I'm reading tom Dieck's book on algebraic topology. For a sphere $S^n$ and $N = e_{n + 1} = (0,...,0,1)$ (usually referred to as the north pole), he talks about a stereographic projection $\phi_N\colon S^n\setminus\{e_{n + 1}\} \to \mathbb{R}^n$ given by $(x_1,...,x_n,x_{n + 1}) \mapsto \dfrac{1}{1 - x_{n + 1}}(x_1,...,x_n)$ and its inverse $\pi_N\colon \mathbb{R}^n\to S^n\setminus\{e_{n + 1}\}$ given by $(x_1,...,x_n) \mapsto \dfrac{1}{1 + |(x_1,...,x_n)|^2}(2x_1,...,2x_n, |(x_1,...,x_n)|^2 - 1)$.
There maps are mutually inverse homeomorphism.
But then tom Dieck tells that it follows from this that $S^n\setminus\{y\}$ is homeomorphic to $\mathbb{R}^n$ for any $y \in S^n$. This means that there must be a reasonable homeomorphism $S^n\setminus\{x\}\to S^n\setminus\{y\}$ for $x \neq y$. I tried to build a more general homeomorphism $\mathbb{R}^n\to \mathbb{R}^n$ which maps $x$ to $y$, $y$ to $x$ and $z$ to itself for $z \neq x,y$, but it seems this map is not continuous, so one needs something more sophisticated.
For any $x,y\in \mathbb{R}^n$ with the same length there's an isometry $f$ of $\mathbb{R}^n$ taking $x$ to $y$. (One such is reflection in the hyperplane perpendicular to $x-y$.) In the case where both $x$ and $y$ have length $1$, the restriction of $f$ to $S^n \setminus\{x\}$ is a homeomorphism onto its image, which is $S^n\setminus\{y\}$.