Let $f : M\to N$ be a submersion with $M$ compact and $N$ connected. The $f$ is surjective.

Question

I have no idea how to do this. I tried think in, once $N$ is connected and locally path connected it has to be path connected, but, this does not help. Any hints, solutions, will be very appreciate...

Now I realized that $f(M)$ will be also compact... But, how to conclude the claim?

Thanks

Answer 1

Hints: The image set $f(M)$ will be compact, so closed. And the rank of $f$ will be $= \dim N$ for any $p\in M$, so $f(M)$ will be...