I'm working through the first chapter of Morris Hirsch's "Differential Topology". On Chapter 1, section 3 exercise 11, I encountered the following question.
"Regarding $S^1$ as the equator of $S^2$, we obtain $P^1$ as a submanifold of $P^2$ (Hirsch uses $P^n$ to denote real projective n-space). Show that $P^1$ is not a regular level surface for any $C^1%$ map on $P^2$. Hint: no neighbourhood of $P^1$ in $P^2$ is separated by $P^1$."
I attempted to solve this by contradiction, supposing there was a $C^1$ function of $P^2$ such that $P^1=f^{-1}(y)$ and that $T_pf$ is surjective for each $p\in P^1$. Then the inverse function theorem would imply that every open neighbourhood of $P^1$ is diffeomorphic to some open neighbourhood of $y$. I have a feeling that I'm meant to apply the hint here, and arrive at a contradiction by showing a topological property is not preserved under homeomorphism (path connectedness hopefully).
Unfortunately, I am having trouble visualising why the hint it true. I don't think I can go further without understanding that.