MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Return to Answer

3 added 97 characters in body

Brown representability is true for arbitrary pointed (CGWH) spaces, as long as you take into account cofibrant replacement (use the Quillen model structure instead of the Hurewicz model structure). This means that $[A,B]$ does not necessarily represent homotopy classes of maps, but represents homotopy equivalence classes of spans $A\leftarrow Q(A)\to B$. That is, $[A,B]=[QA,B]$ for a fixed cofibrant replacement (iirc all spaces are Serre fibrant).

2 added 8 characters in body; added 7 characters in body

Brown representability is true for arbitrary pointed (CGWH) spaces, as long as you take into account cofibrant replacement (use the Quillen model structure instead of the Hurewicz model structure). This means that $[A,B]$ does not necessarily represent homotopy classes of maps, but represents homotopy classes of spans $A\leftarrow Q(A)\to B$.

1

Brown representability is true for arbitrary spaces, as long as you take into account cofibrant replacement (use the Quillen model structure instead of the Hurewicz model structure). This means that $[A,B]$ does not necessarily represent homotopy classes of maps, but represents homotopy classes of spans $A\leftarrow Q(A)\to B$.