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Comment by nikofeyn | Hacker News Reader

you will understand very little of the math in this book without a substantial background in mathematics. however, and i think the author would agree, you should jump in anyway. you'll pick up pictures, applications, vocabulary, and some ideas, all of which could help pave the way as you learn the material more traditionally. the book sets goalposts so to speak, especially since it is written in a more expository manner with lots of example applications. i don't understand anywhere near everything in the book (much less than that probably), and i still get something from it.

in terms of learning some topology and geometry, i recommend topology by james munkres and an introduction to manifolds by loring tu. both are accessible to junior and senior undergraduates and beginning graduate students.

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I strongly second the recommendation of Munkres' Topology. It's basically the gold standard. However, I think don't think it's accessible to junior/senior undergraduates unless they've taken a prior course in analysis. Technically Munkres' Topology has no formal prerequisites, but it's a rough go of it if you haven't had a rigorous exposure to continuity and metric spaces already. The second part of the book also assumes knowledge of elementary abstract algebra.

If the person you're replying to is a junior or senior undergrad in a math major they might be alright (presumably they'll have taken analysis by then). If not, Topology is not what I'd recommend as the first place to learn about metrics and continuous functions (Calculus doesn't cut it for that).

Thank you. I am actually a self- learner and have only take calculus, linear algebra, probability, statistics. Currently learning bayesian statistics and reading “first course in abstract algebra” by John B Fraleigh.

I am guessing I should do Real Analysis and then the Munkres book to get an idea of topology ?

Right now, I'm self-studying analysis with Stephen Abbott's "Understanding Analysis". I strongly recommend it. It has clear explanations that help to develop the intuition about each concept and what I think are well chosen exercises.

there is a book by jay cummings called real analysis: a long-form mathematics textbook that i highly recommend checking out. used with a more "advanced" book, it could be a nice complement.

Real Analysis: A Long-Form Mathematics Textbook https://www.amazon.com/dp/1724510126/

I personally really enjoy Gamelin and Green's Topology, and (if you need it) the first chapter goes through topology of metric spaces. The rest of the book is well written; I couldn't quite find myself to like Munkres.

agreed. i had it in mind the person was a math major. although i think topology is approachable without analysis with some dedication, it certainly helps provide context for the abstraction.

i listed another nice book in another comment, but metric spaces: iteration and application by victor bryant is a pretty fun introduction to metric spaces and analysis at a very approachable level.

I know Munkres is the go-to standard for topology, but I think that Essential Topology by Martin D. Crossley is a better book. It's only 200 pages, packed full of great examples, and covers all the (as the title says) essential topics in undergrad topology (topological spaces, continuity, connectivity, compactness, Hausdorff property, homeomorphisms, homotopy, Euler number, homotopy groups, simplicial homology, etc.)

+1 on munkres. Also recommend Algebraic Topology by Hatcher as a followup (free on his website) and Introduction to Smooth Manifolds by Lee.

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