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?
Best Answer
The answer is: No and Yes.
If you take a particular construction of a generalised cohomology theory that makes sense for all topological spaces then there is no guarantee that it will be representable in the homotopy category of topological spaces. For example, the various flavours of ordinary cohomology are only guaranteed to coincide for CW complexes (okay, and stuff with the homotopy type of such). That they disagree elsewhere shows that at least one of them can't be representable. Another example is K-theory. It's great that for compact Hausdorff spaces (and some others) that K-theory is exactly what you get when you take vector bundles and group complete, but it's not true for other spaces.
That's the "No". Now for the "Yes". The point about the "Yes" is that Brown representability is so good to have that when we move outside the realm of CW-complexes, we often define our cohomology theory to be (homotopy) homs into the representing object (found by restricting to CW-complexes). That is, we take our "natural construction" of whatever cohomology theory it is, use Brown representability to find the representing object for CW-complexes, and then define the extension of the theory to all topological spaces to be $[X,\underline{E}]$. Then it is representable, but by construction rather than by any fancy theory.