I would like to prove that any point on a (regular) surface of dimension $k$ is either a boundary point of the surface or an interior point. I have the definitions as follows:
A regular surface with boundary of dimension $k$ is a set $S$ where for each point p in $S$ we have an open neighborhood $V \subset \mathbb{R}^n$ containing $p$ and an open set $U$ with injective $C^1$ mapping $\phi:U \to \mathbb{R}^n$ ($U \subset \mathbb{R}^k$) such that one of the following is true:
- If $p$ is an interior point we have that $\phi(U) = S \cap V$
- If $p$ is a boundary point, we have that $\phi (U\cap \{u_k \geq 0\}) = S \cap V$ and that $p \in \phi(U \cap \{u_k = 0\})$
In the second line, the first part says that the image of $\phi$ under the half plane is equal to the intersection of some open ball with the manifold, and the second part says that our point lies on the very bottom of the half plane.
I was hoping that someone could help me prove that there are no points that are both boundary and interior points at once.