Best book of topology for beginner? 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..
 A: 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.
A: You might look at the answers to this previous MSE question, which had a slightly different
slant: "choosing a topology text".  Apparently the poster was also interested in self-learning,
but with less preparation than you.
A: 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.
A: I own Bert Mendelson's "Introduction to Topology" and it looks good. I bought Alexandroff's "Elementary Concepts of Topology" too - believe me, it's not good for an introduction.
A: 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!
http://people.albion.edu/ram/TopologyNow!/ 
A: I recommended Viro's Elementary Topology. Textbook in Problems.
This book is very well structured and has a lot of exercises, the only thing is it do not talk about uniform structure, I think for this part you can read Kelley or Bourbaki.
A: The introduction of Topologie Générale of Bourbaki is a must-read.
Furthemore, the book is brilliantly written and covers almost everything. One of the best books of the Bourbaki series.
A: 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. 
A: Please look at the review of "Topology and Groupoids" 
http://www.maa.org/publications/maa-reviews/topology-and-groupoids
See also 
http://groupoids.org.uk/topgpds.html
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. 
A: There was another version of this question posted today, and it inspired me to write another MSE-themed blog post. So I have collected most of the topology recommendations from MSE (and a few from MO and a few other sources) and written up a post at my blog, mixedmath.
A: 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. 
A: Priniples of Topology, by F.H. Croom 
Foundation of Topology by C.W. Patty
A: 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.
A: 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.
A: 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.
A: *

*I also like Bourbaki's treatise, but some times it is a bit too logical. 

*Introduction to Topology and Modern Analysis by G F Simmons

*Also, "A topological picture book" by George K. Francis.

*K Jänich Topology.

*J. Kelley General topology.
A: For general topology: my preference among commercial boos is Steve Willard‘s General Topology.  For something free, try googling Freiwald, Introduction to Set Theory and Topology.  I like it because I wrote it, but students seem to like it a lot (and I earn absolutely 0 if you use it).
A: $1.$If ur basic is weak try this Topology with Diagram explaination see here for  Pdf
$2.$Basic concept of topology see here for  Pdf
$3.$General Topology by Stephen Willard pdf
Solution of stephen willard
For short Note see this
 topology blog
Personally  I don't recommend Topology  without tears ( That is not good book for beginner , personally i have read  for 1 week   after  that i leave it  because  concept  is not given systematics, ( too many concept is missing in that book)and they exclude many concept   in Topology without tears
A: Topology For Beginners. Dr Steve Warner
A: Neuwirth's "Knot Groups" (Princeton University Press, 1965) explains how to stroll around covering spaces of knot complements by walking through walls (like Superman). It's a bit short on pictures, but if you can't fill that gap from your own intuition you probably shouldn't be studying topology in the first place. For the more mature student, we also recommend Kerekjarto's classic "Vorlesungen uber Topologie I" (Springer, 1923), especially its introductory illustration opposite page 1: a "Zerlegung der Kreisscheibe in drei Gebiete mit demselben Rand." --Ken Perko, lbrtpl@gmail.com.
A: I looked at James Dugundji’s book Topology online. It is extremely thorough. It gives many proofs and tons of exercises. The whole text has been digitized at www.archive.org
Topology Allyn and Bacon Series in Advanced Mathematics by James Dugundji 
Dugundji states the book is self contained and just needs a strong background in analysis. This probably means an introductory course
plus a real analysis course or something similar.
$298 is rather expensive for a text.
