TITLE: Brown representability beyond CW complexes
QUESTION [14 upvotes]: Brown representability states that any contravariant functor from the homotopy category $CW_*$ of pointed CW complexes to the category of pointed sets is representable if it turns coproducts into products and satisfies a type of Mayer-Vietoris gluability axiom, which I like to think of as a weak version of "the functor sends push-outs into pull-backs" as any representable functor must. The proof very much relies on the fact that CW complexes can be built up in a steady and predictable manner, as it uses Whitehead's theorem that a weak homotopy equivalence is automatically a homotopy equivalence. Namely, one shows that an element which is "universal" for the spheres is actually a "universal element" for this functor (in the sense of Yoneda's lemma). 
Brown representability has many interesting consequences, e.g. that there is a "universal" principal $G$-bundle for pointed CW complexes (where $G$ is a topological group) or that the Eilenberg-Maclane spaces represent the cohomology functors. However, in the former case, it's actually true that the universal bundle exists for any topological space, not just CW complexes. I don't know whether the cohomology functors are representable on the category of all pointed topological spaces (even if one restricts to  non-pathological ones: say Hausdorff, with nondegenerate basepoint), though I would imagine that a CW complex couldn't do it. This leads me to ask:

Is there a version of Brown representability for arbitrary pointed topological spaces?

There is a version of it on the nLab in more generality, but I don't know enough about categorical homotopy theory to understand anything. 
Could someone perhaps translate some of that into the special case of topological spaces?

REPLY [6 votes]: Let me try again. The original question

Is there a version of Brown representability for arbitrary pointed topological spaces?

is interesting, and I don't see how it has been fully answered. That Brown's theorem, as stated, does not hold beyond CW complexes doesn't necessarily mean that there is no reasonable version of this theorem for arbitrary (or almost arbitrary) spaces, generalizing an equivalent formulation of Brown's original theorem. On the other hand, versions coming with disclaimers like

Then it is representable, but by construction rather than by any fancy theory.

or

any "generalization" you're going to get for general spaces is really an illusion

are truly admirable for honesty of the disclaimers, but leave some doubt as to whether this is the end of the story. 
More to the point. Given a homotopy invariant contravariant functor $h$ on the category of polyhedra, Watanabe shows that $h$ admits a unique (up to natural equivalence) extension $\check h$ to all topological spaces that is homotopy invariant and also continuous in the sense of satisfying two rather simple axioms (Theorems 5 and 6 here). This extension is also known as the Cech extension, and Lee and Raymond show that if $h$ is representable, then so is $\check h$ (See also Dold's Lectures in Algebraic Topology, Appendix, Example 3.12.) I understand that if $h$ is a (generalized) cohomology theory, then so is $\check h$. 
Let me summarize: if we restrict our attention to functors that are cohomology theories and in addition are "continuous", then yes, we do have Brown's theorem for arbitrary topological spaces, by a certain fancy theory. (Assuming that I'm not missing any fine points concerning e.g. non-Hausdorff spaces. Note that in Dold's Appendix, some things work better for paracompact spaces.)
In case Watanabe's axioms of a "continuous" functor cut no ice with you, then there are the axioms mentioned in my previous reply in this thread (strong excision, strong homotopy invariance, additivity and multiplicativity) EDIT: which do suffice for locally compact separable metric spaces. Doing something similar for more general spaces  could involve a fancier theory, which is probably not developed as yet. Certainly, doing something similar with homology (and other covariant functors) would involve a fancier theory. Going in the direction of such fancier theory are the following uniqueness theorems for generalized homology theories on metric compacta: Theorem 5.3 in Bauer and Corollary 2.17 in Kaminker-Schochet.