theorem of Algebra is really a half-baked conjecture about point set topology.

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Here's what people are doing now: see Langlands program and Quadratic reciprocity. This is part of an interview with Edward Frenkel.

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