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.

Solution 1:

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

Solution 2:

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:

• Viro, et al. Elementary Topology Problem Textbook (Very nice! Also available in hardcover).
• Hatcher's notes

Solution 3:

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.

Solution 4:

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.