Why is $\mathbb{R}^3\setminus \{0\}$ homeomorphic to $S^2\times \mathbb{R}_{>0}$?

Question

One can define a quotient map $q:\mathbb{R}^3\setminus \{0\}\to \mathbb{S}^2$ by $$q(x)=\frac{x}{|x|}.$$ But I don't understand why $\mathbb{R}^3\setminus \{0\}$ is homeomorphic to $\mathbb{S}^2\times \mathbb{R}_{>0}$?

Answer 1

Define $f:S^2\times \mathbb{R}_{>0}\rightarrow\mathbb{R}^3-\{0\}$ by $f(x,c)=cx$, it inverses is $x\rightarrow ({x\over{\|x\|}},\|x\|)$.