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I'm currently in my 3rd year of my undergrad in Mathematics and moving onto my 4th year next year. I took a course in Real Analysis I, but the professor was very confusing and we didn't use a textbook for the class (it was his lecture notes) which was also very confusing as well. Although I got a good mark in the course, I don't think I learned anything from the class since I felt like I memorized how to do the questions rather than actually understood the questions.

So I was wondering what would be a good textbook for real analysis. I'm okay with a textbook with rigorous proofs, as long as everything is explained in good detail.

At the same time, I also want to teach myself Topology. I wanted to take it this year, but there was a conflict and hence I could not take the course. So I was also looking for a textbook for an introduction to Topology.

Any kind of recommendations would be great! I really do want to learn analysis and topology.

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This book covers analysis and topology for the 3rd university year but it's in french. – whatever Apr 19 '14 at 18:14
At a similar point in my mathematical career I enjoyed Kaplansky's book Set Theory and Metric Spaces. It's not really topology and it's not really analysis; it's sort of on the border of the two, and it will give you a good grounding to go either direction in the future. – MJD Apr 19 '14 at 18:49
Real and Complex Analysis by Rudin, Topology by Munkres – Cure Apr 20 '14 at 1:36
x2 for MJD's recommendation. Beautiful book. – user59083 Apr 20 '14 at 7:11

You might try "Principles of Mathematical Analysis" by Walter Rudin.

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I don't understand the downvotes. Rudin is a must read for any mathematician worth his salt. – Joseph DiNatale Apr 19 '14 at 19:45
+1 for Rudin. I agree that it is a good book that is appropriate for Analysis. I'm not sure it's a great self-study book, though. Personally, I'm not an analyst, and I find it hard to weed through Rudin. It doesn't hold your hand, which is appropriate for the course. I think it's a good book to use in a classroom, but not on its own. That's just my $0.02. – ml0105 Apr 19 '14 at 19:53
In my opinion, Rudin has many excellent qualities, but is not a good choice for self-study. – MJD Apr 19 '14 at 21:00
@MJD Could you please elaborate? Baby Rudin was essentially the first book about "fun mathematics" I've read (after a bit of logic and set theory). I've worked through it and found it excellent. – Marcin Łoś Apr 20 '14 at 7:33
Baby Rudin is fine for self study if you're bright. – bwv869 Jun 25 '14 at 1:16

My favorite analysis book is that by Pugh. It's similar to Rudin, but readable and with tons of fantastic problems. I also like Zorich (2 volumes) and Amann (3 volumes). I haven't read a lot of the latter, but it looks really cool and has interesting problems.

So far, I really like Runde for topology. It's rigorous, and short enough that you'd want to read it front to back. It has what you need if you want to continue on to more advanced topics, and is enough even if you don't! Another nice book to read while you're learning topology (as a supplement) is Janich's. In my opinion, these books are far better than Munkres.

For topology, you might also be interested in these free options:

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Oh yeah, and Pugh covers a decent amount of topology on its own! – user59083 Apr 20 '14 at 2:55
I haven't read much of Pugh, but from what I did, it seemed absolutely great - the approach seemed a lot more intuitive at some places compared to Rudin. – Maroon Apr 20 '14 at 5:36
Reading Pugh and Rudin simultaneously is not a bad idea, either! – user59083 Apr 20 '14 at 6:02
I second Runde's book. He covers some interesting material on point-set topology that I haven't seen in other works, such as the Zariski topology and Stone-Cech compactification. – SpamIAm Apr 20 '14 at 19:58
second for Amann, very complete, rigorous and modern covering of analysis. – Xingdong Dec 8 '14 at 23:15

Analyis: Spivak - Calculus or Abbott's Understanding Analysis. Might also want to pick up Gelbaum's Counterexamples in Analysis.

Topology: Munkres - Topology, as well as Steen's Counterexamples in Topology to go with it.

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Spivak is only good for univariate calculus. Definitely a great textbook though. – Maroon Apr 20 '14 at 5:35
@hungerartist As far as I know, Spivak's Calculus on Manifolds is one of the most liked short introductions to (mature) multivariable calculus and differential geometry. It's a great book to run through. – snarski Apr 20 '14 at 18:07
Oh sorry, I was thinking of his univariate text titled "Calculus". – Maroon Apr 20 '14 at 18:29

I consider Folland's “Real Analysis: Modern Techniques and Their Applications” as the best textbook ever written, on any subject. I warn you, though, it is extremely dense: If you want to be thorough and check everything for yourself (as many really easy results are just mentioned in passing without proof and there are a lot of exercises, some quite easy, some extremely difficult), then you may well need to be reading some pages literally for days.

I have found, for what it's worth, that it's worth the effort, though. For me, it has been a costly investment in terms of time and effort but the returns have been even more enormous: before starting to read this book a couple of years ago, I had known next to nothing about measure theory, topology, and functional analysis. Now I feel quite comfortable about having a fair working knowledge of these topics. I definitely suggest at least giving it a try.

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Notice that Folland's book has a prerequisite of at least one year of analysis, at the level of Rudin's "Mathematical Analysis" or Strichartz "The Way of Analysis." – Graduate May 10 '14 at 3:09
@Graduate You're right, it is not intended to be an introductory textbook and prior exposure to less advanced topics (especially calculus) is needed. The author explicitly recommends the same textbooks as you do and enumerates the following prerequisites: limits and continuity, differentiation and Riemann integration, infinite series, uniform convergence, metric spaces, basic arithmetic of complex numbers, elementary set theory, and a bit of linear algebra. In my experience, if you have these, reading this book will still be an intellectual challenge, but a wonderful one. – triple_sec May 12 '14 at 2:29

I have heard good things about Abbott's "Understanding Analysis", though I have only glanced at it myself. For self-learning topology I used (many years ago, but a situation like yours) George Simmons' "Introduction to Topology and Modern Analysis" and recommend it highly.

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The book by Simmons is a didactic masterpiece. You can even read it like a novel having lots of fun. – Cure Apr 20 '14 at 2:10

I learned a lot of this material (real analysis, and some of the basic ideas of point-set topology) from Elements of real analysis by Bartle.

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I would recommend Introduction to Topology by Gamelin and Greene for a few reasons:

  1. Covers the Point-Set Topology that will be very helpful to know when studying Real Analysis. The authors do an excellent job of covering applications of metric space topology to Analysis.
  2. The authors provide solutions or at least guidance on a large set of the problems in the book. When you are self-studying as you are - it is nice to know when you are on the right track or not.
  3. It is priced under $12 which is a steal.
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This is a nice book and my second favorite to Runde. I know other people who dislike it, but I don't see why... I still much prefer it to Munkres. – user59083 Apr 23 '14 at 5:31

The book I would recommend for an introductory course to real analysis is Real Analysis by Bartle and Sherbert. I found it perfect for a first course in real analysis. As for topology, the book I prefer is Topology by J. Munkres.

Another book that I would recommend for real analysis is Mathematical Analysis by T. Apostol.

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Dear Shalin Doctor, Do you know, is Bartle and Sherbert the same book (in a new edition) that was once published under Bartle's name alone? Regards, – Matt E Apr 20 '14 at 2:20
@MattE They aren't the same book. The older book is a bit more advanced. – JasonJones Apr 20 '14 at 3:25
@JasonJones: Dear Jason, Thanks. I added an answer recommending the older book. Cheers, – Matt E Apr 20 '14 at 4:27

You can try Modern Analysis and Topology (Universitext). Here's the table of contents:

part 1

part 2

part 3

part 4

part 5

part 6

part 7

part 8

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This looks really cool. Thanks for posting it! But this is probably rough going for someone without sure-footing in elementary analysis/topology. – user59083 Apr 23 '14 at 5:36

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