Show that the identity $i:\mathbb{R}P^1 \longrightarrow \mathbb{R}P^1$ can not be lifted.

Question

I need help with question.

Let $ p: S^1 \longrightarrow \mathbb{R}P^1,\, z\mapsto \zeta= \{z,-z\}$ be the canonical covering of the real projective line and $i:\mathbb{R}P^1 \longrightarrow \mathbb{R}P^1, \, i(\zeta)=\zeta$.

Show that $i$ can not be lifted to $(S^1,p)$.

Thank you.

Answer 1

Suppose it can be lifted, there exists $f:S^1\rightarrow S^1$ such that $p(f(x))=p(x)$, this implies that $f(x)=-x$ or $f(x)=x$. Let $C_1=\{x:x\in S^1: f(x)=x\}$ and $C_2=\{x:x\in S^1: f(x)=-x\}$ $C_1$ and $C_2$ are disjoint ($0$ is not in $S^1$) closed subsets and $C_1\bigcup C_2=S^1$. This is impossible since $S^1$ is connected.