The biggest obstacle for me to learn geometry and topology is the haziness of textbooks.

I took algebraic topology last semester and the textbook we used in class was Rotman's "An introduction to algebraic topology".

In the text, Rotman frequently mentioned Eilenberg and Steenrod's masterpiece "Foundations of algebraic topology", where the authors developed algebraic topology in a purely axiomatic way. Before starting to read Eilenberg's book, I would like to hear some comments on Eilenberg's book(from those who have read it, of course).

Eilenberg's book was published in 1952, more than sixty years ago. I am wondering if there are some other books for algbraic topology that introduces axioms at the begining, develops the theory based on homological algbra, includes some modern techniques for this subject(and self-contained if possible).

I know professional mathematicians may have one thousand reasons for the importance of intuition in learning geometry and topology. However, personally, I am not that kind of student who want to receive intuitive things first and then the abstract theory. I want to start with the abstractness at the very beginning and the understand the intuitive examples in a rigorous way.

I am not asking for a long list of recommended books for algebraic topology. I only ask for comments on Eilenberg's book and other similar books. Thanks in advance!

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    $\begingroup$ I'm still a novice, but I think if you want an abstract, axiomatic and conceptual way towards algebraic topology, maybe homotopy-first books are more appropriate. Perhaps J.P.May's A Concise Course in Algebraic Topology works for you. It's free online. $\endgroup$ – Yai0Phah May 9 '15 at 7:25
  • $\begingroup$ This isn't quite answer-worthy but Munkres' "Elements of Algebraic Topology" gets good reviews as far as rigorous texts go. $\endgroup$ – JustAskin May 9 '15 at 7:28
  • $\begingroup$ And I don't know to what extent by homological algebra you mean. Just that of Cartan & Eilenberg era or of more modern stage such as derived category? Related: MO thread and this. $\endgroup$ – Yai0Phah May 9 '15 at 7:33
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    $\begingroup$ I haven't studied it , but it seems like you might have fun with tom Dieck's Algebraic Topology . It's online, if you want to have a look at it. $\endgroup$ – Balarka Sen May 9 '15 at 7:43
  • $\begingroup$ If you want comments on E&S specifically, you probably ought to change the title of the question to say so. Otherwise, people will just keep recommending different books to you, which is not what you want, you said. $\endgroup$ – bubba May 9 '15 at 9:58

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