Spaces which Invert Weak Homotopy Equivalences

Abstract

It is well known that if X is a CW-complex, then for every weak homotopy equivalence f : A → B, the map f: [X, A] → [X, B] induced in homotopy classes is a bijection. In fact, up to homotopy equivalence, only CW-complexes have that property. Now, for which spaces X is f: [B, X] → [A, X] a bijection for every weak equivalence f? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.