Is it possible to make this map differentiable?

Question

Let $p:S^n\to\mathbb{R}P^n$ be the projection map from the $n$-sphere to the real projective space of dimension $n$. My question is, is it possible to define a map $p^{-1}:\mathbb{R}P^n\to S^n$ by choosing a pre-image of a point? That is, given, $x\in\mathbb{R}P^n$, $p^{-1}(x)$ has two possible choices. Can we make a choice such that the map is differentiable? My intuition tells me that this is possible by choosing the preimage on the north pole, if that makes sense, but this is not well defined by points on the equator. Is there a way to fix this?

Answer 1

No, there is no continuous such map. Its image would have to be both open and compact in $S^n$. Since $S^n$ is connected, the image can only be the whole space, and obviously that won't work.