Let $U$ be an open set. If $f\colon U\subset{R}^m\to N$ is a local homeomorphism and exists $y\in N$ such that $\operatorname{card}\big(f^{-1}(\{y\})\big)$ is infinite prove that $f$ is not closed.
I proved that $f^{-1}(\{y\})$ is closed and discrete, I proved also that $f$ must be continuous and open. But I don't know how to use the fact that $U$ is open in $\mathbb{R}^m$ and the fact that $\operatorname{card}\big(f^{-1}(\{y\})\big)$ is infinite.