Construct $\mathbb{RP}^n$

Question

Has anyone an idea of how to make $\mathbb{RP}^n $ using an action of the multiplicative group of nonzero integrers acting on $\mathbb{R}^{n+1}-\{0\}$ that gives you $\mathbb{RP}^n $ (must be the orbit space)?

Answer 1

I’m going to be bold and contend that no such construction exists. There’s no way the quotient of an $n-1$ dimensional manifold by a discrete group is going to create an $n$ dimensional manifold.