I am a graduate student of math right now but I was not able to get a topology subject in my undergrad... I just would like to know if you guys know the best one..
As an introductory book, "Topology without tears" by S. Morris. You can download PDF for free, but you might need to obtain a key to read the file from the author. (He wants to make sure it will be used for self-studying.)
Note: The version of the book at the link given above is not printable. Here is the link to the printable version but you will need to get the password from the author by following the instructions he has provided here.
Also, another great introductory book is Munkres, Topology.
On graduate level (non-introductory books) are Kelley and Dugunji (or Dugundji?).
Munkres said when he started writing his Topology, there wasn't anything accessible on undergrad level, and both Kelley and Dugunji wasn't really undergrad books. He wanted to write something any undergrad student with an appropriate background (like the first 6-7 chapters of Rudin's Principles of Analysis) can read. He also wanted to focus on Topological spaces and deal with metric spaces mostly from the perspective "whether topological space is metrizable". That's the first half of the book. The second part is a nice introduction to Algebraic Topology. Again, quoting Munkres, at the time he was writing the book he knew very little of Algebraic Topology, his speciality was General (point-set) topology. So, he was writing that second half as he was learning some basics of algebraic topology. So, as he said, "think of this second half as an attempt by someone with general topology background, to explore the Algebraic Topology.
I would suggest the following options:
Topology by James Munkres
General Topology by Stephen Willard
Basic Topology by M.A. Armstrong
Perhaps you can take a look at Allen Hatcher's webpage for more books on introductory topology. He has a .pdf file containing some very good books.
Singer and Thorpe, Lecture Notes on Elementary Topology and Geometry.
A slim book that gives an intro to point-set, algebraic and differential topology and differential geometry. It does not have any exercises and is very tersely written, so it is not a substitute for a standard text like Munkres, but as a beginner I liked this book because it gave me the big picture in one place without many prerequisites.
Seebach and Steen's book Counterexamples in Topology is not a book you should try to learn topology from. But as a supplemental book, it is a lot of fun, and very useful. Munkres says in introduction of his book that he does not want to get bogged down in a lot of weird counterexamples, and indeed you don't want to get bogged down in them. But a lot of topology is about weird counterexamples. (What is the difference between connected and path-connected? What is the difference between compact, paracompact, and pseudocompact?) Browsing through Counterexamples in Topology will be enlightening, especially if you are using Munkres, who tries hard to avoid weird counterexamples.
Note: This answer was also posted here, on a question which is now closed.
You might consider Topology Now! by Messer and Straffin. Their idea is to introduce the intuitive ideas of continuity, convergence, and connectedness so that students can quickly delve into knot theory, the topology of surfaces and three dimensional manifolds, fixed points, and elementary homotopy theory. I wish this book had been around when I was a student!
I know a lot of people like Munkres, but I've never been one of them. When I read sections on Munkres about things I've known for years, the explanations still seem turgid and overcomplicated.
I like John Kelley's book General Topology a lot. I find the writing stunningly clear. It has been in print for sixty years. You should at least take a look at it.
Remark: This answer was also posted here, on a question which is now closed.
Hope I didn't miss this above: Gamelin & Greene "Intro to Topology." When I was looking for a text, I noticed as an endorsement, that it was used by Terry Tao. But don't think of it as nepotism (the authors and T. Tao are at UCLA), as Prof. Tao said in the syllabus that the text will be followed closely.
The first half is point-set topology and the second is algebraic topology.
Actually the book is replete with examples as each section is followed by questions which are answered at the back of the book.
And a special consideration - it is (as a noted mathematician coined the term) Doverised. At $10+ it is a gift.
Please look at the review of "Topology and Groupoids"
This book is the only topology text in English to deal with the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, and so deduce the fundamental group of the circle, a method dating back to 1967. See this mathoverflow discussion.
Here are a couple of my favorites:
- Gaal, Stephen. Point Set Topology.
- Wilanksy, Albert. Topology for Analysis.
- Laures and Szymik. Grundkurs Topologie.
Gaal has an excellent section on connectedness. Very concise and clear.
Wilansky has an excellent section on Baire spaces and induced topologies. It's a little wordier than Gaal, but has many excellent exercises.
Laures and Szymik write an excellent book on topology that incorporates category theory seamlessly. The proofs are also very different from the typical presentations I see in American books. It's good for a second pass through for topology---that is, if you read German.