Why is the unit square a 2-manifold?

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SUMMARY

The unit square, [0,1]^2, is classified as a smooth 2-manifold in \mathbb{R}^2, but it is not a smooth submanifold due to the presence of corners at points like (1,1). This distinction arises because smooth manifolds require the existence of tangent spaces at all points, which the unit square lacks at its corners. The unit square can be treated as a manifold with boundary, homeomorphic to the unit disc, but it does not inherit a smooth structure from \mathbb{R}^2. Understanding the difference between smooth manifolds and manifolds with corners is crucial for proper classification.

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Sorry for the delay; I've been kind of busy recently.
Yes; I can only think of variations of the fact that a function whose graph contains a corner is not differentiable at the corner, since the tangent space would be generated by the line y=f'(x)x+b.
 

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