> in a similar way, it's very hard to understand the "point" of category theory until you've noticed that many proofs across group theory, linear algebra, real analysis, etc. are all suspiciously similar)
Would you be able to give an example of such a proof cluster?
One example I'm aware of is Lawvere's fixed point theorem, of which Gödel's (first) incompleteness theorem, the undecidability of the Halting problem, and Cantor's theorem are all special cases.
Not really related to "group theory, linear algebra, real analysis, etc.", but interesting nevertheless.
It's quite a wide generalisation which really just captures the nature of diagonalisaion arguments, but it does formally tie together various proofs/theorems which "smell the same".
Re:Topology Illustrated (2015)
by hollowcelery on Thursday July 16, 2020 @09:24AM
(#23858718)
Not the previous poster, and not a maths expert by any means, but for example there are parallels between the rank-nullity theorem in linear algebra, the orbit stabiliser theorem in group theory, the splitting lemma in homological algebra, and the Fredholm alternative in functional analysis. They can all be seen as manifestations in some form or another of the first isomorphism theorem in abstract algebra.
However, I'd say it's very difficult to see the "point" of topology without taking a proof-based real analysis class (where you use epsilons and deltas to prove things about sequences, limits, derivatives, integrals, etc.). The concepts in topology generalize everything that is typically covered in a real analysis class, and many of the standard examples rely on some familiarity.
(in a similar way, it's very hard to understand the "point" of category theory until you've noticed that many proofs across group theory, linear algebra, real analysis, etc. are all suspiciously similar)
As for how to study: Topology is great because you really can build it all from the ground up. Find a book with good exercises, and do all the exercises! That way, you'll be forced to learn for yourself how all the pieces fit together. Unfortunately I don't have a book to recommend since I learned from a set of unpublished lecture notes.
EDIT: I dug up the old lecture notes, here [1] you go! Authored by Harrison Bray (no relation to me afaik) for University of Michigan, MATH 490, Fall 2016.
[1] https://github.com/benrbray/benrbray.github.io-source/blob/m... [2] http://www-personal.umich.edu/~hbray/