This question is an exercise from Lee's book Introduction to smooth manifold. Given $M$ is a manifold with boundary ( that is $M$ is Hausdorff, second countable and every point of $M$ has a neighborhood homeomorphic to either an open subset of $\mathbb{R}^n$ or an open subset of $\mathbb{H}^n$); then $M$ is a topological manifold ( that is $M$ is Hausdorff, second countable and every point of $M$ has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$.) if and only if the boundary of $M$ is empty, $\partial M = \emptyset$. Assuming the fact that a point of $M$ cannot be both a boundary point and an interior point I argue like this:
Suppose $M$ is a topological manifold, if possible let $\partial M \neq \emptyset$, let $p \in \partial M $ that is $p$ is in the domain of some boundary chart, but as $M$ is a topological manifold, $p$ is also in the domain of some chart homeomorphic to $\mathbb{R}^n$, which is to say $p$ is an interior point of $M$ a contradiction.
The confusion: my professor says that this argument is not correct he say that you cannot guarantee that if $p$ belongs to the domain of some chart homeomorphic to an open subset of $\mathbb{R}^n$ then $p$ is an interior point. I could not follow his explanation ( his explanation goes something like this: the chart when considered as manifold with boundary are fixed, now if you choose another chart, it may not be those chart from that former case)
question: is my professor correct? if so why? or are my arguments ok?