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 Unfortunately I don't have a book to recommend since I learned from a set of unpublished lecture notes.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/
Would you be able to give an example of such a proof cluster?
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".
> A rigorous course in linear algebra is an absolute necessity. In the early chapters, one may be able to avoid the need for a modern algebra course but not the maturity it requires.
This is a requirement that is rather unusual for a first book in topology, many of which have not a drop of linear algebra. The reason for this is presumably that the book dives immediately into homology which is often left for an algebraic topology course. It's an interesting choice.
I think we gained a really good intuitive understanding of homeomorphism, neighborhoods, and homology groups. Knowing about these topics has been super useful in my studies and career.
On the other hand, some of the algebraic components of topology went totally over my head. So I do think there’s a limit to how much you can understand without diving into other fields.
Also, we had the benefit of a trusty excellent mathematician as our teacher, which helped quite a bit. Good luck diving into this field, I hope you find it rewarding!
That said, a background in some mathematical topics is useful for learning topology. The subject is rooted in a particularly abstract notion of distance, so it's useful to have some experience reasoning with somewhat less abstract notations of distance like metrics, the most familiar of which is the quite numerical Euclidean metric involving the familiar square root of the sum of squares. Familiarity with metric spaces in general (that is, spaces equipped with any axiomatically acceptable notion of a metric distance) is even more useful because it requires a similar sort of axiomatic reasoning as does topology.
Kolmogorov & Fomin's classic text on analysis, the Dover edition of which is under $12 on Amazon [0], has a good (albeit austere) introduction to topology, leading up to it via axiomatic set theory and metric spaces in the book's first chapters. On the one hand, I hesitate to recommend this book, because it was the text for a course that mercilessly exposed the shortcomings in my mathematical education to that time; on the other hand, I recommend it for precisely that reason. Having later assimilated the material, I do consider the book a very good introduction for anyone who either already has the mathematical maturity to do the book properly or for anyone who wants to gain that ability in the way that everybody who has done does: by staring at the same page for hours on end while working everything out on paper, down to the axioms if necessary, until you stop misunderstanding, and then start understanding.
That said, there surely are gentler introductions to the subject if you just want to get a rough idea of it.
[0] https://www.amazon.com/Introductory-Analysis-Dover-Books-Mat...
Did you mean without? Algebraic topology would be a pretty standard 4th year undergrad course in the UK.
Introductory texts in topology tend to divided into two main subareas - point set topology (the low level axioms of how you define a space and its connectivity properties), and algebraic topology, which focuses on the global mathematical structures of these topological spaces. I found point set topology extremely dry when I first learned it, but it's important to understand those low level basics before moving on to the "cool stuff" in algebraic topology. I recommend "Topology" by Munkres to learn the fundamentals.