I'm reading Lee, 'Introduction to Topological Manifolds', 2011. After he introduces $n$-manifold with a boundary
An $n$-manifold with a boundary is a second countable Hausdorff space in which any point has a neighborhood which is homeomorphic either to an open subset of $\mathbb R^n$ or to an open subset of $\mathbb H^n = \{x\in \mathbb R^n:x_n\geq 0\}$ endowed with a Euclidean topology.
The interior chart is then $(U,\phi)$ where $U$ is open in $M$ and $\phi:U\to V\subseteq \mathbb R^n$ is a homeomorphism (here $V$ is open in $\mathbb R^n$) and boundary chart as (U',\phi') where U' is open in $M$ and \phi':U'\to V'\subseteq \mathbb H^n is a homeomorphism (here $V$ is open in $\mathbb H^n$ and $V'\cap \partial \mathbb H^n\neq\emptyset$).
Then he writes:
If M is an $n$-manifold with boundary, a point $p$ in $M$ is called an interior point of $M$ if it is in the domain of an interior chart; and it is called a boundary point of $M$ if it is in the domain of a boundary chart that takes $p$ to $\partial \mathbb H^n$.
The natural question is - can be $p$ both interior and boundary point of $M$? In 2011 version of the book Lee writes:
Every point of M is either an interior point or a boundary point: if $p \in M$ is in the domain of an interior chart, then it is an interior point; on the other hand, if it is in the domain of a boundary chart, then it is an interior point if its image lies in $\operatorname{Int}\mathbb H^n$, and a boundary point if the image lies in $\partial \mathbb H^n$.
I cannot understand his argumentation. Moreover, in the very same book on the next page he writes explicitly that it is not possible to prove that $p$ is either interior or boundary at that stage. (Theorem 2.59). I have two question:
Do I miss something, or he states two contradictory facts?
I tried to prove that any point is either boundary interior or boundary and come to the fact that any subset V' open in $\mathbb H^n$ such that V'\cap \partial \mathbb H^n\neq\emptyset is not homeomorphic to any open subset of $\mathbb R^n$ and couldn't prove that fact. Can you help me with that proof?