Now that we have the powerful tool of Brown representability, let us use it to prove several basic results in homotopy theory. The first one is that any space admits a “CW approximation,” i.e. a CW complex which is weakly homotopy equivalent to it.

In the theory of model categories, which I hope to say more about later, any object has a “cofibrant replacement,” which is such a CW approximation when one uses the standard Quillen model structure for topological spaces. One of the consequences of this is that the homotopy category of {CW_*} is equivalent to the homotopy category of all pointed topological spaces (where homotopy category means something slightly different than it usually does, namely what you get by localizing at weak equivalences).


Proposition 10 Let {X  } be any pointed space. Then there is a pointed CW complex {Y \in CW_*} and a weak homotopy equivalence {Y \rightarrow X}.

For simplicity, let us just assume {X} is path-connected. Else one can do this for each path component.

Proof: Indeed, we have a functor {F} on {CW_*} sending {Z \rightarrow  \left\{\mathrm{pt \ homotopy \ classes} \ Z \rightarrow X  \right\}}. This is a contravariant functor to {\mathbf{Sets}_*} on the homotopy category. Now the claim is that it satisfies the two axioms of coproducts and Mayer-Vietoris. But we basically checked this right before beginning the proof of Brown representability, and is essentially the gluability of homotopy classes of maps (instead of just functions).