In the book of Int. Smooth Manifolds by Lee, at page 4, it is stated that
it follows easily from these two exercises that any open subset of a topological n-manifold is itself a topological n-manifold (with the subspace topology, of course).
However, I cannot understand exactly what and how things go wrong when that subset is closed ? i.e why does a closed subset of a top. n-manifoldis not again a top. n-manifold ?
To be clear, the closed subset is again Hausdorff and second countable, but why are they not homeomorphic to $\mathbb{R}^n $ locally ?
Edit:
I'm looking mainly for a mathematical explanation, rather than only an intuitive one.
Take a closed disk in $\mathbb{R}^2$, for instance. No neighborhood of a point $p$ of its boundary is homeomorphic to $\mathbb{R}^2$.