Compact submanifolds of $\mathbb{R}^n$ without boundary
Solution 1:
Submersions are open maps; but the image of $M$ is compact in a Hausdorff space, and hence closed as well. So it's a clopen nonempty set. Since $\mathbf{R}^n$ is connected, it's the whole thing. But then $\mathbf{R}^n$ is the quotient of a compact space, so it's compact, which is not true.