Quotient by an action of a group on a topological space I'm asked to prove that $(\mathbb{R}^{n}-\{0\})/G\cong S^{n-1} \times S^{1} $ where G is the group of homeomorphisms $\{\Theta^{i}, i\in \mathbb{Z}\} $ and
$$\Theta: \mathbb{R}^{n}-\{0\} \longrightarrow \mathbb{R}^{n}-\{0\}$$
$$ x \longrightarrow 2x $$
with $n\geq 2$.
So I'm having trouble even visualizing $(\mathbb{R}^{n}-\{0\})/G$. Any help would be great!

Hint:Remark that $S^1=R-0/<h(x)=2x>$. Consider $f:R^n-\{0\}\rightarrow S^{n-1}\times S^1$ defined by $f(x)=(x/\|x\|,[\|x\|])$ where $[\|x\|]$ is the class of $\|x\|$ in $R-0/<h>$ show that $f$ induces a homeomorphism from $R^n-0/<\Theta>\rightarrow S^{n-1}\times S^1$.
Let $(x)\in R^n-0$, remark that $f(2x)=f(x)$ thus $f$ factors by a map $F:R^n-0/<\Theta>\rightarrow S^{n-1}\times S^1$. $F$ is injective: suppose that $F([x])=F([y])$ implies that $x={{\|x\|}\over{\|y\|}}y$ and $\|x\|=2^n\|y\|$. This implies that $x=2^ny$ thus $[x]=[y]$, $F$ is obviously surjective.
