Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ s.t. $f|:U\rightarrow V$ is a homemorphism. $M$ is compact and $N$ is connected and Hausdorff.
- $f$ is surjective
- $f$ is finite-to-one
- $f$ is a covering map.
I'm trying to work out the first part. I know that $f(M)$ is compact and hence closed. Is $f(M)$ open? If so, why? And how does this imply surjectivity?