Topology of level surfaces

Question

I have a level surface of the form $f(x,y,z,w)=0$ and also $g(x,y,z)=0$.

Here f and g are differentiable!

I need to decide if they are compact or not.

Is there any criteria, theorem or anything?

Thanks for the help!

Answer 1

Consider $g:\mathbb {R} ^3 \to \mathbb R$ by $g(x,y,z)=x$. Then the surface described by $g=0$ is just the $yz$-plane, which is not compact in $\mathbb R^3$ as it is not bounded. The criterion you are after is the Heine-Borel Theorem, which says a subset of $\mathbb R ^n$ is compact $\iff$ it is closed and bounded.