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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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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). – Keenan Kidwell Oct 22 '10 at 17:29
all right! thank you for your comment :) – jgg Oct 22 '10 at 17:33
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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. – Ryan Budney Oct 22 '10 at 20:58
Another standard entry-point 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! :) – jgg Oct 24 '10 at 16:30
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13 Answers

up vote 11 down vote accepted

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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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. – Paul VanKoughnett Oct 23 '10 at 6:00
Here are links to Allen Hatcher's homepage and to the free PDF of his Algebraic Topology textbook. Enjoy! – John Tobler Feb 13 '12 at 20:04
up vote 15 down vote

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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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. – Bey Oct 22 '10 at 17:25
@Chandru: thank you for the suggestions. – jgg Oct 22 '10 at 17:34
@bey: thanks too for the comment. I'll that in mind. – jgg Oct 22 '10 at 17:35
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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.. – Brandon Carter Oct 22 '10 at 17:38
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@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
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up vote 8 down vote

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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ok thank you. I'll try to see this one! – jgg Oct 24 '10 at 16:31
+1 for a classic-but 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
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Singer & Thorpe is a poor introduction if you’re really interested in point-set topology. – Brian M. Scott Feb 13 '12 at 23:33
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  1. I also like Bourbaki's treatise, but some times it is a bit too logical.

  2. Simmons General topology

  3. Also, "A topological picture book" (I don't remember the author though).

  4. K Jänich Topology.

  5. J. Kelley General topology.

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I don’t consider Simmons a particularly good text if one is interested in point-set 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. Feb 14 '12 at 6:33
BTW Kelly is a bit oldish in style - which I don't really like. – AD. Feb 14 '12 at 6:35
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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!) – Brian M. Scott Feb 14 '12 at 6:37
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
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up vote 3 down vote

You might look at the answers to this previous MSE question, which had a slightly different slant: "choosing a topology text". Apparently the poser was also interested in self-learning, but with less preparation than you.

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up vote 3 down vote

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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up vote 1 down vote

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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thank you for suggesting this. Ill try to find this. – jgg Oct 24 '10 at 16:34
up vote 1 down vote

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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For both of your answers on this thread, I edited to add in the link to the originals. – Eric May 28 '12 at 13:40
up vote 1 down vote

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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up vote 0 down vote

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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up vote 0 down vote

Please look at

http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=69421

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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. – Michael Greinecker♦ Apr 27 '12 at 11:30
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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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up vote 0 down vote

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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