What is the universal covering space of the projective line $\mathbb{RP}^1$?

Question

I´m thinking about universal covering spaces. I´ve seen a lot of examples and authors ever say "the sphere $S^n$ is the universal covering space of the $n$-dimensional projective space $\mathbb{R}P^n$ for $n \geq 1$.

So my question is: and what about the real projective line $\mathbb{R}P^1$? Has it universal covering space?

Thanks!

Answer 1

$\mathbb{R}P^1$ is homeomorphic to $\mathbb{S}^1$. To see this, first note that more generaly $\mathbb{R}P^n\simeq\mathbb{S}^n/_{\pm id}$ and in the case $n=1$ you have $ \mathbb{S}^1/_{\pm id}\simeq \mathbb{S}^1$ (just factorize the map $z\mapsto z^2$).

From here you can conclude.

Answer 2

The real projective line is just a circle, so the universal covering space is the real line.