Example

Let X be the complement of the origin in $\mathbf{B}_n$. The mapping of $X\times \mathbf{I}$ into X given by $(x, t)\mapsto ((1-t) +t\frac{1}{\|x\|})x$ is a strong contraction of X onto $\mathbf{S}_{n-1}$. The retraction of X onto $\mathbf{S}_{n-1}$ associated with it is the mapping given by $x\mapsto x/\|x\|$.