differential geometry – Gauss map from an $n$ -sufrace is onto.

I was reading differential geometry from a lecture note, there I found the following theorem,

If $S$ is a compact connected oriented $n-$ surface in $mathbb{R}^{n+1}.$ Then the gauss map is surjective.

Now in the proof , the picture of the surface $S$ sketched as if $S$ is homeomorphic to $S^n.$ I am wondering that, is it true?

Is there any characterisation of $n-$ surcaces?

Definition: An $n-$ surface $S$ is a level curve of a smooth function whose gradient is non-zero on $S.$

Thank you for your time.