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..

4I'm not sure if there's such a thing as "the" best (general, I'm assuming) topology textbook. I learned the basics from the first (general) half of Munkres, which I liked. I found that later, when I took abstract real analysis, I really liked the concise but still relatively comprehensive treatment in Folland's text on real analysis (Chapter 4). Of course it's not Bourbaki's General Topology or anything, in terms of coverage, but I still really like it. Incidentally, I also like Bourbaki's General Topology (at least the first volume, which I'm more familiar with). – Keenan Kidwell Oct 22 '10 at 17:29

all right! thank you for your comment :) – gg1 Oct 22 '10 at 17:33

4Do you know what kind of "topology" you want to learn? Topology is a wide subjectarea and there are many entrypoints. Other than pointset topology (which most of the comments below are addressing), differential topology is also a nice entrypoint. Texts by Guillemin and Pollack, Milnor and Hirsch with that (or similar) titles are all very nice. – Ryan Budney Oct 22 '10 at 20:58

Another standard entrypoint might be a knot theory textbook. Like say Adams's book "The knot book" or something similar. – Ryan Budney Oct 22 '10 at 21:05

i definitely don't know what is the scope of my future topology class... I am now only looking for good books. thanks anyway! :) – gg1 Oct 24 '10 at 16:30

I agree with Ryanthis is kind of like asking for a good analysis book. WHAT KIND of analysis? Complex? Functional? Classical Real Analysis/Calculus? You have to be more specific in a question like this. – Mathemagician1234 Feb 13 '12 at 20:34

jgg, you should talk with your professor to see which direction you are taking as this will impact what you should review. For a general (pointset) topology course, I recommend Munkres. – Tyler Clark Oct 04 '13 at 16:14
20 Answers
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 selfstudying.)
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 (nonintroductory 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 67 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 (pointset) 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.
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2For what it's worth, Munkres's algebraic topology only goes into the fundamental group and the theory of covering spaces. If you're interested in the subject, I recommend Allen Hatcher's book, which is available for free on his webpage. Munkres is great for pointset, but not so good for algebraic. – Paul VanKoughnett Oct 23 '10 at 06:00

2Here are links to [Allen Hatcher's homepage](http://www.math.cornell.edu/~hatcher/) and to the [free PDF of his Algebraic Topology textbook](http://www.math.cornell.edu/~hatcher/AT/ATpage.html). Enjoy! – John Tobler Feb 13 '12 at 20:04

3Sorry to revive this. But where did you get those comments by Munkres? – Aloizio Macedo Aug 26 '16 at 04:12

1lolz, I picked up Dugundji as my first book in topology; not working, wayy to hard. I CAN read it, but I am spending so much time on each page that I came here looking for a book with more words in these poofs. Sometimes what he says is $obvious$ takes me 2 days of deep contemplation to figure out. Definitely gives an otherworldly perspective though. – Tsangares Jul 21 '17 at 05:03

4haha; is that "topology without tears" like crying or like ripping paper? – Tsangares Jul 21 '17 at 05:04

Munkres indeed mostly talks about fundamental groups and covering spaces, but his explanations are much better than the ones in Hatcher IMO. – Cronus Aug 20 '17 at 21:41

can you tell me how do I download the Topology without tears? when I am clicking your link it says connection refused. – Abhay Dec 12 '19 at 22:17

1Topology For Beginners by Steve Warner. Requires no prerequisites but mathematical maturity,or the ability to communicate ideas mathematically There is pdf and paperback version both $56 – Feb 19 '21 at 18:16

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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.
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19A note about Munkres: For me, there was very little in the way of intuition in using that book. Also, many counterexamples were quite pathological when simpler counterexamples sufficed. – Bey Oct 22 '10 at 17:25



3I will second the suggestion for Munkres. It is the book I used in my undergraduate topology class, and contains both trivial and nontrivial examples (@Bey, I find some of the more obscure counterexamples to be more interesting in the end, as they provide a perspective I may have not seen myself). You will ultimately want a more advanced book (as Keenan mentioned above), but for the basics Munkres is a great book.. – Brandon Carter Oct 22 '10 at 17:38

11@Bey: I think the way one builds intuition using Munkres is by doing lots of exercises (at least that worked for me when I took his class) rather than having it spoonfed to you. And the pathological nature of the counterexamples is part of the intuition one builds, in the sense that it tells you just how bad the situation can be. – Qiaochu Yuan Oct 22 '10 at 19:29

I think anonymous has listed what areto methe 3 best and most useful textbooks on topology for beginners. Munkres is the best book to learn general and combinatorial topology from,Willard is the best REFERENCE book on general topology and does a great job filling in the blanks of Munkres in the subject,and Armstrong is probably the best allaround introduction to all aspects of the subject,analytic and geometric. – Mathemagician1234 Feb 13 '12 at 20:43

1I would like to second the recommendation for "Basic Topology". Despite having completed an undergrad course in topology, I had no idea why anyone would want to study general topology (instead of just metric spaces) until I read Armstrong's book. – awkward Mar 20 '20 at 13:24
Singer and Thorpe, Lecture Notes on Elementary Topology and Geometry.
A slim book that gives an intro to pointset, 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.
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+1 for a classicbut as you said,the fact it has no exercises is a major hinderance to using it as a textbook. – Mathemagician1234 Feb 13 '12 at 23:16

1Singer & Thorpe is a poor introduction if you’re really interested in pointset topology. – Brian M. Scott Feb 13 '12 at 23:33
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.

For both of your answers on this thread, I edited to add in the link to the originals. – Eric Naslund May 28 '12 at 13:40
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.
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1I don’t consider Simmons a particularly good text if one is interested in pointset topology itself. – Brian M. Scott Feb 13 '12 at 23:35

@BrianM.Scott Maybe not (I borrowed it to someone and forgot it  so I miss it anyway haha). Well, I do remember I liked (1) his treatment on $C(X)$ where $X$ is locally compact, (2) the short introduction $C^*$ algebras, (3) Gelfand theory on Banach algebras and (4) the historical notes. – AD  Stop Putin  Feb 14 '12 at 06:33

BTW Kelly is a bit oldish in style  which I don't really like. – AD  Stop Putin  Feb 14 '12 at 06:35

1That’s why I qualified my objection: I consider at most the first of those important for an introductory text in pointset topology. (I agree with your additional comment on Kelley: I felt a bit that way when I first encountered the book in the late 60s!) – Brian M. Scott Feb 14 '12 at 06:37

I don't like Jänich in gneral, he is to cursory and tries to present the big pcture to people that may not have the background knowledge for making the connections. But for OP, that may actually be a good tourist guide to topology. Not a users guide though. – Michael Greinecker Apr 27 '12 at 11:23

@MichaelGreinecker I don't agree  I think it is a fairly good mix of pointset topology and algebraic topology assuming almost nothing. – AD  Stop Putin  Apr 27 '12 at 14:37

@AD.: I don't think the connections with differential geometry, ODEs etc. are helpful to someone not already familiar with these fields, which may not be the case for beginners in topology. The chapter on topological vector spaces cotains a single result without proof. I actually don't think Jänich is all bad. I think it is useful as a refresher course for people with brad mathematical exprience. But beginners would do well to use something more structured. – Michael Greinecker Apr 27 '12 at 18:11

*A Topological Picture Book* is by George K. Francis. It's a wonderful book but I think it wouldn't be good for learning topology. – MJD Sep 14 '14 at 15:06

@MJD Thank you for the author. All I remember from the book are the pictures. – AD  Stop Putin  Sep 17 '14 at 13:09
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 pathconnected? 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 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 selflearning, but with less preparation than you.
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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.
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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.
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1This is a really awesome book! Best of all, it's provided free (but without any solutions). There is a print version, which comes with hints and some solutions. Moreover, the print quality is fantastic (something I feel lacks in a lot of newer books). – Aug 04 '14 at 03:48
The introduction of Topologie Générale of Bourbaki is a mustread.
Furthemore, the book is brilliantly written and covers almost everything. One of the best books of the Bourbaki series.
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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!
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 pointset 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.
There was another version of this question posted today, and it inspired me to write another MSEthemed 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.
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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 topologythat is, if you read German.
Please look at the review of "Topology and Groupoids"
http://www.maa.org/publications/maareviews/topologyandgroupoids
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.
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2This is a great book for those who want to get into the algebraic or geometric side of topology. The book is quite readable with many great illustrations. It is not as elementary as Munkres, but for a graduate student it would make a nice guide. The only downside is that the geometric viewpoint might be less useful to functional analysts, who need to learn about things like nets, filters and infinite product spaces. – Michael Greinecker Apr 27 '12 at 11:30

What book are we taking about? The link takes me to nowhere. – Rudy the Reindeer May 10 '14 at 09:52

+1 for a remarkable and underrated book that will be particularly helpful to students preparing for a serious algebraic and/or geometric topology course. – Mathemagician1234 Apr 28 '16 at 17:54
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).
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@Leucippus What makes you come to that conclusion? The question is "Best Books for Topology". What else do you expect for an answer? – Brian Mar 07 '20 at 18:43

My comment based on noting over 9000 downloads worldwide. Probably some didn’t like it, but...RF – Ronald Freiwald Mar 11 '20 at 23:26
$1.$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
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
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Topology For Beginners. Dr Steve Warner

It has complete solutions for all the problems. He doesn’t explain the concept of “well defined “ clearly . Not a bad text. It goes through a lot of material – Apr 13 '21 at 13:22
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.
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