Theorem 2.3.3 (Whitehead’s theorem, [Hatcher (2002), Theorem 4.5]). Let \(f\colon X \to Y\) be a weak homotopy equivalence between spaces having the homotopy type of cell complexes. Then \(f\) is a homotopy equivalence.
Proof. Choose homotopy equivalences \(u\colon X' \to X\) and \(v\colon Y' \to Y\) with \(X'\) and \(Y'\) cell complexes, and let \(w\colon Y \to Y'\) be a homotopy inverse to \(v\). The composite \(wfu\colon X' \to Y'\) is a weak homotopy equivalence, hence a homotopy equivalence by the usual Whitehead theorem. Since \(u\) and \(w\) are homotopy equivalences, so is \(f\). □
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