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I am sorry, if this is a repetition of previous questions. But my case is sightly different. I am a physics undergrad who wants to shift to pure maths, and I want to study topology. The supreme reference is apparently Munkres , but I think it would be too much of a time-investment to study point-set topology from it. I want a shorter treatment of point-set topology, so that I can quickly move on to algebraic topology. Two of the books which I am thinking about are Armstrong's 'Basic Topology' and Lee's 'Topological manifolds'. Do you think it could give a shorter more effective treatment. I am not completely new to topology, and have been exposed to it before in Physics. I understand the intuitive meaning of quotient spaces, compactness, topological groups, definition of fundamental group and homology.

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Trying to rush your education in mathematics is a universally bad idea. Just read Munkres, and do it slowly and completely. – Potato Nov 24 '12 at 8:17
This seems - to some extent - related: Topology needed for differential geometry. (The OP studies physics, too.) – Martin Sleziak Nov 24 '12 at 8:36
But the OP also mentions that he wants to study algebraic topology. – Rankeya Nov 24 '12 at 8:42
Munkres does not take as much as time as you think. Reading chapter 2,3,4,5 should be enough to move on. – user27126 Nov 24 '12 at 9:12
@Sanchez: That is good advice. Thanks. I will think about it. – user23238 Nov 24 '12 at 9:26
up vote 7 down vote accepted

My case was similar to yours: I graduated in theoretical physics but then made the transition to pure mathematics, so I had to quickly get a good grasp of required background material. The fastest path I know to the essential point-set/metric topology needed to start algebraic topology is the recent textbook:

  • Runde, V. - A Taste of Topology.

It develops all elementary concepts and proves all standard theorems in just ~165p. in a course-like set of rigorous lectures with exercises. I think it is the best supplement of, or starting point before, Bredon's "Topology and Geometry", as this last title is geared towards algebraic topology and develops general and differential topology in a very succinct manner (although very complete!). That couple of books would make a quick route to what you want. You can check out other suitable book collections at my Amazon listmanias.

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Thanks. I had at Runde, and it seems interesting and also concise. – user23238 Nov 24 '12 at 9:10

I think the perfect book for you would be Glen E. Bredon's "Topology and Geometry". It has a clear introduction to smooth manifolds as well. I use this book a lot and highly recommend it.

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Thanks. I had a look. Bredons book looks too advanced to me, lacking examples and problems for a first read of topology – user23238 Nov 24 '12 at 9:13
I will take this opportunity to advertise my own book "Topology and Groupoids" since it marries the geometric and categorical viewpoints, has lots of exercises and pictures; it covers some aspects, e.g. the use of groupoids for covering spaces and orbit spaces, not dealt with elsewhere. The e-version is available via kagi at £5. This new edition is published privately to cut the cost. Available from amazon. – Ronnie Brown Jan 19 '14 at 11:06

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