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 Unfortunately I don't have a book to recommend since I learned from a set of unpublished lecture notes.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... [2] http://www-personal.umich.edu/~hbray/