For algebra, Dummit and Foote is as easy and gentle an introduction as you could ask for. It's not quite as easy for topology.
A MSE Collection: Topology Book References
To be clear, this is a compilation of much (not all) of the discussion in the following questions (and their answers): best book for topology? by jgg, Can anybody recommend me a topology textbook? by henryforever14, choosing a topology text by A B, Introductory book on Topology by someguy, Reference for general-topology by newbie, Learning Homology and Cohomology by Refik Marul, What is a good Algebraic topology reference text? by babgen, Learning Roadmap for Algebraic Topology by msnaber, What algebraic topology book to read after Hatcher’s? by weylishere, Best Algebraic Topology book/Alternative to Allen Hatcher free book? by simplicity, and Good book on homology by yaa09d. And I insert my own thoughts and resources, when applicable. Ultimately, this is a post aimed at people beginning to learn topology, perhaps going towards homology and cohomology (rather than towards a manifolds-type, at least for now).
One cannot mention references on topology without mentioning Munkres Topology. It is sort of ‘the book on topology,’ and thus I’ll compare all other recommendations to it. When I was blitz-preparing from my almost-no-topology-outside-of-what-I-learned-in-analysis undergraduate days to Brown’s (and the rest of the world) topology-is-everywhere days, I used Munkres. In fact, Munkres once said that he explicitly wrote his book to be an accessible topology book to undergrad. I thought it was alright, although it wasn’t until much later that I began to see why anything was done the way it was done. If the goal is to immediately go on to learn algebraic or differential topology, some think that the first three chapters of Munkres is sufficient. Munkres serves primarily to give an understanding of the mechanics of point set topology.
Willard’s General Topology is a more advanced point-set topology book. The user Mathemagician1234 says in a comment that it is practically the Bible of point-set topology, and extremely comprehensive. It has a certain advantage over Munkres in that it’s published by Dover for cheap. In a comment on a different thread, Brian M. Scott suggests that for a mathematician of sufficient maturity, Willard is superior to Munkres and Kelley’s General Topology. Sean Tilson wrote up an answer discussing Willard as well. In short, he thinks it is superior to Munkres, but that it is necessary to do the exercises to learn some key facts. In Munkres, this is not the case, but one needs to do exercises to develop any sort of intuition with Munkres.
If Willard is much harder than Munkres, it would seem that Armstrong’s Basic Topology is comparable. It’s part of the Undergraduate Texts in Math series. I know many people who learned point set from Armstrong rather than Munkres. I have personally only cursorily glanced through Armstrong, and I found it roughly comparable to Munkres. When I read the reviews on Amazon for Munkres and Armstrong, however, there is a strange discrepancy. Munkres far outscores Armstrong (whatever that really means). Perhaps this is due to the recurring theme in the top reviews of Armstrong, which say that Armstrong deviates from the logical (and some would say, intuition-less) flow of Munkres to instead ground the subject in a diverse set of subjects. On the other hand, this review says that Munkres is simply more extensive. This ultimately leads me to believe that it’s a matter of taste between the two (although Munkres is much longer). But the autodidact’s guide recommends this book as an undergraduate text.
On a different note, there is a freely read (but not printed) book by S. Morris called Topology without tears. This is a link to the pdf on his website. I should note that the version linked is not printable. There is a printable version, but you will need the author’s permission to print it. I think this book is very readable, and is perhaps the gentlest introduction text here (for better or worse).
While all the texts mentioned so far have sort of been aimed at developing the skill set necessary to learn algebraic/differential topology, there is a book Introduction to Topology and Modern Analysis by Simmons that is instead aimed at the topology most immediately relevant to analysis. It comes very highly recommended for those interested in that niche.
If, on the other hand, you are gung-ho towards algebraic topology, then Massey’sA basic course in algebraic topology has sufficient introduction to the material that it might be used as a first topology text (vastly helped by a course in analysis). But the homology theory section is… lacking.
Finally, John Stillwell wrote a book Classical Topology and Combinatorial Group Theory which has a very interesting presentation and selection of material is atypical. But it seems like a pretty interesting angle of approach.
One might consider using Singer and Thorpe’s Lecture Notes on Elementary Topology and Geometry as a supplement to one of the above texts, as it is known to give a big picture. It actually introduces point-set, algebraic, and differential topology as well as some differential geometry. But it’s very short, very terse, and it has no exercises. So it is absolutely not a replacement for learning point-set topology. One might also use Viro’s Elementary Topology (mentioned below) or Counterexamples in Topology (mentioned below) as a supplement. Counterexamples might be particularly nice, because Munkres has a tendency to give really pathological counterexamples sometimes.
It’s also good to know what books to not use as an intro text, but that might be great supplements or advanced texts. This includes:
- Alexandroff’s Elementary Concepts of Topology: It is simply too hard.
- Viro’s Elementary Topology, Textbook in Problems (this is a link to his website, where it’s freely available): although it would be a good supplement.
- Seebach and Steen’s Counterexamples in Topology (a Dover book): it doesn’t serve to teach topology, but it’s full of exactly what it says – counterexamples.
- Dugundji’s Topology or John Kelley’s General Topology: these are graduate texts, and they remind me of the difference between, say, Atiyah-Macdonald (an intro-t0-commutative-algebra text) and Matsumura or Eisenbud (more advanced texts, both more useful and much more difficult). In other words, these are good ‘next’ books on topology, but not introductions. For someone who already knows their stuff, though, these seem great.
- Engelking’s General Topology: this is the reference on topology, but not an introduction. This is to general topology what Lang is to Algebra. It is not aimed at differential nor algebraic topology, but is instead designed to be a reference. If interested, make sure to get the 1989 second edition, which is expanded and updated.
- Bourbaki: this is another reference, not meant for an introduction to the subject. This answer on MathOverflow has comments to it that speak of some of the unfortunate pitfalls of Bourbaki as well.
- Spanier: It has a tremendous amount of material, but I’ve almost never heard or read any other redeeming aspect of the book. Chicago’s mathematics bibliography mentions that “… glad I have it, but most people regret ever opening it.
- Almost any book with “Algebraic Toplogy” or “Differential Topology” in the title.
Now, away from the intro-to-general topology and instead towards intro-to-algebraic topology:
The classic choice here is now Allen Hatcher’s Algebraic Topology (this is a link to his webpage, where he has the book available for free download). Some might say that Peter May’s freely available book is the other ‘obvious’ choice. But for this, it’s easy to say that since they’re both freely available, you might just try them both until you find one that you prefer.
Hather’s book has a large emphasis on geometric intuition, and it’s fairly readable. I learned from this book, and my initial dislike (I had a hard time with some of the initial arguments) turned into a sort of respect. It’s a fine book. May’s book (called A Concise Course in Algebraic Topology) is, in May’s own admission, too tough for a first course on the subject. But it has a very nice bibliography for further reading.
Tammo Tom Dieck has a new book, Algebraic Topology, which just might be loved (even by Hatcher himself). And Rotman’s book is generally well-regarded.
Interestingly, the most common response at MSE when asked “What is the best alternative to Hatcher?” was along the lines of “No, use Hatcher. It’s lovely.”
There are also some additional free sources of notes. There are complete lecture notes of K. Wurthmuller and Gregory Naber (available at their websites and elsewhere).
I want to point out the autodidact’s guide and Chicago’s undergraduate mathematics bibliography again, because they’re good places to look in general for such things.
I learned from Herstein for algebra and Munkres for topology, and they both require you to do exercises to grasp the material. And that's how learning almost anything else in math from a text will be, so that's okay.