Hello and thank you for being such a helpful community, being a self taught person, communities like this one are an essential source of (complementary of course..) knowledge.
Background (very resumed): I always used to be a self taught learner, mostly programmer, interested (mostly) in topics of computer science. Some years ago I started to focus into formal logic and mathematics. My biggest difficulty in doing so is to find a good path, i.e. to understand the order in which things should be learned.
Now I'm reading Bert Mendelson's "Introduction to Topology" which in the back cover says "It is directed to undergraduate students whose studies of calculus sequence have included definitions and proofs of theorems"
Al least for chapter one, I understand the book and the proofs, but doing the exercises is annoying. I understand what the exercises mean, and I understand what should I prove. Problem is, the logic I used and still use to learn is completely abstract and formal, without any relation to mathematics so here I don't know how to proceed in the style of the author, without any formal system or specific set of rules. Everything I can think of feels very loose and arbitrary. Should I read Velleman's "How to prove it" or any other book oriented to mathematical proofs? Should I skip exercises and keep going..?
PD (if relevant): I also did a first course in calculus, but it wasn't a rigorous one.
Thank you.
You should definitely look into proof writing. A good example of this would be to take a rigorous course (or read a rigorous book) in linear algebra and abstract algebra. Here you can learn to prove lots of different things that are more closely related to topology than calculus. If you know nothing about linear or abstract algebra, you can still learn a good amount of topology, but the proofs in those algebra courses are very helpful.
Lastly, set theory is a very helpful subject to learn about proofs, and this would set you up perfect for topology.
I hope I helped!