I'm working on Lee's Introduction to Smooth Manifolds Problem 16-8:
Suppose $M$ is a smooth manifold with corners, and let $\mathcal{C}$ denote the set of corner points of $M$. Show that $M \setminus \mathcal{C}$ is a smooth manifold with boundary.
I know that from the topological point of view, manifolds with corners and manifolds with boundaries are the same thing but I can't understand why this (the problem) has to be proven.
In a manifold with corners we also have boundary charts, ¿do these work for boundary charts looking at $M$ as a manifold with boundary?
¿Does the boundary need to be non-empty?
Any idea to solve the problem is useful.