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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!

Community
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henryforever14
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    I'm pretty sure this question was asked here already... probably multiple times. You should really use the search bar, imo. – Sam May 28 '12 at 00:03
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    Take your pick: #1 #2 #3 – bzc May 28 '12 at 00:04
  • 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? – Antonio Vargas May 28 '12 at 00:04
  • 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. – William May 28 '12 at 00:20
  • @William Is the book you recommend about point set topology or algebraic topology? or it covers both? – henryforever14 May 28 '12 at 00:22
  • Most of the book is general point set topology. I would would recommend this book for point set topology. There is a little at the end about algebraic topology. Again his treatment of algebraic topology is well-written but only barely touches algebraic topology. For a concrete treatment of algebraic topology, I would suggest Hatcher $\textit{Algebraic Topology}. – William May 28 '12 at 00:27
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    You need very little general topology for differential geometry. – André Nicolas May 28 '12 at 00:59
  • @AndréNicolas: I'm not sure how much I agree with that. If the differential geometry course focuses on abstract smooth manifolds (rather than just surfaces in $\mathbb{R}^3$), then the student ought to know about separation axioms (Hausdorff), countability axioms (2nd countability), and compactness properties (paracompactness), and also about subspace, product, and quotient topologies. – Jesse Madnick May 28 '12 at 03:59
  • Since you intend to study differential geometry, this question might be interesting for you: Topology needed for differential geometry. – Martin Sleziak Aug 07 '12 at 07:58

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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.

ncmathsadist
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  • For pure point-set topology, Wilansky's book is impossible to beat. – ncmathsadist May 28 '12 at 00:24
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    As you are the only person who answered, I will pick your answer. Thank you for all of you who commented as well. – henryforever14 May 28 '12 at 00:38
  • 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). – GeoffDS May 28 '12 at 01:26
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    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. – Mathemagician1234 May 28 '12 at 02:28
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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.

MJD
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    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. – 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.

Austin Mohr
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    +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! – Mathemagician1234 May 28 '12 at 02:29
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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

Sharanjit
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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.

MJD
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