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In my university, the "official" book for topology is https://www.amazon.com/Topology-2nd-James-Munkres-dp-0131816... It's quite advanced and assumes you have taken a few math courses, but it not centered in curves.

I'm a big fan of SICP. I'd describe it as "hard-core": requiring hard work, but extremely rewarding for the people who take the time and effort to go through it.

I'd recommend the following three books to similarly cover the three main areas of higher mathematics (analysis, algebra, and topology):

1. Principles of Mathematical Analysis by Rudin (http://www.amazon.com/Principles-Mathematical-Analysis-Inter...)

2. Algebra by Artin (http://www.amazon.com/Algebra-2nd-Featured-Titles-Abstract/d...)

3. Topology by Munkres (http://www.amazon.com/Topology-2nd-James-Munkres/dp/01318162...)

Working through all three (including the exercises!) will give you a solid understanding of the basis of modern mathematics. If you don't have experience with proofwriting, you might find them difficult at first - the activity is very different from performing calculations or solving equations. It's also best to have someone trained in mathematics talk to you about the proofs, until you develop a feel for the needed level of logical rigor.

Rudin, in particular, leaves a lot of work to the reader; going through that book is the most intellectually difficult work I've ever done. If you find it hard-going (which is completely natural), you might want to try Artin first, especially since you have some background in algebra.

(Incidentally, these are the three books used to teach analysis, algebra, and topology to MIT mathematics majors. You can look up the assignments and exams for the three courses - 18.100B, 18.701, and 18.901 - for a good list of exercises to work through.)

⬐ swordswinger12

Personally I have heard nothing but bad things about Artin. Most people I know recommend Dummit and Foote instead; its increase in difficulty is supposedly offset by superior presentation of the material.

⬐ radmuzom

Great list. Another addition to the list would be

4. Topics in Algebra by I.N.Herstein.

It has a slightly better (and tougher) set of problems than what you would find in the book by Artin.