Pullbacks of spaces are not homotopy invariant

Example 2.1.2. Given two continuous maps \(f\colon X \to Z\) and \(g\colon Y \to Z\), their pullback \(X \times _Z Y\) is given by the subspace \[ X \times _Z Y := \{(x,y) \in X \times Y \mid f(x) = g(y)\} \; \subseteq \; X \times Y. \] To see that pullbacks are not homotopy invariant, suppose that \(Z\) is a topological space and \(p\colon [0,1] \to Z\) a path whose endpoints \(z_0 := p(0)\) and \(z_1 := p(1)\) differ, and consider the following two pullback squares in \(\Top \):

These two diagrams are homotopy equivalent, since the maps \(z_0,z_1\colon * \to Z\) are homotopic via \(p\), yet their pullbacks \(*\) and \(\emptyset \) are not homotopy equivalent (there is not even a map \(* \to \emptyset \)).