Consider the equivalence relation \(\simeq_w\) generated by weak homotopy equivalence: \(X \simeq_w Y\) if there are spaces \(X = X_1, X_2, \ldots, X_n = Y\) with weak homotopy equivalences \(X_i \to X_{i+1}\) or \(X_i \gets X_{i+1}\) for each \(i\). Show that \(X \simeq_w Y\) if \(X\) and \(Y\) have a common CW approximation.
Solution
Let \(K\) be a common CW approximation for \(X\) and \(Y\). By definition this means there are weak homotopy equivalences \[ K\xrightarrow{u}X, \qquad K\xrightarrow{v}Y. \] Hence there is a zigzag \[ X\xleftarrow{\ u\ }K\xrightarrow{\ v\ }Y \] consisting entirely of weak homotopy equivalences. This is exactly one of the zigzags allowed in the equivalence relation generated by weak homotopy equivalence. Therefore \[ \boxed{X\simeq_w Y.} \]