
Topology Illustrated (2015) - Koshkin
https://calculus123.com/wiki/Topology_Illustrated
======
elipsey
What are the prerequisites for learning the basics, or developing a
rudimentary understanding of this subject? I find the idea of it (as far as I
can understand what that even might be) fascinating, but also exotic and
forbidding. I had a couple of friends who reconsidered their major after
taking topology :(

~~~
benrbray
There aren't really any mathematical prerequisites, except for a basic
understanding of writing proofs. Most of topology is about defining a set of
axioms (specifically, a topology is a set X together with a collection T of
subsets such that...) and studying the consequences of those axioms. Studying
topology is a great way to build up mathematical maturity.

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learned from a set of unpublished lecture notes.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...](https://github.com/benrbray/benrbray.github.io-
source/blob/master/content/static/files/umich_math490_f16_sbray.pdf) [2]
[http://www-personal.umich.edu/~hbray/](http://www-personal.umich.edu/~hbray/)

~~~
Myrmornis
> 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?

~~~
p0llard
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".

------
wegs
I've been looking for a good calculus book. I was curious if this worked well,
and I found:

1) It costs $300 for a paper copy 2) There are no previous. I have no idea if
it's actually illustrated

I'm looking for something which is intuitive, and illustrated. A lot of
pictures, good example, and perhaps less math and text (in this case, assuming
the educator will be talking through most of this, and the learner doing
things together).

~~~
iamcreasy
Calculus Made Easy is exactly what you are looking for.
[http://calculusmadeeasy.org/](http://calculusmadeeasy.org/)

~~~
wegs
Thank you!

