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).
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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 $A\leftarrow Q(A)\to B$. |
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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 $A\leftarrow Q(A)\to B$. |
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