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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!

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marked as duplicate by Brandon Carter, Sam, Benjamin Lim, 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.

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 0:03
Take your pick: #1 #2 #3 – Brandon Carter May 28 '12 at 0:04
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 0:04
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 0:20
You need very little general topology for differential geometry. – André Nicolas May 28 '12 at 0:59
up vote 8 down vote accepted

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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For pure point-set topology, Wilansky's book is impossible to beat. – ncmathsadist May 28 '12 at 0:24
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 0:38
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). – Graphth May 28 '12 at 1:26
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 2:28

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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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

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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+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 2:29

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.

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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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That's a fabulous book. I like it a lot. – ncmathsadist May 28 '12 at 1:29
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. – Brian M. Scott May 28 '12 at 1:46

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