Weak Homotopy Equivalence

A weak homotopy equivalence is a continuous map which induces a bijection on path components and an isomorphism on every homotopy group,

for every base point and every . Every homotopy equivalence is a weak homotopy equivalence, but the converse is false for arbitrary spaces. Whitehead's theorem states that a weak homotopy equivalence between CW-complexes is a homotopy equivalence.