I found this exercise I'm not able to solve.
EXERCISE : Let's call $X$ the topological space obtained quotienting $\mathbb{R}^{n}$ by the equivalence relation $\sim$ :
$x\sim y \Leftrightarrow x=y$ or $||x||=||y||$ or $||x||\cdot ||y||=1$.
Is $X$ an Hausdorff space ($T2$)?? Is $X$ compact?? Can anyone help me?
The only idea i had is this one but I had plenty of doubt about its correctness.. Using hyperspherical coordinates (http://en.wikipedia.org/wiki/N-sphere#Hyperspherical_coordinates) can I say that $\mathbb{R}^{n}$ is omeomorph to $\mathbb{R}\times \left[ {0,\pi } \right] \times \cdots \times \left[ {0,\pi } \right] \times \left[ {0,2\pi } \right[$ ??? And if that was true, using the fact that $\sim$ act only on the norm and not on the angles, can I say that $\mathbb{R}^{n}/\sim {\rm{ }} = \left( {\mathbb{R}\times \left[ {0,\pi } \right] \times \cdots \times \left[ {0,\pi } \right] \times \left[ {0,2\pi } \right[} \right)/\sim {\rm{ }} = \mathbb{R}/\sim$ ??? Thanks in advance.