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Are there textbooks on algebraic topology which first starts with the Eilenberg-Steenrod axioms and then derives consequences and applications directly out of the axioms. Only at the end they show there are indeed theories which satisfy the Eilenberg-Steenrod axioms.

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  • $\begingroup$ You could start at chapter 3 of Vick, and then read chapters 1 and 2. :) For books like this, I'd search for publication dates in the late 1960s to late 1970s. But I have no particular instance in mind. $\endgroup$ Dec 21, 2015 at 14:46

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How about Foundations of Algebraic Topology, by none other than Eilenberg and Steenrod.

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"A Concise Course in Algebraic Topology" be J. P. May does just that. In chapter 13 May defines homology via the Eilenberg-Steenrod axioms. He then proceeds to show that these axioms fully determine your homology theory on CW-complexes, and then, by the "weak equivalence" axiom, to all spaces. The textbook is available on May's website: http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

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Bredon does this if you start at chapter IV, section 6. Of course, it's hard to get very far with only the axioms.

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