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Possible Duplicate:
choosing a topology text
Introductory book on Topology

I'm a graduate student in Math. But I never learnt Topology during my undergraduate study. Next semester, I am going to take Differential Geometry. I assume this course would require a background of Topology. So I would like to take advantage of this summer and learn some topology myself.

I don't need to become an expert in Topology. All I need is that after this summer, my topology knowledge will be enough for my Differential Geometry course.

So can somebody please recommend me a textbook? I'd be really grateful!

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marked as duplicate by Brandon Carter, Sam, user5783, t.b., Jesse Madnick May 28 '12 at 4:01

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ I'm pretty sure this question was asked here already... probably multiple times. You should really use the search bar, imo. $\endgroup$ – Sam May 28 '12 at 0:03
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    $\begingroup$ Take your pick: #1 #2 #3 $\endgroup$ – Brandon Carter May 28 '12 at 0:04
  • $\begingroup$ I assume you've done some cursory research on common topology texts already. Do you have any specific questions about the plethora of advice already available on the internet? $\endgroup$ – Antonio Vargas May 28 '12 at 0:04
  • $\begingroup$ I would recommend $\textit{Topology}$ by Munkres. I am not at all interested in topology, but I would say it is my favorite math textbook. It is very well-written. I don't think you need much point set topology for differential geometry or algebraic topology. You probably just need to know about continuous functions, compactness, and connectedness. $\endgroup$ – William May 28 '12 at 0:20
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    $\begingroup$ You need very little general topology for differential geometry. $\endgroup$ – André Nicolas May 28 '12 at 0:59
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Munkres Topology is a magnificent book. It is well written and covers the basics of point set and elementary geometric topology extremely well. I agree with William.

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  • $\begingroup$ For pure point-set topology, Wilansky's book is impossible to beat. $\endgroup$ – ncmathsadist May 28 '12 at 0:24
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    $\begingroup$ As you are the only person who answered, I will pick your answer. Thank you for all of you who commented as well. $\endgroup$ – henryforever14 May 28 '12 at 0:38
  • $\begingroup$ This is the book that my graduate course in point-set topology used. I found it to be a good book for someone who had never had topology (because I had not). $\endgroup$ – Graphth May 28 '12 at 1:26
  • $\begingroup$ Munkres is a classic for good reason,but Wilansky is indeed a great book for students already familiar with the elements of point-set topology from real analysis. We should all be very grateful to Dover for making it available again for a very low price. $\endgroup$ – Mathemagician1234 May 28 '12 at 2:28
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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.

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    $\begingroup$ These counterexamples can shed insight though. A great example involves showing that first countable and separable do not jointly imply second countable. This is achieved via the "bubble topology", an ingenious piece of mathematical craftsmanship. $\endgroup$ – ncmathsadist May 28 '12 at 18:56
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I entered my graduate general topology course with no previous background in the field (save what I knew about the real line). Despite this, I had great success with Stephen Willard's General Topology.

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    $\begingroup$ +1.Willard is the Bible of point-set topology,the single most comprehensive text ever written on the subject. Again,Dover has done a huge service to mathematics students by making it available again in a cheap edition! $\endgroup$ – Mathemagician1234 May 28 '12 at 2:29
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Crossley's Essential Topology gives a slightly more elementary introduction than Munkres, and driven more by examples than by theory. I found it useful when I got stuck with Munkres.

http://www.amazon.com/Essential-Topology-Springer-Undergraduate-Mathematics/dp/1852337826

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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.

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  • $\begingroup$ That's a fabulous book. I like it a lot. $\endgroup$ – ncmathsadist May 28 '12 at 1:29
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    $\begingroup$ De gustibus: I much prefer Munkres to Kelley. Come to think of it, I also prefer Dugundji to Kelley. If one has the necessary maturity, Willard is perhaps a better choice than any of these. $\endgroup$ – Brian M. Scott May 28 '12 at 1:46

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