Reddit reviews Introduction to Metric and Topological Spaces (Oxford Mathematics)

I study topology and I can give you some tips based on what I've done. If you want extra info please PM me. I'd love to help someone discover the beautiful field of topology. TLDR at bottom.

If you want to study topology or knot theory in the long term (actually knot theory is a pretty complicated application of topology), it would be a great idea to start reading higher math ASAP. Higher math generally refers to anything proof-based, which is pretty much everything you study in college. It's not that much harder than high school math and it's indescribably beneficial to try and get into it as soon as you possibly can. Essentially, your math education really begins when you start getting into higher math.

If you don't know how to do proofs yet, read How to Prove It. This is the best intro to higher math, and is not hard. Absolutely essential going forward. Ask for it for the holidays.

Once you know how to prove things, read 1 or 2 "intro to topology" books (there are hundreds). I read this one and it was pretty good, but most are pretty much the same. They'll go over definitions and basic theorems that give you a rough idea of how topological spaces (what topologists study) work.

After reading an intro book, move on to this book by Sutherland. It is relatively simple and doesn't require a whole lot of knowledge, but it is definitely rigorous and is definitely necessary before moving on.

After that, there are kind of two camps you could subscribe to. Currently there are two "main" topology books, referred to by their author's names: Hatcher and Munkres. Both are available online for free, but the Munkres pdf isn't legally authorized to be. Reading either of these will make you a topology god. Hatcher is all what's called algebraic topology (relating topology and abstract algebra), which is super necessary for further studies. However, Hatcher is hella hard and you can't read it unless you've really paid attention up to this point. Munkres isn't necessarily "easier" but it moves a lot slower. The first half of it is essentially a recap of Sutherland but much more in-depth. The second half is like Hatcher but less in-depth. Both books are outstanding and it all depends on your skill in specific areas of topology.

Once you've read Hatcher or Munkres, you shouldn't have much trouble going forward into any more specified subfield of topology (be it knot theory or whatever).

If you actually do end up studying topology, please save my username as a resource for when you feel stuck. It really helps to have someone advanced in the subject to talk about tough topics. Good luck going forward. My biggest advice whatsoever, regardless of what you study, is read How to Prove It ASAP!!!

TLDR: How to Prove It (!!!) -> Mendelson -> Sutherland -> Hatcher or Munkres

> Axler + Evan Chen for linear algebra

> Sutherland + Evan Chen for topology

> Tao analysis I and II for analysis

> Pinter + Evan Chen for abstract algebra

> Evan Chen for complex analysis


Thanks :)


Is this what you're referring to
? link, re the napkin project? That's
a neat idea.


How long have you spent going over them?

At what level? Sutherland's Introduction is good. I also recommend Korner's lecture notes.