Given the group action of the orthogonal group \(\O(n)\) on \(\mathbb{R}^n\) . The orbit space \(\mathbb{R}^n/\O(n)\) is homeomorphic to \([0,\infty)\).
Proof
This is a consequence of the identification theorem .
The map \(x\mapsto \lvert x\rvert\) and the natural projection \(\mathbb{R}^n\to \mathbb{R}^n/\O(n)\) makes the same identification (see (0x67cca4f8) ).
It only remains to show that \(x\mapsto \lvert x\rvert\) is a quotient map .
One should be able to prove that \(x\mapsto \lvert x\rvert\) is open and surjective. Then, Proposition (0x67a4b848) implies that it is a quotient map .