I'm reading the book Differential Geometry of Curves and Surfaces written by do Carmo. And there is one theorem I'm trying to prove. Here is the statement:
If $S$ be a compact, connected, regular surface with constant Gaussian curvature $K$, then $S$ is a sphere.
In the proof, do Carmo claims that $S$ is open in the sphere $\Sigma$. But I'm quite vague about his argument. Why is $S$ open in $\Sigma$ if $S$ is to be a regular surface? Thanks.