Generalising a homeomorphism Show that $\mathbb{R}^n\backslash\mathbb{R}^k$ is homeomorphic to $S^{n−k−1}× \ \mathbb{R}^{k+1}$.
I have already shown that $\mathbb{R}^2\backslash(0,0)$ is homeomorphic to $S^{1}× \ \mathbb{R}$. I think that I can generalise this to all $n$ and $k$ such that $n-k=2$, that is, I can show that
$\mathbb{R}^n\backslash\mathbb{R}^{n-2}$ is homeomorphic to $S^{1}× \ \mathbb{R}^{n-1}.$
However, for each $n$, there are $k$ cases to consider :
$k=0,1,...,n-1.$
These seem to me to require different arguments. It would be much appreciated if someone could give me a tip as to how these cases can be covered by one argument.

Notice that $\mathbb{R}^n \backslash \mathbb{R}^k = \{(x_1,...,x_n) \mid (x_{k+1},...,x_n) \neq (0,...,0) \} = \{(x_1,...,x_k,x) \mid x \in \mathbb{R}^{n-k} \backslash \{0\} \}$, so $\mathbb{R}^n \backslash \mathbb{R}^k = \mathbb{R}^k \times (\mathbb{R}^{n-k} \backslash \{0\})$. 
Therefore, it is sufficient to prove that $\mathbb{R}^n \backslash \{0\}$ is homeomorphic to $\mathbb{R} \times S^{n-1}$.
But it is just a polar decomposition, consider $\phi : \left\{ \begin{array}{ccc} \mathbb{R}^n \backslash \{0\} & \to & \mathbb{R} \times S^{n-1} \\ x & \mapsto & \left( ||x||, \frac{x}{||x||} \right) \end{array} \right.$.
