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I want to study the basis of topology. I know functional analysis and very basic topology. I need to learn about homologies and homotopies but it seems that all the books (mostly of Russian authors) that I've found do not explain everything in detail and don't bring many examples so that it would be easy to understand.

Can anyone suggest me a good 'heavy' and intelligible book to begin with?

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Start with the second half of Munkres' Topology, then move to Hatcher's Algebraic Topology. –  Neal Apr 23 '12 at 13:58
    
@Neal Don't you need point-set topology before you can do algebraic topology. OP knows very basic topology. Would it be better to start at Chapter 2 (skipping Chapter 1 on Set Theory and Logic)? –  Graphth Apr 23 '12 at 14:06
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I thought by "very basic topology", OP meant a graduate course in point-set. If they haven't even had that, they should start at the beginning of Munkres, as you say. –  Neal Apr 23 '12 at 14:24
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I'm a fan of Lee's "Introduction to Topological Manifolds" but assuming that 'heavy' means covering a ton of stuff you might want to go with Hatcher. I don't think Hatcher is easy to understand so maybe you want to have a look at both. Lee only covers homology but he's pleasant to read. –  Matt N. Apr 23 '12 at 14:32
    
Neal and Graphth btw, I know point-set quite well, and also 'theory of measure' as a part of functional analysis ))) –  superM Apr 23 '12 at 15:39

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There is an Algebraic Topology Book by James R Munkres other than the only Topology by the same author, I suggest you to start reading fundamental group from the Topology book and then read upto chapter 13th, then start from the Algebraic topology book by the same author where he explained everything in an interactive way with the reader I found it nice for the first reading, then you can look on James W Vick, allan hatcher, spaniers book also. good luck

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I've just found the book and looked through it. it seems really nice. thank you very much –  superM Apr 23 '12 at 15:41
    
@Makuasi The problem with Munkres' algebraic topology book is that it's very old fashioned and the simplicial complex constructions get very tedious and ugly at times.That being said,it is remarkably clear and forms a very natural sequel to his introductory book. –  Mathemagician1234 Apr 23 '12 at 15:52
    
I agree,but Homological algebra portion are very nice manner and in more detail. –  Bunuelian Trick Apr 23 '12 at 16:02

My book "Topology and groupoids" (available on amazon) covers results related to general notions of homotopy, does no homology, but is the only available text which takes seriously the notion of the fundamental groupoid on a set of base points, thus allowing a version of the Seifert-van Kampen theorem which computes the fundamental group of the circle, and more. It also uses groupoids for a base point free approach to covering spaces, and gives results on orbit spaces and orbit groupoids not available elsewhere.

Here is a link to an MAA review:

http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=69421

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thank you very much –  superM Apr 26 '12 at 17:57

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