Étale function between a compact manifold and a connected one.

Let $M$ a compact manifold, $N$ a connected manifold and $f\colon M\longrightarrow N$ an injective étale function. Why $f$ is a diffeomorphism?

I know that since $M$ is compact, $f(M)$ will be compact too, but I don't know why if $N$ is connected, I will get the diffeomorphism.

• What is an etale function in the smooth category? – levap Nov 30 '16 at 23:32
• From the book where I'm reading, an étale function is one that is between manifolds of the same dimension, and is an immersion if and only if is a submersion. – MonsieurGalois Nov 30 '16 at 23:33

An etale function is a local diffeomorphism. If $N$ is connected, $f$ is surjective: to see this remark that the image of $f$ is a closed subspace since $M$ is compact, $f(M)$ is compact. It is also open since $f$ is a local diffeomorphism. So it is a union of connected components of $N$, so it is $N$ since $N$ is connected. A local diffeomorphism which is bijective is a diffeomorphism.