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An n-manifold with corners is topologically an n-manifold with boundary, but with a smooth structure that makes it locally diffeomorphic to $[0,\infty)^n$ instead of $[0,\infty) \times \mathbb{R}^{n-1}$. See also:

The filled cube is naturally a 3-manifold with corners, with every vertex having a neighborhood that is diffeomorphic to a subset of $[0,\infty)^3$. It does not seem possible to me that we can map a neighborhood of the top vertex of the filled square pyramid to $[0,\infty)^3$ since it has four adjacent faces instead of three.

Can we conclude that the filled square pyramid is not a manifold with corners? If yes, what is the point in excluding this seemingly useful structure by such a strict definition?

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