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?