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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..

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    $\begingroup$ I'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). $\endgroup$ Oct 22, 2010 at 17:29
  • $\begingroup$ all right! thank you for your comment :) $\endgroup$
    – gg1
    Oct 22, 2010 at 17:33
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    $\begingroup$ Do you know what kind of "topology" you want to learn? Topology is a wide subject-area and there are many entry-points. Other than point-set topology (which most of the comments below are addressing), differential topology is also a nice entry-point. Texts by Guillemin and Pollack, Milnor and Hirsch with that (or similar) titles are all very nice. $\endgroup$ Oct 22, 2010 at 20:58
  • $\begingroup$ Another standard entry-point might be a knot theory textbook. Like say Adams's book "The knot book" or something similar. $\endgroup$ Oct 22, 2010 at 21:05
  • $\begingroup$ i definitely don't know what is the scope of my future topology class... I am now only looking for good books. thanks anyway! :) $\endgroup$
    – gg1
    Oct 24, 2010 at 16:30

21 Answers 21

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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.

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    $\begingroup$ For 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 point-set, but not so good for algebraic. $\endgroup$ Oct 23, 2010 at 6:00
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    $\begingroup$ Here are links to Allen Hatcher's homepage and to the free PDF of his Algebraic Topology textbook. Enjoy! $\endgroup$ Feb 13, 2012 at 20:04
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    $\begingroup$ Sorry to revive this. But where did you get those comments by Munkres? $\endgroup$
    – Aloizio Macedo
    Aug 26, 2016 at 4:12
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    $\begingroup$ lolz, 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. $\endgroup$
    – Tsangares
    Jul 21, 2017 at 5:03
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    $\begingroup$ haha; is that "topology without tears" like crying or like ripping paper? $\endgroup$
    – Tsangares
    Jul 21, 2017 at 5:04
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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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    $\begingroup$ A 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. $\endgroup$
    – Bey
    Oct 22, 2010 at 17:25
  • $\begingroup$ @Chandru: thank you for the suggestions. $\endgroup$
    – gg1
    Oct 22, 2010 at 17:34
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    $\begingroup$ I will second the suggestion for Munkres. It is the book I used in my undergraduate topology class, and contains both trivial and non-trivial 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.. $\endgroup$
    – bzc
    Oct 22, 2010 at 17:38
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    $\begingroup$ @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. $\endgroup$ Oct 22, 2010 at 19:29
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    $\begingroup$ I 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. $\endgroup$
    – awkward
    Mar 20, 2020 at 13:24
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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.

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  • $\begingroup$ For both of your answers on this thread, I edited to add in the link to the originals. $\endgroup$ May 28, 2012 at 13:40
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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.

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  • $\begingroup$ ok thank you. I'll try to see this one! $\endgroup$
    – gg1
    Oct 24, 2010 at 16:31
  • $\begingroup$ +1 for a classic-but as you said,the fact it has no exercises is a major hinderance to using it as a textbook. $\endgroup$ Feb 13, 2012 at 23:16
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    $\begingroup$ Singer & Thorpe is a poor introduction if you’re really interested in point-set topology. $\endgroup$ Feb 13, 2012 at 23:33
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  1. I also like Bourbaki's treatise, but some times it is a bit too logical.

  2. Introduction to Topology and Modern Analysis by G F Simmons

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

  4. K Jänich Topology.

  5. J. Kelley General topology.

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    $\begingroup$ I don’t consider Simmons a particularly good text if one is interested in point-set topology itself. $\endgroup$ Feb 13, 2012 at 23:35
  • $\begingroup$ @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. $\endgroup$ Feb 14, 2012 at 6:33
  • $\begingroup$ BTW Kelly is a bit oldish in style - which I don't really like. $\endgroup$ Feb 14, 2012 at 6:35
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    $\begingroup$ That’s why I qualified my objection: I consider at most the first of those important for an introductory text in point-set 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!) $\endgroup$ Feb 14, 2012 at 6:37
  • $\begingroup$ 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. $\endgroup$ Apr 27, 2012 at 11:23
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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.

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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.

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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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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!/

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  • $\begingroup$ thank you for suggesting this. Ill try to find this. $\endgroup$
    – gg1
    Oct 24, 2010 at 16:34
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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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    $\begingroup$ This 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). $\endgroup$
    – user59083
    Aug 4, 2014 at 3:48
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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.

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  • $\begingroup$ Do more recently printed editions have more modern notation? $\endgroup$
    – Nethesis
    May 8, 2015 at 23:05
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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.

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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.

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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.

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    $\begingroup$ This 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. $\endgroup$ Apr 27, 2012 at 11:30
  • $\begingroup$ What book are we taking about? The link takes me to nowhere. $\endgroup$ May 10, 2014 at 9:52
  • $\begingroup$ +1 for a remarkable and underrated book that will be particularly helpful to students preparing for a serious algebraic and/or geometric topology course. $\endgroup$ Apr 28, 2016 at 17:54
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Priniples of Topology, by F.H. Croom

Foundation of Topology by C.W. Patty

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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 topology---that is, if you read German.

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$1.$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

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  • $\begingroup$ I agree,layout is beautiful, some problems aren’t explained well $\endgroup$
    – user837396
    Apr 13, 2021 at 13:31
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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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  • $\begingroup$ @Leucippus What makes you come to that conclusion? The question is "Best Books for Topology". What else do you expect for an answer? $\endgroup$ Mar 7, 2020 at 18:43
  • $\begingroup$ My comment based on noting over 9000 downloads worldwide. Probably some didn’t like it, but...RF $\endgroup$ Mar 11, 2020 at 23:26
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Topology For Beginners. Dr Steve Warner

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  • $\begingroup$ 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 $\endgroup$
    – user837396
    Apr 13, 2021 at 13:22
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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, [email protected].

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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.

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