$f: X \to Y$ is a submersion, $X$ is compact and $Y$ is connected - why $f(X)$ is open?

Question

$f: X \to Y$ is a submersion, $X$ is compact and $Y$ is connected - why $f(X)$ is open?

Assume I have proved that for an open set $U \subset X$, $f(U)$ is open.

Thank you.

Answer 1

For any topological space $X$, $X \subseteq X$ is open.