Boundary points of a manifold

Question

I'm reading about Riemannian Geometry and my question is regarding Manifolds with Boundary.

I want to show a point of a manifold with boundary is either an interior point or a boundary point, so no overlap.

Let $\mathbb{H}_n$ be the closed half space of dimension n. In order to prove the above my aim is to show an open neighborhood U of a point p, in $\mathbb{H}_n$\ $\partial \mathbb{H}_n$ can't be homeomorphic to an open subset U of $\mathbb{H}_n$ when we identify p with a point in $\partial \mathbb{H}_n$.

How should I proceed?

Answer 1

As this thread mentions in the comments,

Interior and boundary points of $n$-manifold with boundary

you need the "invariance of domain" theorem:

http://en.wikipedia.org/wiki/Invariance_of_domain

to prove what you want, which is nontrivial, and requires the use of algebraic topology. As far as I can tell, there doesn't seem to be a much simpler proof that's easily accessible.