I was reading in the book Morse Theory and Floer Homology by Audin and Damian (translated in english) that the boundary of a manifold is not always a submanifold. I cannot see why that is true. Any explanations or an example of such a case would be great.
I am using the definitions of manifold and submanifold from Jeffrey Lee's Manifolds and Differential Geometry.
I don't have the book in front of me, but the authors might mean that the boundary (in the topological sense) of a submanifold need not be a submanifold. For example, an open square $(0,1)\times(0,1)$ is a submanifold of $\mathbb{R}^2$, but its boundary is not a smooth submanifold, as it has corners (you can easily get topological counterexamples, by for example removing $\{0\}\times(\frac 12,1)$ from the open square).