I need a homeomorphism between $\mathbb{R}^2 \setminus \{0\}$ and $S^{1}\times\mathbb{R}$
So i took $f$ and $h$, and need to demonstrate $h\circ f$ is the homeomorphism that i need.
$f: \mathbb{R}^2 - \{0\} \rightarrow S^{1}\times\mathbb{R}_{+}$
$f(x) = (\frac{x}{\|x\|}, \|x\|)$, is injective, surjective and continuous.
The inverse is: $f^{-1}(x,y) = xy$ and it is also continuous. Therefore $f$ is a homeomorphism.
$h: S^{1}\times\mathbb{R}_{+} \rightarrow S^{1}\times\mathbb{R}$
$h(x,y) = (x, \log(y))$ and $h^{-1}(x,y) = (x, e^y)$ both continuous.
so $h \circ f: \mathbb{R} - \{0\} \rightarrow S^{1}\times\mathbb{R}$ is a homeomorphism.
is this all correct?
Can i generalize to $\mathbb{R}^{n+1} \setminus \{0\} \rightarrow S^{n}\times\mathbb{R}$?