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?

areall representable if you restrict to those that are trivialised on a numerable cover. Of course, for paracompact spaces this implies bundles trivialisable over all open covers. $\endgroup$