TITLE: For which spaces is homology (or cohomology) determined by the Eilenberg-Steenrod axioms
QUESTION [36 upvotes]: This is a spinoff of Can anyone give me a good example of two interestingly different ordinary cohomology theories? .  By an ordinary homology theory, I mean a functor on topological spaces which satisfies the usual set of axioms (e.g., excision, homotopy, infinite coproducts, & dimension axiom (with integer coefficients)).
It is well-known to the sort of people who know these sorts of things that any two such homology theories are isomorphic on any space with the homotopy type of a CW-complex.  (If I remember correctly, one proves that there is a natural map from singular homology to another theory defined on all spaces, and that this map is an isomorphism for spaces homotopy equivalent to a CW-complex.)
The question is: are spaces with the homotopy type of CW-complexes the largest class of spaces for which this is so?  A similar question is: given a space X not homotopy equivalent to a CW-complex, are there two homology theories which have different values on X?
(I'd like to add the tag "shape-theory" to this, since that's the area where an answer is mostly likely to be found.  But I can't create new tags!)

REPLY [6 votes]: Clark Barwick suggested to me that this should be true for any Δ-generated space, and that this is a strictly larger class than spaces homotopy equivalent to a CW-complex.  I haven't attempted to verify either of these statements.