I have a question about a stetement in wiki's article about real projective space:https://en.wikipedia.org/wiki/Real_projective_space
There is said that the covering map $S3 \to \mathbb{RP}^3$ is a map of groups $Spin(3) \to SO(3)$, where $Spin(3)$ is a Lie group that is the universal cover of $SO(3)$.
What does it mean? Why $S^3 \to \mathbb{RP}^3$ equals $Spin(3) \to SO(3)$?
I suppose that $\mathbb{RP}^3 \cong SO(3)$ holds (by the way, do anybody know a nice argument that it's not only a bijection, but also homeomorphism?).
But does $S^3 \cong Spin(3)$ also hold?