I have the following question,
If $S\subset\mathbb{R}^3$, is a regular surface, then interior of $S$, can be empty?
My approach: I have a list of ideas for this problem.
- The definition, that I know, for regular surface is for each point in the surface, therefore the interior cannot be empty.
- $A$ is a open subset of a regular surface if and only if, $A$ is a regular surface (this prop. is easy to demonstrate); So that, if $A$ is a open set, then it interior is not empty, and $\mbox{int}\{A\}\subset\mbox{int}\{S\}$. Then, interior of $S$ is not empty.
- I try to think in what case I have empty interior of a surface. If the interior of surface is empty, then his points are in the boundary of $S$, but the boundary of this set is a closed set, therefore I think that, I would lose the condition of regularity for the homeomorphism.
- On the other hand, If we consider $S$ as, the "shell" of the sphere, then $S$ is a regular surface, and the interior of $S$ (I have not demonstrated it) will be a empty set.