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Recently I've been trying to decide on some fun math summer reading on some areas of math which I have less experience with. I'm an undergrad studying mathematics with a focus in actuarial science, but my focus has prevented me from taking classes in areas of pure mathematics. I basically just want to get more familiar with various subjects, to get a better sense of what I like, in case I decide to go to grad school, and also just out of interest.

By the time I graduate, I don't expect to have taken any classes in the following areas: topology, (real?) analysis, and perhaps some other subjects that I have no idea about. What I'm looking for is an introductory book in one of these areas which is relatively short (preferably $<200$ pages), and not too highly-technical. Do such books exist? The books that I see suggested elsewhere seem like they will be too long for me to really have a chance to get through.

A couple books that I have read at least partially in my spare time are Lawvere's Introduction to Categories, and Lancaster's Curve and Surface Fitting. Books in a similar style would be great.

I'd prefer books in topology or analysis, but I'm definitely open to other suggestions. I really enjoyed my introductory course on number theory/group theory, a suggestion for a more in-depth book on the subject would also be welcome.

Apologies if this question is a bit broad or in the wrong place.

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closed as too broad by user223391, Claude Leibovici, R_D, JonMark Perry, Siminore Jul 7 '16 at 10:00

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Basic Analysis: Introduction to Real Analysis by Jiří Lebl is an approachable and freely available text on real analysis: jirka.org/ra $\endgroup$ – GPhys Jul 7 '16 at 2:24
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    $\begingroup$ I rather enjoyed reading the knot book by colin adams as a soft introduction to knot theory. $\endgroup$ – JMoravitz Jul 7 '16 at 2:26
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    $\begingroup$ Calculus on Manifolds by Spivak. Galois Theory by Artin. Let's hear it for skinny math books. $\endgroup$ – user4894 Jul 7 '16 at 2:42
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    $\begingroup$ Switch to a pure math major. Listen to your heart. $\endgroup$ – James S. Cook Jul 7 '16 at 3:06
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    $\begingroup$ @user4894 I think Spivak and Artin are way too difficult for students with weak backgrounds and who aren't ready to chew through them. $\endgroup$ – Mathemagician1234 Jul 7 '16 at 4:00
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For topology, John McCleary's A First Course in Topology: Continuity and Dimension could be exactly what you're looking for. It's fairly short, covers the essentials of both point set and low dimensional algebraic topology in a rigorous yet very pictorial way and contains a wonderful historical slant that makes it a pleasure to read. This combined with a series of terrific problems as well as suggestions for further reading make it a great choice I think you'll find very helpful.

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