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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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    Since you intend to study differential geometry, this question might be interesting for you: [Topology needed for differential geometry](http://math.stackexchange.com/questions/159787/topology-needed-for-differential-geometry).2012-08-07

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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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    Munkres is $a$ classic for good reason,$b$ut 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.2012-05-28
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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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    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.2012-05-28
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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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    +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!2012-05-28
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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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    *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.2012-05-28