Could some help me understand how $\mathbb{R^{n}} - \{x\}$ is homeomorphic to $S^{n-1} \times \mathbb{R}$.

Also, if the one point set is replaced by any finite set, how does the argument work.

This is in Hatcher's book, example 1.15, p.35, where he states this as a fact. I could not find an explanation either before or in the example.

Thank you.