# Two technical questions about cohomology theories

1) Milnor proved that singular cohomology is the unique additive cohomology theory satisfying the dimension axiom on the category CW of pairs (X,A) such that X and A have the homotopy type of a CW-complex. Moreover, Eilenberg and Steenrod proved that for any admissible category containing triangulable spaces, a cohomology theory satisfying the d.a. coincides with the singular cohomology on pairs of triangulable spaces.

My question is: is the uniqueness result true for any admissible category contained in CW and containing finite triangulable spaces? For example, I suppose it is true for pairs homotopic to finite CW-complexes, but I cannot find a reference.

2) K-theory can be defined in the usual way on finite CW-complex, and in the paper of Atiyah and Hirzeburch "Analytic cycles on complex manifolds" it has been extended to spaces homotopic to finite CW-complexes. Nevertheless, if I'm not wrong, in the book of Atiyah "K-theory" it is defined in the usual way for any pair of compact Hausdorff spaces, even not homotopic to a CW-complex. Is there any problem to define a cohomology theory in this case? Does the extension work for any pair of spaces homotopic to compact Hausdforff ones? (I think the Chern character does not work properly outside of the category of finite CW, but I suppose that K-theory is a cohomology theory anyway).

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homotopic = homotopy equivalent? –  Fernando Muro Mar 29 '13 at 17:14
yes, I mean homotopy equivalent. –  Fabio Mar 30 '13 at 13:21