Good Textbooks for Real Analysis and Topology. 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. 
 A: 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.
A: 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. 
A: You might try "Principles of Mathematical Analysis" by Walter Rudin.
A: 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.
A: I would recommend Introduction to Topology by Gamelin and Greene for a few reasons:


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*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. 

*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. 

*It is priced under $12 which is a steal.

A: I’m a student in the subject myself and based on all recommendations, I’ve read Rudin, Abbott, Bartle, and Tao.
My personal favorite is Rudin, while I do refer to the other books.
For the general reader, a beginner in the subject, my recommendation goes completely to Understanding Analysis by Abbott. Not only does it guide you through theorems and proofs, it also offers, usually right after the use of one technique, the chance to practice it. 
Compared to older books such as Rudin or Bartle, Abbott’s book is available for most university students with their school’s online library as a typeset pdf, so anyone with an Ipad wouldn’t worry about carrying it around.
I did say that my personal favorite is Rudin, but Rudin’s proofs usually fail to provide the thinking behind them—try reading his proof on Bernoulli’s Rule, you wouldn’t know how to think of it on your own. 
Abbott’s book also covers materials such as $F_\sigma$ and $G_\delta$ sets, so there’s much more to it than being easy to understand—it’s also broader than other books, and deeper in some aspects. It’s really the all-around best stuff.
Bartle and Tao on the other hand, I didn’t really like Bartle that much—I didn’t like its structure. Who on earth does integration first, then Infinite series? Tao’s book is a bit too slow for me. While Rudin’s already full on talking about integrals, Tao’s probably just started on limits.
A: 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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*Viro, et al. Elementary Topology Problem Textbook (Very nice! Also available in hardcover).

*Hatcher's notes
A: 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.
A: 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. 
A: 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
