I'm reading Brown's theorem in Hatcher and Brown's original paper. I am confused by the assumptions concerning the generalized cohomology theory, which is defined axiomatically by the EilenbergSteenrod axioms except the dimension axiom. Brown replaced the dimension axiom by the condition that the nth cohomology group of a point is countable for nonzero n, and Brown said this condition couldn't be weakened. The wedge axiom used by Hatcher to define reduced cohomology theory, however, implies that the nth cohomology group of a point is trivial. This seems contradictory. Can somebody please help me?

Hatcher's Theorem 4E.1 in his 'Algebraic Topology' book is equivalent to Brown's Theorem II version (4.6) in his 'Cohomology theories' paper. Countability of coefficients is only required for Theorem II version (4.5). The first theorem is for cohomology theories defined on all CWcomplexes, and the second one for those defined on finite CWcomplexes. It looks like if the countability hypothesis were necessary if we wanted to represent cohomology theories defined only on finite CWcomplexes, but this hypothesis was removed by Adams in 'A variant of E. H. Brown's representability theorem'. 


The most general reference I know for Brown Representability for cohomology is Neeman's book on triangulated categories. There even fewer of the EilenbergSteenrod axioms are assumed (additivity is not). It's so general that it should cover both cases you list. In this sense, Brown was wrong and his ideas could be made to work more generally, without a countability hypothesis. For homology theories the story is more subtle and countability is often needed. In Neeman's book this theorem is one of the big goals, so it's stated in the introduction and proved close to the end (chapter 12 I believe). Check it out on Google books for a preview 

