Exercise 4, § 1

Let X be the interval $[-1,1]$ of $\mathbf{R}$ and let A = X $-\{0\}$. The canonical mapping of X onto $X/A$ is a homeotopy but there does not exist any homotopy $\sigma$ with origin Id$_X$ satisfying the hypotheses of Proposition 10 of III, p. 252.