groups are different (as can be seen from the Künneth formula); thus, X and Y are not homotopy equivalent. The Whitehead theorem does not hold for general topological spaces or even for all subspaces of Rn. For example, the Warsaw circle, a compact subset of the plane, has all homotopy groups zero, but the map from the Warsaw circle to a single point is not a homotopy equivalence. The study of possible generalizations of Whitehead's theorem to more general spaces is part of the subject of shape theory. Generalization to model categories In any model category, a weak equivalence between cofibrant-fibrant objects is a homotopy equivalence. References J. H. C. Whitehead, Combinatorial homotopy. I., Bull. Amer. Math. Soc., 55 (1949), 213–245 J. H. C. Whitehead, Combinatorial homotopy. II., Bull. Amer. Math. Soc., 55 (1949), 453–496 A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. xii+544 pp. and (see Theorem 4.5) Theorems in homotopy theory