There are different axioms for a (co)homology theory: Homotopy invariance is always there, but for the rest there is or isn't:
long exact sequence for pairs of topological spaces
exact sequence for cofibrations
additivity (for coproducts of spaces)
Now my question is: Which of these axioms define really a (co)homology theory and which subsets of these axioms are equivalent?
Edit: I have the impression that "long exact sequence for pairs of topological spaces + additivity + excision" is one possibility, and "exact sequence for cofibrations + additivity + suspension isomorphism" is another. See http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, p. 110.
Oh, and is it obvious that the cohomology theories coming from Brown representability satisfy excision? See http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, bottom of p. 111.