Universal covering space of the real projective line?

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

$\mathbb{R}P^1$ 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.