Can the interior of a (non-empty) smooth manifold with boundary be empty

Question

Let $M$ be a smooth manifold with boundary.

If Int$M=\emptyset$, then $M=\partial M$ would be a smooth manifold without boundary, and so every point of $M$ is an interior point. Thus we have $M=$Int$M=\emptyset$.

Is my argument correct? In other words, if I have a non-empty smooth manifold $M$ with boundary, am I authorized to assume Int$M\ne \emptyset$?

Answer 1

Your argument is unusual, but correct. You invoke two well-known facts:

1. Each manifold with boundary $M$ can be written as $M = \text{Int}M \cup \partial M$, where $\text{Int}M \cap \partial M = \emptyset$.

2. $\partial M$ is always a manifold without boundary.

However, to prove 1. and 2. you need to consider charts as in IEm's comment. This makes clear that no boundary point can exist without interior points.