Setting:
I consider a two times continuously differentiable function $H:\mathbb R^{2n}\to \mathbb R$ and a regular value $\beta\in\mathbb R$ such that $N=H^{-1}(\beta)$ is compact and connected.
Claim:
$\mathbb R^{2n}\setminus N$ has a bounded component $B$ and an unbounded component $A$.
My attempt:
It is well known, that $N$ is a $2n-1$ dimensional submanifold of $\mathbb R^{2n}$, i.e. using the implicit function theorem. And since $N$ is connected, we can see it as one piece.
But I'm curious why $\mathbb R^{2n}\setminus N$ has two components. Why can't $N$ be a manifold with boundary like $[0,1]\times\{0\}$ for the case $n=1$? Or some strange construction, since $N$ is just connected but not path connected?
We can also consider a compact connected $m-1$ dimensional submanifold of a $m$ dimensional space. Can we conclude that it has to separate the space in a bounded and an unbounded set? Since it is used, without any explanation or arguments, in a paper it seems to be obvious for the author, but not for me.