Ever since we started talking about manifolds, we’ve said that they locally “look like” the Euclidean space . We now need to be a little more flexible and let them “look like” the half-space .
Away from the subspace , is a regular -dimensional manifold — we can always find a small enough ball that stays away from the edge — but on the boundary subspace it’s a different story. Just like we wrote the boundary of a singular cubic chain, we write for this boundary. Any point that gets sent to by a coordinate map must be sent to by every coordinate map. Indeed, if is another coordinate map on the same patch around , then the transition function must be a homeomorphism from onto , and so it must send boundary points to boundary points. Thus we can define the boundary to be the collection of all these points.
Locally, is an -dimensional manifold. Indeed, if is a coordinate patch around a point then , and thus the preimage is an -dimensional coordinate patch around . Since every point is contained in such a patch, is indeed an -dimensional manifold.
As for smooth structures on and , we define them exactly as usual; real-valued functions on a patch of containing some boundary points are considered smooth if and only if the composition is smooth as a map from (a portion of) the half-space to . And such a function is smooth at a boundary point of the half-space if and only if it’s smooth in some neighborhood of the point, which extends — slightly — across the boundary.