Is the inverse of a quotient map continuous?

Question

I tried my best looking this up, but could not find anything. Very sorry if this is a duplicate.

We can define a quotient map $q:S^\infty \to \mathbb{R}P^\infty$ by identifying all antipodal points. We can possibly define an inverse to this map by choosing one of the two points. Can we choose the points in such a way that this inverse map is continuous, or does the structure of $\mathbb{R} P^\infty$ holds us back in creating such a continuous inverse?

Thank you!

Answer 1

No, there are no continuous $\mathbb{RP}^\infty\to S^\infty$ that is right-inverse to the quotient $S^\infty\to\mathbb{RP}^\infty$.

Indeed, it fails at the $1$-skeleton level already: there are no continuous $\sqrt{}$ on the unit circle $S^1\subset\mathbb{C}$.