Remark 1

Let X be a topological space whose arcwise connected components are open. Let A be a closed subspace of X; let $\iota : A\rightarrow$ X denote the canonical injection, let $X/A$ denote the space deduced from X by contracting A to a point $o$, and let $p: X\rightarrow X/A$ denote the canonical mapping. Suppose further that the pair $(X,A)$ has the homotopy extension property. The canonical mapping $\overline{\rho}:$ Cone$(\iota )\rightarrow X/A$ is then a homotopy equivalence (III, p. 255, Remark 1), and one deduces from Prop. 4 the calculation of the Poincaré group of $X/A$ at its base point $o$. In particular, if the arcwise connected components of X are simply arcwise connected, the group $\pi_1(X/A, o)$ is a free group.