How to show the orbit space $\mathbb S^n/\mathbb Z_2$ is $\mathbb RP^n$?

Question

How to show the orbit space $\mathbb S^n/\mathbb Z_2$ is $\mathbb RP^n$? Here $\mathbb Z_2=\{0, 1\}$ is the additive group and the group action considered induces the aplications $\psi_0=Id$ and $\psi_1=-Id$.

Answer 1

The map $S^n\to \mathbb P^n(\mathbb R): s\mapsto \mathbb R\cdot s$ is continuous and induces a bijection $S^n/\mu_2\to \mathbb P^n(\mathbb R)$ ($\mu_2=\{Id, -Id\}$).
That bijection is continuous by the universal property of quotient topological spaces and is a homeomorphism, like all continuous bijections from compact spaces to Hausdorff spaces.
This proof can be adapted to show that the above homeomorphism is also a diffeomorphism of smooth manifolds.