Corollary (Whitehead's Theorem for Topological Spaces). Let $X$ and $Y$ be topological spaces having the homotopy type of CW complexes, and let $f: X \rightarrow Y$ be a continuous function. Then $f$ is a homotopy equivalence if and only if it satisfies the following pair of conditions:

  • The induced map of path components $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is a bijection.

  • For every point $x \in X$ and every $n \geq 1$, the map of homotopy groups $\pi _{n}(f): \pi _{n}( X, x) \rightarrow \pi _{n}(Y, f(x) )$ is an isomorphism.