Example 11.1:

The sphere $S^{n-1}$ is a deformation retract of $\mathbb{R}^n - \{0\}$. A retraction $r:\mathbb{R}^{n} - \{0\} \longrightarrow S^{n-1}$ is given by the formula $r({\bf x}) = {\bf x}/\Vert{\bf x}\Vert$. The homotopy between $j\circ r$ and the identity map on $\mathbb{R}^n - \{0\}$ is provided by the convex combination

$$F({\bf x}, t) = t{\bf x} + (1-t)\frac{{\bf x}}{\Vert{\bf x}\Vert} \eqno(11.6)$$

The student must however check that $F({\bf x}, t)$ omits the zero vector. From this we get the following important result.